Motion along a straight line. Motion about an axis of rotation. These two things can happen simultaneously. Molecules can both rotate and translate move along a straight line. The wheels of a car or a bicycle can do the same thing. How do we understand these motions? And how do we describe the energy that's present when both of these things can happen? Not only this, but what directional quantity can we associate with rotational motion? After all, there was a directional quantity of motion for linear motion, momentum. Is there an equivalent for rotational motion? We'll begin to explore these questions in this lecture. We're going to take a look at the final concept that we have to revisit now in the context of rotational motion, and that is angular momentum. But before we do that, we're going to take a look at a motion that combines both linear motion and rotational motion in one situation. The key ideas that we will explore in this section of the course are as follows. We will explore rolling as an application of angular ideas, combining old ideas about linear motion with the new ideas as they connect to rotational motion. We will then revisit the concept of momentum, but now in the context of rotational motion. And we will also revisit the conservation of momentum in rotational situations, and from this we will draw some conclusions about rules that we can use to understand the universe. To begin, let's talk about a motion that combines both translation, that is motion along a line, and rotation, that is motion about an axis. This motion is known as rolling. So what is rolling? Well, let's consider something like a wheel, for instance, a bicycle wheel. It is a ring-shaped object, much of its mass is concentrated on a circumference away from the axis of rotation, which is the hub or center of the wheel. It has a radius r, for instance. Now, a wheel usually makes contact with the ground. And so the ground touches a point on the wheel that is a distance r, the radius of the wheel, away from the center of mass. The wheel is designed in such a way that it has a uniform distribution of mass about its center, and so one would expect its center of mass to be located dead center, and this is usually where the axis, that is, the place where the wheel is affixed to the rest of the bicycle is attached. Now, at the point of contact, the wheel might have a material like rubber, for instance, that makes up its outer edge, so we expect that there's going to be friction between the wheel and the ground. And in fact, the only way to make a wheel work the way we understand wheels is to make sure that there is contact friction between the wheel's outer edge and the surface on which it is resting. So we have a uniform distribution of mass in a ring shape. That's the wheel. Its center of mass is located at its physical center, but it makes contact with the ground on its edge, and that would be a distance r, the radius of the wheel, away from the axis of rotation. So for instance, as you now push on the axis of rotation of the wheel, the center of the wheel, if it's free to rotate about that line, then friction will act to oppose your push. But you're pushing on the center of mass of the wheel, and friction is acting on the edge of the wheel. So while friction is opposing your attempt to move the object, this creates a torque, because the force of friction is applied not at the axis of rotation, but away from the axis of rotation on the edge of the wheel. So static friction generates a torque. Torque is a force that causes a rotation, and this sets the wheel into rotating. So this is rolling, and to roll without slipping means that the wheel is never pushed so hard that that point of contact overcomes static friction and begins to slide. In English, there's a colloquial term for when your wheel spins so fast that the point of contact exceeds the static friction force, and you get kinetic friction, and this is known as peeling out. And it makes for a very dramatic video, because when you suddenly start to slide with the contact point between the wheel and the ground, this can generate a tremendous amount of thermal energy which can actually melt the tire rubber. This is not a good situation to be in. You never want to be in a situation where you have a wheel that's normally functioning, but then suddenly is put in a situation where static friction becomes kinetic friction. This usually leads to accidents and other problems. So we're going to talk about rolling in the context of rolling without slipping, which means that the object that can rotate and move linearly is never pushed so hard that it slides at the point of contact with the ground. Now this interplay of a force that pushes on the wheel in a specific direction, so for instance a force applied to the center of mass of the wheel, its axis of rotation, combined with static friction generating a torque that then sets the wheel into a rotational motion is what is meant more precisely by a rolling motion. Now in rolling without slipping, which is what I'm going to refer to henceforth, the wheel translates. The center of mass moves along a straight line, but the wheel itself also rotates. The edge of the wheel cycles around the center of mass, which is where the axis of rotation is located. And if you study this motion extremely closely, rolling without slipping, what you find is that the wheel, the center of mass of the wheel will translate, it will move along a straight line, a distance that is equal to the amount of circumference that the wheel rotates through on the outer edge. So in other words, if we translate the center of the wheel by a distance delta x, then we will find that the edge of the wheel moves through an arc length of delta s that's exactly equal to the translation distance of the center of the wheel. Again, this only is true when you roll without slipping. Now what's nice about this is with this condition and with the relationship between arc length and angle in radians, the angle that the bit of arc length delta s subtends, we can write that this delta x is also equal to the radius of the wheel times the angular displacement delta theta that the wheel experiences when its center of mass translates a distance delta x. So this allows us to relate the linear motion, for instance, of a bicycle wheel, so moving forward 10 meters, with its angular motion and what relates them is the radius of the wheel. So if you know the radius of a wheel and you know how far the wheel's center of mass moves forward, then you can also tell me what its angular displacement is and this is great because it lets us understand this kind of motion very carefully. Now, since we can relate rotation and translation in this special kind of rolling motion, rolling without slipping, it gives us the ability to do this analysis I mentioned in a great amount of detail and then what's particularly interesting about this is that we have an object with a certain mass, m, and it can roll, but it can also translate. It can rotate, it can translate. So that means it possesses two kinds of energy of motion. It has linear kinetic energy associated with the motion of its center of mass at a certain speed, which I've denoted as V with a subscript COM. So it's going to have linear kinetic energy and it's also going to have rotational kinetic energy and that rotational kinetic energy will be given by the product of the moment of inertia of the wheel about its center of mass, times its angular speed squared, times a half. So we have a total energy of motion, a total kinetic energy that is the sum of two kinds of kinetic energies. The translation kinetic energy of the center of mass of the rolling object and the rotational kinetic energy. So what's interesting about this is that means that to get a wheel started rolling, you have to take whatever energy you have available and split it into two buckets. One bucket is the translational kinetic energy and the other bucket is the rotational kinetic energy and they're related to each other by this fact that the translation distance of the center of mass is related to the angular displacement of the wheel. So there's all kinds of interesting relationships that enter into this specific kind of motion that allows you to connect the linear velocity of the center of mass to the angular speed of the wheel as it rotates. Now, let's consider a special case and this will help us to understand how this motion can be quite complex but nonetheless, describable using these concepts. So for instance, let's consider as an example of rolling a round object that can roll down an inclined surface. We've looked at inclined surfaces before. This is just a flat surface that has been lifted up at one end through an angle theta so that gravity when it acts on anything on the surface is not generating a weight that points exactly perpendicular to the surface. It points at an angle theta, in fact, relative to the surface of incline. So gravity will exert a force on the center of mass. This is the weight that we have been studying as a force for quite a while in this course now. And that means that gravity provides both a normal force. The weight is perpendicular to the inclined surface and if you look back at problems where we've attacked this question before we see that the component of weight perpendicular to the inclined surface is given by the mass of the object times the acceleration due to gravity times the cosine of the inclined angle. Now, the weight also generates another force. It generates a component of force that's parallel to the incline moving and translating, moving in linear motion along the surface of the incline. That parallel component, which I've denoted here W with two parallel lines is the mass of the object times the acceleration due to gravity times the sine of the incline angle theta. But now that we have an object that can rotate and translate, that is roll, gravity has to do work and that work is split into these two pots, rotation and translation. And so we have two activities now to divide the same energy between. So, for instance, we can answer questions like what will actually be the linear acceleration of the center of mass as a result. It won't just be G sin theta. G sin theta, if the object was sliding without friction, that would be its acceleration parallel to the incline. But now that gravity has to put work both into rotating and translating the object, we would expect that the acceleration will be reduced by some amount because some of the energy has to go into rotational kinetic energy, not just linear kinetic energy anymore. So we already have some intuition here. There's a fixed amount of energy from gravity doing work, but now we have to split that energy into two places and we're just not going to get as much linear acceleration as we used to when an object could, for instance, slide without friction. Well, let's attack this problem using Newton's second law. Newton's second law tells us that the sum of the forces will be equal to the mass times the acceleration of this rolling object, but what acceleration are we talking about here? Well, the sum of the forces, what are the forces acting on this? Well, we've got friction at the point of contact of the object and the incline surface. And we have the force of gravity parallel to the incline surface that's acting at the center of mass of the object. So as I've said before with wheels this situation is no different. We have two forces. They act in opposite directions. One is pushing on the center of mass of the object. The other one is pushing on the edge of the object. So they're attempting to cancel each other out, but the result of this is that because one of the forces is acting far from the center of mass, it generates a torque and this induces rolling. So we have the static friction force at the point of contact that's acting up the incline so I'll call that the positive direction and we have the component of the weight parallel to the surface acting at the center of mass of the object that's pointing down the incline which I'll denote as the negative direction and so these two get deducted from each other but gravity may win out over the static friction force and in fact in the case of an object that can roll it will. And the result of this is that acceleration of the center of mass times the mass of the object. Well, let's go further. This doesn't look very useful so far. So let's take information we know about forces applied on the edges of objects that can rotate about a center and see if we can start whittling this down to a useful formula. Well, I've rewritten the Newton's second law equation up here. Again, we have the friction force acting up the incline trying to oppose the motion of the rolling object. We have the component of the object's weight acting at its center of mass pointing down the incline. These are the two forces that are trying to oppose each other and the net effect of this will be to tell us what the mass times the acceleration of the center of mass will be. Now the friction force acts at a point in such a way that it's exactly tangent to the wheel. So if we draw a line from the center of mass of the wheel out to where the friction force acts, the friction force acts perpendicular to that incline. So it's tangent to the wheel. This will generate a torque and the magnitude of that torque, due to friction tau sub f, is given by the friction force times the distance from the center of rotation. That's the radius of the wheel. So friction force times r. Now, this in turn because it's a torque will generate an angular acceleration and that angular acceleration can be determined from Newton's second law for a rotating system. So writing that down we have the torque due to friction is the friction force times the radius of the wheel or the rolling object and this must be equal to the moment of inertia about the center of mass times the angular acceleration. Now, angular acceleration is not independent of linear acceleration. We know that the angular acceleration is related to the acceleration of a point on the edge of the rotating object tangent to, again, this line that goes from the center of mass out to the edge. So we can write alpha as the tangential acceleration divided by the radius of the object. Now, keep in mind that because this is an object whose center is accelerating one way, that the friction force is generating a torque that pushes the bottom of the wheel in the opposite direction that the tangential acceleration at the point of contact will wind up being opposite to the acceleration of the actual center of mass. So we have a relationship we can use to our advantage here. We can actually wedge in the acceleration of the center of mass into this torque equation. The tangential acceleration of the edge of the wheel which points up the incline is opposite the center of mass's acceleration, which points down the incline. And so finally we obtain that the force of friction, solving these series of equations up here, is equal to the negative of the moment of inertia about the center of mass, times the acceleration of the center of mass, divided by the radius of this rolling object squared. So we have the force of friction in terms of moment of inertia, acceleration of the center of mass, the radius of the wheel. We can plug that back into the original Newton's second law equation and now we basically have the mass of the wheel, the moment of inertia of the wheel, the radius of the wheel, the acceleration of the center of mass winds up being the only unknown. So we can now finally figure out what the acceleration of the center of mass is going to be. So substituting that back into our original Newton's second law equation, we get something that doesn't look very pleasant here. I've just substituted in for the force of friction. Here I still have the component of the weight parallel to the incline surface, and again I have the total mass times acceleration of the center of mass. We have acceleration of the center of mass in two places, so we can do some algebra. I'm going to skip that here, but ultimately what this algebra allows us to do is to solve for an equation that tells us about the motion of the center of mass, and what we find is that the acceleration of the center of mass of this rolling object is given by the negative of g sin theta divided by this term. So if this object couldn't roll, that is if it didn't have a moment of inertia, moment of inertia was zero for instance, then in that case we would recover the old relationship between the acceleration the object would experience, and the angle of the incline. We would find that the acceleration of the center of mass would just be related to g sin theta. But in this case we have an object that can roll because of the torque exerted by friction on the edge of the object, and so this acceleration g sin theta is modified by this denominator. So we see that this number, this denominator can only ever be bigger than one, and so we're always dividing by a number bigger than one. We're always taking g sin theta and we're dividing it by a number that's one or greater, and that means that we get less of an acceleration of the center of mass when rolling is present than we would if an object could just slide. So we see here how this motion can get quite complicated, but nonetheless it admits itself to an analysis based on rotation and linear motion, translation. With this background having looked at an application of both linear motion and rotational motion in one scenario, let's turn our attention to the last concept in linear motion that we encountered, momentum, but now try to place it in a rotational context. So we've focused on the kinetic energy of rotation. We've taken a look at how if we want to rotate an object, and we put a certain amount of work into rotating that object, that if that object can roll some of the energy goes into linear motion and some of the energy goes into rotational motion. But what about a directional quantity of motion? After all, linear motion had a directional quantity momentum. It had a scalar quantity, kinetic energy, and it had a directional quantity, momentum. Is there an equivalent directional quantity for rotational motion? Indeed, we know that all the points on a rotating body rotate, and so there must be some equivalent for rotational motion to momentum. After all, in a rigid rotating structure there's clearly some sense of directional motion, right? If we use our right-hand rule to figure out where the angular velocity points, there is a sense that going counterclockwise is different from going clockwise, that the angular velocity flips direction. So we should be able to use these sort of intuitions to now think a little bit more about what it means to define momentum in the context of a rotating situation. So let's consider a single point of mass m, and I've denoted that here in the picture by this red dot. And it's going in circular motion around some axis of rotation, some center of motion. So that path that it's taking as it goes around the center of motion is indicated by the blue circle with the arrows. And it's a counterclockwise motion, so it has a positive angular velocity. Now it's doing this rotational activity a distance r from the center, and I can denote this distance by an actual vector in space. It starts on the center of rotation, the axis of rotation, it points out to where the mass m is at any given moment in the motion. So at the moment I have frozen this motion in this picture, it points down into the left from the center out to the mass m, the red dot. Our vector then is the vector that describes the length and direction of this green arrow that I have drawn here. Now at a given moment in time this rotating object has linear velocity, v, that's tangent to the circular path. And that means it has linear momentum, p equals m times v. And that linear momentum will also, if this was exactly circular motion, be tangent to the circular path. Now of course it might be possible that in a rotational motion a point on a rotating body has a component of momentum that points parallel to the radius that goes back to the center of motion. But that component won't contribute to the rotational motion, it will cause some translation of the object, but it won't cause more rotation of the object. It's only the momentum perpendicular to the r vector, that is tangent to the rotational motion that actually does anything involving rotation itself. But we can imagine that there's an analogy here. Torque was a force applied perpendicular to this radial line causing a rotation to begin. We can imagine defining angular momentum in a way that similarly we saw that torque was defined. Torque was vector r cross vector f. So angular momentum could be written down as vector r cross vector p, the momentum vector. And since we have a point mass we can write explicitly what the momentum is for this point. It's the mass of the point times its velocity. Mass is a scalar quantity, it's just a number so I can pull it out in front of all this vector multiplication that's happening here. I can't change the order of the vector multiplication without consequence. If I swap the order of the vector multiplication I pick up an extra minus sign, go back and look at the cross product if that doesn't feel familiar to you at this point. As long as I preserve the order of the cross product I'm okay and I really have to make sure that if something is in the order of A cross B I leave it in the order A cross B as I'm playing around with an equation. However, numbers like mass which are just scalars, they're directionless, they can be moved around anywhere at no consequence to the vector multiplication that's going on.  Okay, so I have a definition here of angular momentum. It's just a vector and it's cross product with the vector that goes from the axis of rotation out to where the mass is sitting at a given moment in its rotational motion. And again, I'm considering a point mass so I was able to write this as m times the quantity r cross v. Let's look at the magnitude and the direction of this quantity L, angular momentum, L vector. Well, the magnitude of angular momentum will be the same as the magnitude of any cross product. So if I want to know the magnitude of A cross B, it's just going to be the length of A times the length of B times the sign of the angle between them, theta. So if we imagine that there's indeed an angle theta between the vector r and the vector p, then it must be true that the magnitude of angular momentum is r times p times the sign of the angle theta that lies between the two vectors. Now, again, p sine theta, that's just the component of momentum that's perpendicular to r. So I could write this in this notation again, where we say that the angular momentum is just the magnitude of the radial line out to the mass m times the component of that momentum of mass m that's perpendicular to r in the first place. That's the only thing that contributes to the angular momentum because of this cross product definition for the magnitude. What about the direction? Well, the direction will be given by the right hand rule for a cross product. So what do we do? We get out our right hand, we take our index finger, that's the finger closest to the thumb on your right hand, you point it in the direction of r, that's going to go from the center of motion, the axis of rotation out toward the mass m. You then reorient your right hand so that your middle finger can point comfortably in the direction of the momentum vector itself.  And where your thumb points indicates the direction of L vector, the angular momentum vector. So it's, again, very similar to figuring out the direction of a torque, given a force that's acting a distance r away from a center of rotation. It's the same activity because it's just a cross product and the rules for figuring out the directions of a cross product don't change. So we've been talking about a point mass m that's circulating around in some rotational motion about some axis, some center of rotation, but what about a rigid rotating body that's made from many such little points of mass? Well, in that case, it's just like what we did when we summed up all the kinetic energies of a rigid, extended rotating body, we have to add up all the momenta of all the points in the body. Well, momentum is a vector sum, so now we have to do a vector sum of all the individual angular momenta of all the little masses, m with a subscript i, where i is the index that labels all the masses, one up to n, and could be Avogadro's number, could be a big number, could be a small number, could be three or four masses, could be Avogadro's number worth of masses, but it doesn't matter, this is a generic principle. So if we want the total angular momentum of this system of masses, we take the angular momentum of one of them and add it to the angular momentum of the second of them in a vectorial body, and so on, all the way up to the last mass, the nth mass. So we can write this sum more compactly as the sum from i equals one to n of the individual angular momenta l sub i for all the little points of mass m, i. Now, we can then explicitly put the cross product into this, the original definition of the angular momentum, r cross p, for each of those little masses, m sub i. So let's do that. It's going to seem at first to make this more complicated, but what we're going to find out is that very quickly we're going to reduce this seemingly complex sum down to a fairly nice compact definition. So even having a really complicated rotating body, if we just know a couple of things about it, we can very quickly solve for the angular momentum. So the total angular momentum is the sum of all the individual angular momenta. Each of those can be written as the cross product of r i vector. That's the distance from the center of rotation out to the mass point m i and the momentum of that mass point. So p sub i vector. But again, the momentum of that mass point is its mass m sub i times its velocity v sub i vector. So let's put that in. Now we can write this term in the sum as this quantity here. Just like I did on the last couple of slides for a single point mass. Nothing really exciting is going on here yet. But it's the next step where things get more interesting. Okay, so here again is that sum. The total angular momentum of all these n masses is written out this way with the cross product of the mass of each point. Now what I'm going to do is I'm going to do something that harkens back to the original definition of a vector. A vector is a quantity with length and direction. So for instance the vector r has a length r. That's the radius of the circular motion for instance that these points might be executing. But it also has a direction. That direction points from the center of rotation out to where the mass point we're looking at is located. So it has a direction. That direction can be written using a unit vector that simply indicates the direction I just stated, r hat. So I can replace r vector and v vector with the product of their magnitudes and their unit vectors that indicate the direction where they point. So let's do that. So if I do that I pull the lengths of these vectors out in front. They don't affect the cross product but I've left the cross product of these unit vectors intact. The order is preserved. The unit vector for r precedes the unit vector for v in this cross product. I have preserved the order of the multiplication. Nothing has changed. All I've done is pull the magnitudes of r and v out and put them next to the mass. Now again as we did with kinetic energy recall that all points on a rigid rotating structure rotate with the same angular speed. That hasn't changed. That principle still holds true here. So that means they all move at some angular speed omega and it will generically be true that for every point on this rigid rotating body the quotient of v i and r i the ratio of those two things will give you omega. Doesn't matter which mass point you're looking at. They all rotate in the same sense. They all go through the same angular displacement in the same amount of time and so they all move at the same angular speed. Well awesome I can replace v in this equation with omega. V i is omega times r i and if I do that I get this the total angular momentum of this system of masses is the sum of these terms where each term is the mass times the radius squared the distance from the center of rotation to where that mass is located times the angular speed of that mass but that's the same for all masses and then I have this this unit vector cross product. Let's take a look at that and see if we can figure out what's going on there. I mean at first it seems like we have a whole bunch of these cross products that we have to do and maybe they're all different for all the Avogadro's number worth of masses but because this is a rigid rotating structure and all the points are rotating in the same angular displacement in the same time something very interesting also applies to this cross product. So do the right hand rule draw a circle and pick a dot on the circle and let's pretend the circle that whole bunch of points that make up the circle are all rotating counterclockwise so use the right hand rule for angular velocity to figure out which way the angular velocity for a single point is pointing so curl your fingers in the direction of the sense of rotation of a given mass point your thumb sticks out and indicates the direction of the angular velocity. Pick another point it too is rotating counterclockwise at an angular speed of omega so it too will have your thumb pointing in the same direction giving you the direction of r cross v what we find out is that this cross product r i hat cross v i hat is the same for every single point on the rigid rotating structure and so as a result of that we can identify this cross product r i hat cross v i hat as being the unit vector that points in the direction of the angular velocity omega hat it's just the unit vector that points in the direction of the angular velocity of this rigid rotating body so let's substitute that in and see where we've gotten ourselves replacing that cross product now just with the unit vector omega hat that indicates the direction of that cross product which is the direction that the angular velocity points we can suddenly make a whole bunch of identifications in this equation I've got the sum over a whole bunch of point masses of m i r that looks suspiciously like the moment of inertia of a point mass about a center of motion and then every single one of them has a term omega times omega hat in it which is constant to every single term in the sum omega times omega hat is just the vector omega it's its length times its direction that is exactly the definition of the vector it's magnitude times its direction that's a vector so we just have omega vector in every term in the sum and then this other thing the sum is the moment of inertia of the rotating body it's the sum over all points of the quantity m i r i squared so thus in compact notation we can actually write the angular momentum of an extended rigid rotating body in a very simple way the angular momentum of a rigid rotating body is just the product of its moment of inertia and its angular velocity omega vector that's it the product of these two things tells us everything about the angular momentum the direction dependent quantity of motion associated with this complex extended rigid rotating structure this equation applies to a wheel that's rotating it applies to the fan blades of a wind turbine it applies to a whole planet going around in circular motion this equation can be universally applied to any rigid rotating body and what's interesting about this is we see a kind of again symmetry emerge between linear quantities and rotational quantities linear momentum was the product of mass and velocity angular momentum is the product of an inertial quantity the moment of inertia and a speed related quantity angular velocity again these are quantities in an angular context so p equals m v l equals i omega the symmetry is beautiful between these things now it's worth noting that we have a unit here for angular momentum the meter kilogram second or standard international unit of angular momentum is the joule second it's the product of energy and time so you can write this as j times s joule seconds or you can write it as kilogram meter squared per second that's what it would be in actual kilograms meters and seconds it's just energy times time now with this new concept of angular momentum in mind let's step back and ask a question what happens if we do the following what is required for angular momentum to vary in time not a very exciting question I've defined this quantity angular momentum what does it mean for it to be varying changing value in time well let's consider a simple picture again of a point mass moving in rotational motion about an axis a single point mass well in that case we can dig back and we can just write down the exact expression for the angular momentum of this point mass going around some center rotation some axis of rotation in that case the angular momentum of this mass is its mass times the cross product of r vector and v vector let's try to answer the question what does it mean for angular momentum to vary in time and to answer questions like that we have to apply calculus we want to know how thing one varies with thing two and that is exactly the question that the first derivative helps us to answer so let's take the first derivative of angular momentum with respect to time well mass is a constant it's not changing and I'm going to hold to that for this exercise so mass is a constant it's not affected by the derivative and when I substitute in for L with m r cross v I can pull m out to the left of the derivative doesn't affect the derivative however I should be careful and not assume that the radial distance isn't changing it could be maybe the object is getting further away from the center of motion over time that's one way you could change angular momentum or maybe it's speed is changing maybe the direction of the speed is changing or maybe the magnitude of the speed is changing so I could have changes in v and I could have changes in r and I've got the product of r and v here well that feels like I need to do the product rule for the time derivative so let's do that let's do the product rule for the time derivative of a product of two functions of time r could be a function of time v could be a function of time the cross product is just another kind of multiplication albeit a special one with an order dependence that matters so I'm going to keep the order of the cross product the same but I'm going to distribute the derivative to the two parts of the product in the way that you're supposed to do this in first semester calculus so I have my mass out in front and then I have the sum of two things r cross the time derivative of v and the time derivative of r cross v that's the application of the product rule and again the cross product is just a multiplication of two things and those things could depend both on time and so even if I have to apply the time derivative to a cross product of vectors the product rule for distributing the derivative is still in effect it holds true you just use it so that's what I did here okay so I've rewritten that expression here and it's not very pretty looking but let's see if we can make some sense of this let's dig way back way back to linear motion what is the time rate of change of velocity now the time rate of change of velocity is just our old friend acceleration what's the time rate of change of position well the time rate of change of position is our old friend velocity so dv dt is just acceleration and dr dt is just velocity it's the velocity vector so let's substitute those in if angular momentum is changing with respect to time that change is given by mass times this thing in parentheses r cross acceleration and v cross v added together ooh that second term looks interesting remember the rules of the cross product the cross product of a vector with itself or any vector that happens to point in exactly the same direction is zero the cross product answers the question what vector is perpendicular to the two vectors you've given me when you have a vector and itself there's no one direction that is exactly an only perpendicular to that pair of vectors there's an infinite number of them so the cross product returns zero the cross product of a vector with itself is zero remember the rules of the unit vectors in the cross product i cross i equals j cross j equals k cross k equals zero so it must be true that v cross v is also zero and so one of these terms just goes away and we're finally left with this if angular momentum is changing in time that quantity dL dt must be equal to mass times r cross acceleration alright well let's rewrite that equation here and let's consider the magnitude of the above vector alright so if we do that what's the magnitude of this vector well the magnitude of the left side is just the time derivative of the magnitude of the angular momentum l the right hand side however we have a cross product the magnitude of the cross product is r a sine theta where theta is the angle between them and a sine theta is just the component of the acceleration that's tangent to the circular path perpendicular to r so we can write this more compactly as dL dt is m times r times the tangential acceleration again only acceleration tangent to the circular path that is perpendicular to the vector r contributes to the cross product anything that's parallel or anti-parallel to r doesn't contribute to the cross product digging back a little bit here recall that the tangential acceleration is related to the angular acceleration we looked at this in rolling earlier in this video alpha the angular acceleration is the tangential acceleration divided by the radius of the rotational motion around the center of motion the axis of rotation so we can substitute in we can eliminate the tangential acceleration and we can put in alpha and r and if we do that we found out that the time rate of change of angular momentum is equal to mr squared alpha this is a point mass moving in rotational motion about a center of rotation an axis of rotation mr squared is its moment of inertia mr squared is the moment of inertia of a point of mass m distance r from a center of rotation going in rotational motion so we can substitute this equation with one more piece recognizing that this mr squared is the moment of inertia we then get the following equation the change in angular momentum with respect to time must be equal to the moment of inertia times the angular acceleration of this mass well what is i times alpha i times alpha is the angular equivalent of mass times acceleration and the angular equivalent of forces is torque so in other words angular momentum can only change in time when a torque is present a torque generates a change in angular momentum in a certain amount of time just like a force generates a change in linear momentum dp in a certain amount of time dt so f forces can be written as changes in angular momentum with respect to time torques can be written as changes in angular momentum with respect to time see the analogy has a beautiful symmetry between these things it's really quite amazing that this is even possible but what this also tells us is if there are no net external torques on a system that is if you have a closed and isolated system external forces can't reach in and accelerate components of the system then we have to draw the following conclusion that the time rate of change of the angular momentum of that system is zero for a closed and isolated system that is to say in a closed and isolated system angular momentum is conserved well this is nice total energy is conserved in a closed and isolated system total momentum in a linear sense is conserved in a closed and isolated system and now we find out that total angular momentum is conserved in a closed and isolated system where no external forces can generate external torques that can affect the angular momentum of the system so if in a closed and isolated system it is true that the time rate of change of angular momentum is zero that is the initial momentum in an angular sense is equal to the final momentum in an angular sense then this has interesting consequences for physical systems that are closed and isolated from external entities first of all if a body is rotating and only forces internal to the closed system can act then angular momentum must remain fixed for the entire system even if it's parts change their orientations for instance by breaking apart or just moving around so that's an interesting consequence of this this is akin to the idea that because in a system where no external forces are acting momentum must be conserved that if that system explodes or breaks apart in some way it will be true that the momentum before the explosion will be equal to the momentum after the explosion and this lets you do all kinds of interesting analysis of the system now since angular momentum is equal to the moment of inertia times the angular speed if a closed and isolated rotating system alters its moment of inertia then angular speed must change an interesting demonstration of this consequence is as follows take a chair that can rotate and sit on it so you can spin yourself around and you have an axis of rotation that goes sort of from under your seat up through your head you can rotate around that axis on the stool hold two masses a certain distance away from your axis of rotation make sure they're really heavy start yourself spinning and then take your feet off the ground so now no external forces can really act on you yes there's friction but if it's a good enough stool it won't matter draw the masses closer to your chest you have now decreased your moment of inertia if you decrease your moment of inertia your angular speed must increase pull the masses further away from the axis of rotation increasing your moment of inertia and your angular speed must decrease draw them in closer to the axis of rotation your speed increases push them out further from the axis of rotation your speed decreases this is a nauseating demonstration and you have to be extremely careful when doing it but nonetheless it's a very quick and effective demonstration of the conservation of angular momentum and its consequences now conversely such a rotating system alters its angular speed in some way for instance by using chemical energy to suddenly speed itself up moment of inertia must change in response you can't alter one without altering the other so in a system where angular momentum is conserved changes in things like angular speed have consequences for the moment of inertia and vice-versa now here's an application of the conservation of angular momentum to a large cosmic phenomena and that is the laws of planetary motion I hinted at these in the lecture on gravitation the person who's credited with establishing the three laws of planetary motion is the astronomer, mathematician and physicist Johannes Kepler and as I mentioned in the earlier lecture on gravitation he used an extensive observational data set to establish these laws of planetary motion the first law is that the shapes of the orbits of planets for instance about the Sun are not circles as was believed but rather ellipses these are elongated circles with a long axis and a short axis the second law is that a planet in orbit around another body like a star like our Sun will sweep out as it moves equal areas and equal times we'll look more closely at what I mean by that in a moment this sweeping out of equal areas and equal times is independent of when in the motion you observe this sweeping effect to occur and then finally the square of the period of any planets orbital oscillatory motion as it cycles around its say parent star will be proportional to the cube of the long axis the so-called semi-major axis of its elliptical orbit so that is to say the period squared of a planet orbiting around a star will be proportional to the length of its semi-major axis cubed now these are empirical laws that Kepler established by studying extremely detailed samples of data on planetary motion but at the time he discovered these laws he didn't have the benefit of Isaac Newton's work he didn't have the benefit of the law of gravitation or the laws of mechanics and so he could not know the reasons for these planetary laws they were clear from the data but the reason they existed was not clear but now we understand their origin its conservation of momentum and angular momentum and the law of gravitation combined together this establishes where the three laws of planetary motion come from as an example of the application of the conservation of angular momentum let's look at the origin of the second law of planetary motion this picture will help you to understand a little bit better what I meant a moment ago in equal times so imagine we follow a planet for instance Mars Mars's orbit could be represented by this exaggerated elliptical shape shown here in purple so this is the planet labeled planet 1 in this drawing now if we let the planet complete one orbit this for instance for Earth would be one year we would see that the planet would start here and then after one cycle about the Sun it would return to that point and we would call that a year on that planet that is one orbit around the parent star now what do I mean by sweeping out equal areas in equal times well let's imagine that we watch the planet from this point to this point shown here so it starts off here over a little higher and a little above into the left on the ellipse and then after some time T1 ends up down here where it is currently indicated in the picture what you would find out is that the area swept out in that time T1 would be A1 okay fine well we wait a little bit and then we wait until the planet gets to this point over here on the ellipse and then again we time it the same time T1 and this time it moves over here in that time T1 so it sweeps out this grey wedge A2 with a seemingly different area so we've observed the planet moving along its elliptical orbit in two times they're equal times both T1 and we see this grey wedge that represents area 1 and we see this grey wedge that represents area 2 and what Kepler discovered is that these areas are the same if you watch the area swept out by planetary motion in equal times they're the same areas now this one is sort of shorter and wider and this one is longer and thinner but the areas wind up being exactly the same this was what having a tremendous data set made Kepler able to do these are sort of wedge shaped things with curved sections at the end they look a lot like slices of pizza with straight line edges and then a curved crust so let's consider a wedge of a planet's orbit so for instance this purple planet planet 1 which you know could be Mars and it's going around the sun and it's in different times say you know T1 here and T1 here it's traveling along the curve of its ellipse now let's consider a very short time DT over which it moves a very short length ds now this ds is so slight that it basically just forms a triangle two sides of the triangle or the radius are of planetary motion and the third side is the arc length s the area of the triangle that represents this wedge that swept out in this very short time DT will be one half times the base times the height of the triangle well the base is the arc length s and the height is the radius are okay so fine during that little time DT we're in an orbital orbital radius are we've swept out in a triangle with area one half s are now since ds is a little bit of arc length it's related to the angle that's subtended by that arc d theta and that's according to our old friend the definition of the radian ds equals r d theta so now we can write the change in the area as a function of time and then think about what it means to change area in some amount of time okay so the change in the area is a function of time sounds like we need calculus again so let's do that let's take the first derivative with respect to time of the area of this triangle well that's the first derivative with respect to time of one half times the little bit of arc length times r okay well I can substitute with r theta for s s equals r theta so I substitute that into this equation I have the first derivative with respect to time of one half r squared theta well one half is a constant and r is a constant over these short times the radius of the orbit is not changing appreciably over very short times but theta on the other hand is not zero theta has changed over time so we pull out the one half we pull out the r squared and we're left with the first derivative of the angular position with respect to time d theta dt what is the change in angular position with respect to time that's just the angular speed so the change in area as a function of time for a planet moving in this elliptical shape is just one half times r squared times omega again over a very short period of time okay well if we treat the planet as a point of mass m and on the scale of a solar system planets are really not that big so we can get away with this approximation I mean Mars is not very big compared to say the whole solar system or certainly to its orbit so we can treat it as a point with a concentration of mass m at that point and we know because it's going in rotational motion around some center that it's got angular momentum and that angular momentum is given by the product of its moment of inertia and its angular speed well if it's a point that's just m r squared omega we had an r squared omega in our equation for the change in area with respect to time so we can make a substitution we can divide both sides of this equation by the mass we can replace r squared omega with l the angular momentum divided by m the mass of the planet so let's do that so now we have that the time rate of change of the area swept out by the planet is one half l over m so we have this thing that you naively might think would be different different areas swept out in the same amount of time in different places in the orbit but in fact the change in area swept out by a planet as it moves around in its orbital path its elliptical path around a star for instance is equal to a constant because planets are closed in isolated systems there's no friction there's no drag there's just gravity and gravity is holding those planets in orbit around the parent star so we have one half times the angular momentum divided by the mass of the planet with the mass of the planet if that's not changing then da dt is not changing if the angular momentum of the planet is not changing then da dt is not changing and so the reason Kepler's second law of planetary motion is true that planets sweep out equal areas in equal time regardless of where you look at them in their orbital path is because the angular momentum is constant in a closed and isolated system which for the most part a planetary system is it's largely free of external forces because solar systems are so far apart from one another and so it must be true that in a planetary system like our own da dt must be constant that is planets will sweep out equal areas in equal times regardless of where they are in their orbital motion it's a little counterintuitive but it's true because angular momentum is conserved in a closed and isolated system so you can begin to see how many regular phenomena in the universe could arise from the application of simple smaller sets of rules and maybe you're beginning to see now what motivates the physicist what motivates physicists is to try to find all those rules that underlie the structure of the cosmos be it material science plasma physics gravitational wave astronomy particle physics my area of study or many others we see regularity in the universe it has underlying causes we've come to understand many but not all of those we want to understand more and that is ultimately what drives the physicist so let's step back and take a big picture look at conservation laws because we've essentially completed a set now we have a set of conservation laws that if we can apply them or at least understand why they're not holding allows us to analyze even seemingly complex systems now that analysis may be very difficult but nonetheless the underlying equations are quite elegant and simple so in any closed and isolated system where either translation linear motion and or rotation are possible then the following statements will be true based on everything we've studied so far in this course the total energy of that system the sum of its linear kinetic potential energy from conservative forces u potential rotational energy rotational kinetic energy internal energy such as chemical energy in the system that could be converted to other forms and thermal energy the energy of the jittering motion of atoms that won't change over time for a closed and isolated system so now where before we might have been hiding the rotational energy in this vague term internal energy we've now busted it out because we know how to write it down the kinetic energy of a rotating body so if you have an object that's spinning and moving along a straight line and doing so in a gravitational field but otherwise this system is completely closed and isolated I can tell you just about everything about the energy of that system and what its energy is going to be in the future it might change from one bucket to another but the total amount won't decline or increase the time rate of change of the total energy is zero so whatever total energy this system has at the beginning at some time zero will be the same total energy it has at some later time again it might get subdivided into different places some rotational energy might get slurped up and put away in thermal energy but none the less if I can account for all of that this statement will hold total energy is conserved in a closed and isolated system so that is one of our laws what's another one that we've looked at? well in a closed and isolated system free from external forces because force is equal to the change in linear momentum with respect to time if there are no external forces acting there is no change in the total linear momentum of a system and so dp total dt must be zero in which case the initial momentum of a system in total will be equal to the final momentum of a system in total now you might have to sum up the linear momentum but if you can do that this statement will hold total momentum will be conserved under these circumstances and finally we have our new law that we have just explored a little bit with the planetary example that is the conservation of angular momentum so if there is rotation possible in the system whatever the initial angular momentum is at the beginning of a system that becomes closed and isolated that angular momentum cannot change in total and so we have a situation where the time rate of change of the total angular momentum of the system will be zero the initial angular momentum in total will be equal to the final angular momentum in total total angular momentum is conserved these three principles together form an incredible foundation for studying the cosmos and perhaps even unraveling some of the seemingly most complicated puzzles that the universe can throw at us we have a basis for a physical analysis of the natural world so these principles combined with Newton's law of motion and the law of gravitation are really the five statements that lie at the heart of everything we have done this semester in this course if a system is closed and isolated all of the above apply we can use Newton's laws to analyze the details of the motion we can write down the different kinds of energy we can see how they might change from one to another if collisions are occurring orientation and a conservative force is changing so we have different potential energy we can do all that it might be complicated but we can do it if a system is not closed and isolated and we can recognize that that's the case then as long as we can account for the fact that energy might enter into the system or momentum might be taken out of the system that analysis can nonetheless continue if we can account for that in our analysis we still know something about what's going on and we can make some predictions about what might happen to that system in the future so let me conclude this lecture by reviewing the key ideas that we have explored in this section of the course we've looked at rolling as an application of angular ideas where both a linear and a rotational motion are possible and they're linked to each other rolling without sliding that is rolling where the point of contact always is acted on by a torque that's given by static friction we then revisited the concept of momentum but now in the context of rotational motion we revisited the conservation of momentum in rotational situations and we've accounted now for a number of conservation laws that can help us to understand the universe and in fact I would argue that you are now armed with five tools that would let you study in detail even complicated systems now it may require computers to do the analysis but if you can faithfully execute that analysis using these five ideas conservation of energy conservation of linear momentum conservation of angular momentum Newton's laws of motion and the law of gravitation in a situation where those are the only things that matter and there are other forces in nature but we're going to ignore them for right now and I argue that you have all the tools you need to pick apart what's actually going on in a situation and not only make predictions but maybe even make discoveries about what's actually happening in a system where there's order and regularity but no pre-existing explanation as to why that might be so taken together all of these ideas linear motion, rotational motion and all of these momentum and energy and force concepts are the basis of a foundation for analyzing all of nature or at least figuring out what we don't know already about the universe as a whole


Everybody forced here ripping 2021 flawless baseball hobby Two bucks case number 69 random team style check out these results at any time a late in sports cards calm And I there is a link in chat for these as well It was going on chat. Let's go man Let's get started everybody flawless baseball. It's been a month. It's been a minute for me Lawless MLB and how much going on tonight outside of the breaks man the NFL game is pretty me There's not a ton of NBA on there's some college basketball, but I didn't even not a lot of that It's pretty met tonight. All the action is here with the brakes. Let's go. Maybe not. Maybe I can't get it open There we go. What's up hook effects? It depends on what World Cup prison you're talking about We did some breakaways earlier It's a break schedule right here for the future breaks. It's a pretty sweet hit and In the briefcase. All right, we have a dual patch with the Cardinals Jack Flaherty one of ten Great patches. That's a spring training patch. I think in the bottom was just a logo patch player worn Orioles triple relic Luis Apparicio 17 of 25 Luis Apparicio Nice rookie dual patch auto rangers Sam Huff seven of 20 Great patches there probably I think like a minor league Rangers patch or maybe an inaugural I forget the name of the park will blive something nice Sam Huff Rangers Ruby foil color match, too What is that? RPA Alec Bohm seven of 25 What is that patch that is sick? I don't know what that is There's a hand What is that design that's nasty Philly's awesome card Spring training maybe or some event.  Yeah, Rorschach test Nice big sexy for the angels Elohim and his patch auto 10 out of 10 white socks Hold an England Patriots logo White Sox patch auto Eloi Ruby RPA blue jays Nate Pearson number to 20 10 of 20 and Twins rookie auto a rookie signatures Alex Kira law 14 out of 20. Hello, Brian. How's it going man? Diamond Jen Miguel Cabrera 16 out of 20 For the Detroit Tigers There's a diamond guarantee that goes with that and this is Disgusting Quad patch show hey ohtani auto two of three boom Player warn a quad patch some disgusting relics. Oh My goodness big big show hey The angels go to Scott F Wow, man That's just The gorgeous card Scott F buddy congrats flawless big show two of three Yeah, it used to be like that newer one the newer one is the newer one is Different yeah, so that was changed that the one he uses now is what he changed from okay It is kind of a cross I think Rory, but that's what it used to be The newer one the more the one you see more now is is is is the new one the acanjiados Oh my god the acanjiados Dude if he ever sounds like that again people go people lose their minds. I bet how about a Bobby door triple relic Red Sox 8 of 20 Retired Red Sox number Bobby door some old pieces of fabric right there Star swatch signatures Mets David Peterson 18 of 20 big Mets watch Phillies again with an Alec boom 19 of 20 horizontal Ruby RPA Phillies with two bums now Jonathan P. There you go, man. Nice break for you Ted Simmons Cardinals auto 7 to 15. I don't think I don't think at this price point there would there would be one Rory unfortunately a Drew blood so congy Lady Tavares RPA Rangers to 15 3 to 15 Guardian Shane Bieber auto 11 to 15 rookie dual-patch auto Marlin's jazz Chisholm 8 of 20 Smooth jazz Last three cards here. Where's the guy right there at 10 at 20 or sorry one of 20 diamond Aaron judge Yankees Diamond guarantee you can buy a lot of diamonds now No, yes Dummy thick Kaleat relic Corey seager Dodgers That is cool. Call we see good numbered 8 out of 10 That's a pretty big piece of a relic player worn Flawless was it called sneakers flawless spikes Call we see go Love the spike relics and on the back Orioles greats auto Cal Ripken Jr. Today, Jr. Nice hit O's That'll do it everybody For flawless baseball. That was fun, man. Good case. I'd say good case on the basketball


Dear residents of Castries Central, I bring you warm Christmas greetings in this holy season. We have reason to be joyful as we welcome Christ Himself into our hearts and our homes. I say we have reason to be joyful and indeed it is so in spite of the many challenges we may face at this time of year. Jesus' birth in a stable reminds us that God always allows what is best for us even though we sometimes do not understand it. Jesus' birth gives us the hope that despite our own poverty in many respects that His glory is yet to be revealed in our lives. It is with this hope that I invite you to thank and praise God with me for our many achievements in Castries Central this year. We launched the Synergy Project which focuses on personal development, training and employment. This is a historic project for Castries Central and for St. Lucia and I thank you for your overwhelming participation in our programs. Your interest, commitment and performance gives me great hope for our success in the future. I hope that together we will tackle the social problems, in particular the crime and violence on our streets. Please pause with me to remember the families and victims of those who have died from violence this year. Let us keep those who are in mourning and sorrow this Christmas in our thoughts and our prayers. As your Member of Parliament, I am very concerned about the poor state of housing which so many families have been enduring. I wish to reassure you that I understand the need for adequate housing and that even as we continue to focus on job creation in the new year, housing will receive particular attention. I will be sharing my thoughts and plans for improving access to housing in the new year and I look forward to your continued support and cooperation. Till we speak again, I thank you for your love, dedication and support and from my family to yours, Merry Christmas and a Happy New Year.


What's going on everybody? Got Kyle here with Layton Sports Cards and we are ripping 2023 Immaculate Collegiate. This one is five box full case number 19, Random Division style. Did I randomize for the break? I'll show you guys the results again. We had Miguel with the NFC East down to Don G with the AFC East. Alrighty guys, good luck on this one. Let's see what we got. Get one of these cheat sheets up in case. Alrighty guys, good luck. I'm down. Real quick. We did get you a glitter, Sam. You, uh, really? Yes. Oh, I didn't know that. Uh, I think it was Brunson. I think. No. Was it not Brunson? I don't think it was Brunson. This is, uh, Jake. This is Jake right here. Hello. Who said that? Alrighty guys, box one. Brew Eye? That's the, I'm trying to get everybody's names down. Hello. Alrighty guys, box one. I'm Activision's. Let's see what we got. It's nice to meet you.  Pick your team style. I got four to go. Yep, that's me. Yes it is. What? He said Mr. Sam Diego. Because when Matt hit the super. That's all I can think of is Dan or CEO. Uh, Tony Pollard to 99. Started us off for the NFC East Cowboys. Tony Pollard on the base. Jumbo Jersey to 99. We got Zach Evans Rams rookie. Um, that one is 29 and 99 for the NFC West. We'll start your dare action on the glitter. Oh, that's. Oh, was it? Yeah, that's not bad. Not bad at all. God, it's a 75. Auto is Dallas Clark. Dallas Clark on the auto. I believe that's going to be Colts to 75. Let me check NFC North or, um, oh, AFC South close though. Dallas Clark. I think it's going to be Colts. I think it has to be. Yeah. Yeah. Very much so. To 75 Immaculate Inc. Colts. AFC South. Behind it we have to 49 on the auto. Going to be Darnell Washington. That's the Steelers new tight end. Um, AFC North. He is gigantic for sure. AFC North there. Nice one here. We got Charbonnet on our signature moves auto to 49. That's interesting. Charbonnet signature moves. That's NFC West. Seahawks. And we got a dual auto. It is pretty cool too. Keyshawn Johnson and Jordan Addison. On the dual USC receivers auto there. Nice one. I think Keyshawn is probably a... What? He's a jet. So that'll be NFC North and AFC East on the dual. We'll do a random at the end here guys. Keyshawn and Jordan Addison. That is sick. The USC receivers. All righty guys. Get that one. Random at the end here. Box two. And guys that is NFC North and the AFC East. Jets and Vikings on that random. It's a cool one though man. I like that dual auto too. That's Raven's A-Bucks legend. That's a matter of fact. I don't remember him as a buck. I bet he's just going to practice while I die as a buck. But he had like one year as a Raven. I also didn't start watching football too. He called it. He called it. Oh yeah? I don't know about the number. He called it. There is Kelsey to 75. Travis Kelsey. Got another random here. It's going to be Tebow and Richardson number to 99. What's Tebow? Broncos AFC West and it'll be AFC South Colts there. We'll do another random. We've got two randoms so far. This one is AFC East. It's going to be Dalton Kincaid to 10 on the Gold Signature Moves Auto. Nice one there. AFC East. The Beals. Looking like a good tight end for them. Nice hit there. Dalton Kincaid. And Auto. This one is going to be introductions. Kaleija Kansi. I think that might be Bucks. I can see that he's on Bucks. This is what Sean deals with, huh? Yep. Every night. Let's go. Love it. He is on the Bucks. He is on the Bucks. NFC South. Kaleija Kansi to 49. We've got a RPA to 99. Xavier Hutchinson. Xavier Hutchinson. I don't know why I can't remember him. Texans. Texans. There we go, Jake. It's a 99 on the RPA. Texans. And last one is going to be a random to all. It's a nice one. His introductions. Auto Arch Manning. Seven of 99. Nice one there, guys. That'll be a random to all. He is still at Texas. He is still at Texas. We'll random that one. We've got three randoms so far. And on to Box 3. Nice arch. We'll mag that up. Hey, what up, Sammy? What's going on, sir? How we doing? Yeah, it does. We're ready to 75. We've got Dan Jones. NFC East. Giants QB. Quad Relics to 99. Lamb Jack. Moments on the Quad Relics. Louisville there for the AFC North. This one is RPA to 99. Going to be Davon at Shane. Out of A&M there. Dolphins rookie. Going to be AFC East on this one. At Shane. This one is Tank Bigsby. Another RPA number to 99. Jags rookie. AFC South. Tank Bigsby. Got another AFC South to 99. And we got Tarheel Nation. Josh Downs. Colts rookie. Another AFC South there. RPA to 99. And nice one on the back. We've got RPA to 65. It's going to be Jordan Addison. Nice one there. That's going to be the Vikings. Vikings. NFC North. Still got a chance at that dual auto as well for the Addison. Nice RPA there, Addison. Two boxes left. Dolphins got four to go, guys. Elite Donnerous. Last break of the day here. Four teams to go. Hey, what's up, Mitchell? Evening, sir. Evening. How you guys doing? Did your new castle play today? Me and Force Cherry's got a tie today. Brickford tied it up right at the right stop each time. Fortunately. It's getting one. Must be a redemption in here. Is a redemption in here. Got another Dan Jones to 75. This one's going to the NFC or NFC East again. Oh, 3-1. Who do you guys play? Who are they up against? 3-1 is a lot against for the... I don't know, for Newcastle. There is another random. This 25. This one is Keishan Bute, AFC East. And Jabbar Chase for the AFC North. Nice one there to 25. The former LSU receivers. Another random. Wow. It's a cool patch. This one is AFC South. Really cool patch here. AFC South is... Jason. With the SEC logo patch. One of two we got. Tank Bigsby. Boom. Nice one there. It's really sick. Bigsby, one of two. The Jag's new is running back. That is sick. Nice SEC big logo there. We'll mag that one for you. AFC South. Mr. Jason. We've got two 99. RPA Jalen Carter for the birds there in NFC East. Not bad at all. Jalen Carter. Got redemption. It is Rookie Auto Redemption, Joey Porter Jr. Now I want to go AFC North just like his dad, the Steelers. Joey Porter Jr. on the Steelers. RPA to 99. We got Jalen Hyatt Giants. Nice one there out of Tennessee. Jalen Hyatt. And one more box. Really Brighton 3-1. That's crazy. Brighton getting the 3-1 win. All right, you guys. Box 5, last one of the break here. Still got four teams to go, guys, in our last break of the day. We got Dolphins. Dolphins, Bucks, Titans, and Commanders. Last break of the daytime here. Then we got Super Fracture Sammy coming in at 5. And Fanatics Force will be live as well over on Fanatics Live at 5. The dual stream is going. Dalvin Cook there, number 35. Number 35 Jets, AFC East. Dalvo. Got dual. This one's going to be another random. It's Davis Mills, Texans, AFC South, and Tanner McKee, NFC East. Dual jerseys there out of Stamper to 99. So we got five randoms now. To 99, we got introductions. Lucas Van Ness on the rookie auto to 99. You know what's one, Jake? It should be Packers. Lucas Van Ness Packers. Packers, NFC North. This one's sick. Three of five, we got Zach Evans on the Nike Swoosh laundry tag. Nice one there, Rams. NFC West on the Zach Evans. Three of five on the laundry tag. It's a sick-looking one there. Nice hit, NFC West. Behind that, we have to 99, Auto Josh Downs. Signature Moves Auto, AFC South Colts. Nice one there. And last hit of the break, we've got Zay Flowers, number to 99. That is gonna be AFC North Ravens. Ravens on these Zay Flowers. Alrighty, that will do it for iMac. Let me get some randoms set up here, guys. Give me just a moment, guys. I got four multi-team randoms, and then we got this Arch Manning to 99. Randoms to all. Randoms to all on this one. Alrighty, guys. Give me just a second. Gonna set up... I got four or five randoms to do here. Do Arch first. Randoms to all. And then we've got McKee. This one will be NFC East. NFC East and AFC South. And we've got Boute and Jamar Chase. That one's AFC East and AFC North. Then we've got one Richardson and Tebow. Richardson and Tebow. That one will be AFC South and AFC West. Hey, good luck, Sammy, on the Star Wars break. I really like the new Star Wars, man. It has been looking really good. The new Galaxy. Good luck on that one, my friend. And then we've got this Duel Auto. Last one here is gonna be Addison and Keyshawn. Johnson, AFC East and NFC North. Hey, guys. Let me roll two dice. See how many times we're gonna go on the randoms here. We've got five randoms to go. Go three times. Two and one. Three times. Good luck here, guys. We'll start with the Arch. Alrighty, guys. Arch Manning here, number to 99 on the Auto. This one's gonna be random to all divisions here. We'll go three times. Good luck. One, two, three, NFC West. Gonna take home the Arch Manning. Nice one there, so 99. Really good prospect out of Texas coming up here. Next one we've got will be the McKee, Tanner McKee and Davis Mills, NFC East and AFC South. We'll go three times on this one. One, two, three, AFC South. Taking home the Tanner McKee and Davis Mills.  We've got Butte and we've got Jamar Chase. One's AFC East and AFC North. We'll go three times. One, two, three, AFC East. Butte there, the Patriots taking home that one. Now we've got the Tebow and Anthony Richardson Duel. AFC West, AFC South. Three times. One, two, three, AFC South. It'll be Colts there taking home the Richardson. And last one we've got Duel Otto, Keeshawn Vaughn and Jordan Addison 25-25, the Duel USC Autos there. Three times on this one. One, two, and three, AFC East. We'll take home that one. That is Keeshawn there for the Jets. Alrighty, guys. That'll do it for iMac Collegiate. Five box number 19 Renvision style. Thank y'all.


गर में जबसे आप बच्छे हो तबसे बात ऐदे हो किक सी चाडी कर हो आप ए, बही मेरे को यह एई निए पता है ने अगली दन क्या करोंगा? गर में जब से बच्छे हो थब से भात औरे बड़े हो किकसे शाधी करूए. अरे बाई मेरेगे तो यह नहीं पता है में अगले दन क्या करूएंगा. शाधी कसे करूए. चवग ते इस दर तो बूल नहीं साधी गदी बाद़्णा. तो ब्द at the problem is finding a life partner ठादाना, तो अप protesters, जो सच्ट ब्छण देखी, the most pressure to get married in India is around marriage in a country like India, बगी निया में, खडिया में देखी। But the hardest place to find the life partner is India, तो बगाले गरा करजा भी थे है. देखी देखी थो और्बीन कुई सैंटे च्र्तॉत, तो आप मिलो तो कैसे मिलो, फिर भी आमारी आबादी है, 140 क्रोट्स, आप नहीं जोग से पाड़, it was a big problem at that point, because people didn't know how to find a life partner. तो आप शादी पीपल आरे बरु तो सोट अग, तो आप शादी पीपल आरे बरु तो सोट अग, तो आप शादी पीपल आरे बरु, नहीं, उनको लगं को, रहा था,कि मैं अं ग़र साथी नहीं भो पीपल आरे बरु शादी पीपल आरे ब्रु तो सोट अग, जिस में एक अ वो रोग इन पीपल किसहो रहा थे, चि बीता, में दप की बादा पूझा रहा, नहीं जोगrecht नती, जो आप बाद दे थो सोथ सादी another thing you were talking about ौ, match 你 yapıyorsun in celebrates ौ Match And you were talking about fireworks You were worrying about the marriage ौament . ौ meaning ौ . तो वेस्ट क्या उता लोगमरुँँँँँँँँँ यागा प्रीज़ दिएह்गासा अब आद प्यक्रीनन टाएम द़ूरनाच मैरेच सब द़ाक्चिय हो जब पुरी फाँरी आद गईनात आपके पीचे लगग जाती में गो एज में दिन तुछग कि देखो थी के दो साँल और देदों... दो साल और देदो कर लेंगे आज वो रबेल्लिएस पीरेद गया आद नाईन्टीस का विन्यों तो मैरी अगेंस त्यो पैरेंस विष्यास तोड़े पीपल से आप विल फाँन दे चोईस आप आप एल एकसेसाचा दे चोईस आप परेरेद से पाँन देखाई से विल ख़ाई आप बचादी कर ले दो अच्छा है दो विल फाँन जी बहुत लेगा दो मैं भी ल्गाउ्� centre of the world कोई मैं मैं political कोई मैं कोई मैं लेए लिए और रहा हैंवेझ जे देख़े भी सब आजुए है आप़ रष्टा दी कुई प्रएटा थी खुडवें जे ऐंब यह आप ची। कोई मैं पच्टिठी क्योसे कि आठ आता मैद़ाक ग़ा की बासके लेगाता. मने नहीं की और स्टाल लाएस लेएः। मैंसार कुह्ती कुथ खुशित है। मैंसार कुछ आगर ठहार मैं वो सोँ हैंनें ऐस मैं बससाल रा गाउ लेएए स्टाला की लेएगी। फर्स ताए मैंन्गद शुक लेएगी। वो कमबि बशिँर बन गगे अगांई और ग़ापानागोंफ्ोावेटाउ तो मुझे हाई सेशा जो गज लगा नेकादू कि तो आबंदिशासे एक ख़ड्दांगा भाजा बागाता है लोगा अप दरालेक.  तो न्ड़िनेड दी नी था. और on the internet business शुल्ड़ कर चकछर गे दे. So it was very very challenging at the point. तो इसकबाद चाएद की आप सब लोग अई आँँ्ट्त्दंपीनोर हैं यह बंना जाते है. So you talked about, that all of you are entrepreneurs and they want to be one. तो वो आप में पश़ागागा वो के नहीं तो सुद्वाईगा आप ज़ादा वो और ओग़्ेद एक और ग़ाईगा आप आप वो और ग़ाईगा आप आप और प्चादाउ, आप ज़ागा इतागा अगर देदा, वो भगी रेगागा और ख़ी एक कहनेगोग ड़ू्द, � अार ग़ा तवोई compared to each other after me and emotionally mentally how do you make yourself strong enough to be able to over come that situation and you know live the next day as if nothing it doesn't matter तथ 갖고 बहुत नहीं की होग़े। that's tough i think you can't read a book or go to a course असे की मुजे इतना दुईत बना दो and say that make me so agat अगर सोने साच़ मैं जा को बहुत बादो। do you know this word deat यत्ना दीट बनाद गो की तो की रत्रे को ग़ा करते प्रो् सीगेएका, तो ईतना दीट बनाद हो की फेलिट दहास नाविक्त है। मैं दाट देउस ग़ा काँटी या। गरे आशका नावे टेग्गा नाविक्ते। ment how I remember back when I used to work as in a job in the U.S. jab mae, vak kmk liye  ghetto ko ndayata tha I think those were the last times I ever had vacations in my life because I didn't care key whop are妻 lak ja hai to mujhe kuch pharag neek. After you become an global entrepreneur, it's very hard to switch off. Because the last few years capital has been available for everyone अड हुई किस लूँ एद instead of the पहństर के कासी बजेंगा के वो नहkahता looks the media prince the success stories about पहने कैंथ क्हो की वो सरता डीई क्या फीरे रूके कि its a 6 Pilot network that goes from 0-1 के कहराद75% कोंगे विअ � vitamins playful के निनने भी दे का डीई Reserve को ऊपमे के दuallyा धि strengthening मवाले शाहो वह सेठारे बॉदिए। rid of the success stories भी रो�排 अध्छ़़ो,ज़ो ब को ईगठाए,ईगठाए,उब क्यो,गे बाज़ाय की,गठाए,ग़ईच़ därगठाए,गठाए,श्थदाय की,गर प्रठ़ोए आप काईस उरेई. तो आप आब लिएगा है, भी बगादान की सिर्ब आप रगादी ट्धाने किस only. पूग हैं, पर पूग क्मवीं सदानाा थरे जागा इस शाकटाऊ्रींक के, आज ऄत और काहे अगब थे कुई अची ठुफुट्ट्ट्मईर की आज अमना है और आद ओर रहे अगगगगगग पूग शाकटाऊंक को आप की अगगगग आम्ःाह आप कुओगग अघटाऊ after doing 250 and probably listening to 1,000 pitches you sort of maybe crack that formula where you know this start up makes sense and you know this is a start up that will work and therefore you should invest in it. So you know and donning both these hats both as an entrepreneur and as an investor how do you now evaluate a start up to say that this looks promising? Do you do it by economic numbers or do you sort of go by emotional aspect की इस आद्मी में पैसा लगान आद? Yeah I think it's a bit of both. But let me break down your question because I think there were quite a few questions in there. So is the question how do I evaluate companies? Yeah and you know honestly I mean what part of it is maths and what part of it is I would say art. Look I think it depends investing really is as much an art as it is a science right and that's not just me but any VC will tell you that. Particularly when you come in at a very early stage because when you come in at a very early stage there are so many variables stacked up against you right. So for me frankly there are only three things that matter at a very early stage right because So first let's break down a couple of things. I'll answer one question very specifically do I look at the finance side more in terms of return or how do I look at things. I am not great at financial engineering. I'm terrible at optimizing you know 18% or 23% or an IRR of 26% that's not me. That's great for private equity investors for chartered accountants and that nature but that's not me at all. So you have to play to your strengths what therefore I've chosen early stage investing because what that allows me to do is play for zero to a thousand returns right. So when you play for that order of magnitude even if you come out at 10x or 20x or 30x or 100x that IRR is going to be very very healthy right. I don't have to super optimize so that's kind of my play field because that's what I understand. I can given the time I've spent in technology and consumer businesses I can look at an idea right or a concept or a business plan and the team and have some sense of whether this will find market acceptance and because I'm coming in so early if I'm right the outcome is 100x. So I don't have to worry too much about you know super optimization so that's one thing in terms of how I evaluate companies and teams this really when it comes to early stage there's really only three things and I'll try to give you an example. I mean one is of course team right investing in a company is really a long term affair. If you ask me particularly in India companies don't get built in three four years it's a ten year journey right. So unless there's chemistry with those investors or with the founders and unless you enjoy meeting and talking to them what's the point. So besides high IQ which is a given you want to ensure that you have chemistry and you like look forward to talking to these people. So I think that's one the other is total addressable market and the reason total addressable market is important is because large markets are very forgiving. If there's a market of lakhs of crores then if the team is smart you will find a nice business but if the market is small no matter what you do you will struggle. And sometimes there is no market right so then you have to find proxies. So how do you determine total addressable market then you have to really get close to the customers and understand the problem. If customers are really you know screaming for this without actually articulating it if you can develop that sense then you know that the time can be big. And the third final vector is timing. I'll give you an example which phone do you think was the first which was the first hand held device which you could operate with touch in the world. Sorry Nokia N95 the first smartphone Samsung Galaxy. All right so clearly I'm tilting the average age here much higher but all of you guys are probably too young to know this but there was a device called the Palm Pilot which was way before Apple. It was a touch screen device right and people used to use it in the US for productivity for managing their work for managing their email. But why did Apple become what it did and Palm Pilot didn't the device existed many years before Apple simply because of timing right. When Apple launched the smartphone ubiquitous internet connectivity and therefore apps were available on your phone. Without the infrastructure of the internet and ability to always be connected to the internet you could not have had an apple phone. So the timing of the launch was perfect given a year or two window here and there. So you know that's just one example but my point is let's take for example creator led businesses or companies building infrastructure for creator led businesses. That was not possible five years back. It's only possible today because you suddenly have thousands and hundreds and millions of creators. So timing plays a very critical role and that is something that comes with the experience understanding the timing is right for this business or for this business model. So yeah team time timing T3 easy. And what about the person the person which is the team. Yeah the team the founder co-founders. Yeah I think I already mentioned that it's critical but everybody says team. Everybody says team. So I think what you want me to do is maybe go a layer below. And what about a team. So here's the key right. Here's what I look for in a team. One is do they there are three things or maybe maybe a little more. But first and foremost they have to be passionate about the problem they are solving. Right. Because as I said India made that's what I thought. And I got passionate. If you're just doing it because your friends have built a company and you also want your name in the newspaper saying you know raises series A of 10 million. You're not going to last very long. So if you're doing it because you're really passionate about solving that problem. I think that's one box that's checked off the other thing that I look for is business acumen. In Hindi it's called Dhanda. I come from a Baniya family. So for me Dhanda is very important. Good products do not necessarily make good businesses. People often mistake good product managers and good storytellers as good businessmen. Not true. And that's a big big red flag particularly in the last you know cycle that we've seen the liquidity and capital has been available. You've seen a lot of storytellers and product builders raise a lot of capital but are struggling to build businesses out of that. So that's something that I care for a lot. I try to examine that and finally co-founder chemistry. For me it's very important that the founders know each other for a number of years and have worked with each other. Either hostel roommates or a job together or have started up before. Because this whole fad of finding a co-founder in the last six months a year and then raising capital is a disaster. Exceptionally it might work. I know some cases where it has. But nine times out of ten it's heading straight for a disaster. You know Shadi ki baat kar rahe the poori zindagi nikal jaati husband wife mein nibaane ke liye. Imagine finding somebody six months and then building a business.  It's very difficult. So these are the things. No I think fair enough. And you're totally right that finding that chemistry outside marriage in a business is also equally important. So you know today we also have a lot of young entrepreneurs who you say you know jise aap kairon dhandegi baat unko samaj haane chige. Lekin they're so young that you know it's probably too much to expect ki un ke paas wo maturity or experience hoga. Now I have these two young daughters in this sitting in this room and listening to the talk. And they want to become entrepreneurs at a young age. So now my point is do and today we also talk about mentors. You know you need a mentor to be able to take you there. So how how should a young entrepreneur find that mentor early on to be able to find that dream that they want to realize. I mean you know I have another guy out there who is in 10 standard wanting to start a business. So how will it work for them. How will that mentorship come from them. Do we need a more formal mentorship infrastructure which might just come from our educational institutions. Or do we need to sort of have it. And you know you're a mentor to a lot of startup founders. So and you know I would want you to end this question by telling us.  But who is your mentor. So once again Mr. Rituji you have asked some 10-15 questions together. So we will try and unbundle them. You might have to remind me along the way. But let's start with mentors. Let's first and foremost debunk the mentor myth. Too many mentors can become tormentors. When there are too many voices telling you what to do. You should find the nearest door and get out of there. I think the biggest I'll come to mentors in a minute. I'll come back to this. So let's park that aside. Why because if you are going to be successful you are going to be successful despite not having a mentor. And if you're not going to be successful if you don't have that DNA. Mentor can make you successful. So we'll come back to that. It's too much. People are giving away equity to mentors and stay away from that. What you must do if you are a new or somebody who wants to become an entrepreneur. Especially somebody who is very young. The one thing that's critical is take up a sales job. If you really want to learn how to do business. Learn how to sell. What do I mean by that it could be anything. Summer internships you could be selling. Back then people used to sell in foreign directories. You could sell anything. It doesn't matter what. But interacting with people trying to sell them something getting money out of them. There is no better way to learn business. To learn how to be an entrepreneur. That's the first point I would make. The second point is doing is really the new thinking. Things change so fast that if you sit down to think of the perfect business plan and the perfect product and getting that right mentor. You're not going to get very far. You have to start doing and if you're doing something doesn't matter if you're embarrassed of it. In fact if you're not embarrassed about your product then you're doing something very wrong. You should be embarrassed about your products and about the business plan that you develop. But if you keep doing what does doing mean what does doing entail doing entails activity. It means momentum. It means moving forward. You will stumble across people who can help you along the way. Somewhere somehow you will meet people. And third today the internet has truly made the world flat. It is amazing but some of the top entrepreneurs in the world if you write an email from your heart and it's not hard to find their email addresses. And if you have a good story to tell you might be surprised that you get a response back from them.  That's not going to happen. The only thing a mentor can do in my view in my experience is maybe multiply your resources by 2-3-4 times. A million dollars and a right mentor. And what is a right mentor? A right mentor is not somebody who tells you what to do. A right mentor is somebody who asks you the right questions and you figure out yourself what you must do. Right? That million dollars with the right mentor will feel like 3-4 million dollars. Because it gives you, you can do in lesser time what you would have done otherwise. For example, a business like mine, let's say Shadi. I would say it's something that should be built in 5-6 years, not 20 years with the right mentor. Why did it take us so long? The time is very different. So that's what a mentor can do. They can accelerate your learning. And if you have the right one, I think it can go a long way. I think that's very well said that times change and we cannot always have the same mentors also. Because you need people on different stages of your business who are probably different for you. People are dying to ask questions from you here. I can almost see a hand going up. So I'll ask you this one final question before I give it to them to ask their questions. We talk about funding winter and everybody in this room wants to for you to invest in their business. Is it real? At what stage is it real for any business? What are your views on a funding winter? Do you think it's not possible to get a funding in today's time? Let me first start by saying this is the best time to build a business. The top companies in the world, whether it was Amazon, whether it's Google, whether it's Flipkart which is the biggest example in India. Our own company. All of these have been started in recessions or what is called the funding winter. When in climates like that you can make one rupee stretch far more than you would otherwise. The cost of employees goes down, attrition goes down, you are not forced to keep up with the next store guy. You can focus on the things that matter and what matters and what is this funding winter. Frankly funding winter, this is a business cycle.  I will come to you. I thought I said something wrong. You are so muscular. But funding winter is just a cyclical thing. You have boom periods and you have bust periods. It's creative destruction and it's absolutely needed. When a new technology emerges, when a new paradigm emerges, nobody knows what exactly will be successful. But it will change the landscape forever. So every technological paradigm will bring with it thousands of companies and more than 70-80% of those will fail. Maybe even 90%. And that's desired. That's not a bad thing. So don't look at funding winter and say oh my god now I can never get funded. That's not true. Companies are still getting funded. I'm going to give you two things to think about which will work for you not just in a funding winter but anytime in the course of building your business. I'm going to change the meaning of these acronyms that what is B2C. For me it is back to customers. Thank you. Somebody got it. But let me explain. What happens when money is readily available? We leave sanity and fall into vanity. We are not solving a problem for the customer anymore. What are we doing? We are seeing what we want to see. And what story the VC wants to hear so the money can come in. And nobody is to blame. Because if you don't raise your competitor, you will be out of the market. So you have to play that game. So everybody's focus shifts on other things. But what really matters as far as your customer is concerned. There is only two questions. Why will they love you? This could be true of your life partner also by the way. Why will they love you? And why will they not leave you? Very simply. Or jitane start-ups. Ye boom period mein start ho hain ha. Eighty percent of them cannot answer these questions. Because ho focus karne ka time ni mila. This is product market fit basically. Agar aapko koi product market fit ka definition puche. If you can answer these two questions well, you found product market fit. Okay. That's B to C. What is B to B mean? Back to basics. Thank you. Somebody is awake. Good. And what is back to basics? Dhe kye jahatak mein hai, I told you I am a Baniya family. Mere ko toh doh line samaj mein hai. Ek hai top line. Aur ek hai bottom line. Profit. Abhi paach different type ke toh profit ho gaye. CM1, CM2, CM3 adjusted Ibita, normalized Ibita, Ibita. These things are a great way. Sure there is some value as far as financial analysts are concerned on these numbers. But for the average person, these things should not hold any value. They are a great way to distract yourself. For example, sales kya hota hai? I asked a lot of founders, aapka sales kya hai? What should be that number? Basically sales is jo ghar mein bacha after returns, after discount, after tax. Tax to government le gaya aapka sales kya se hua? So you can't confuse GMV with sales. Employees. People take a lot of pride and write long LinkedIn post saying, today we have reached 1000 employees. How is that a good thing?  It's the most expensive thing in the world. You know what my definition of a great business is? Go to the office, plug a computer in, write an algorithm and never ever come back to work. Not a single employee. That's the best business in the world, right? Great margins, nobody to manage. But these excessive times shift our focus away from the basics to all the other things that keep you in the race to raise capital but are very rapidly taking your business towards a long, towards a tall, hard wall. So yeah, that's what I'd say in terms of I don't even know if that answers your question but No, I think it did. For a lot of it, probably it made a lot of sense to understand why profits are really important and more than profits, Kesh. If you're profitable, then even if you don't get funding you're good to go.