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1ldohbj4t | physics | work-power-and-energy | work | <p>A small particle moves to position $$5 \hat{i}-2 \hat{j}+\hat{k}$$ from its initial position $$2 \hat{i}+3 \hat{j}-4 \hat{k}$$ under the action of force $$5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$$. The value of work done will be __________ J.</p> | [] | null | 40 | The given expression calculates the work done by a force vector $\vec{F} = 5\hat{i} + 2\hat{j} + 7\hat{k}$ when it acts on an object that moves from an initial position vector $\vec{r}_i = 2\hat{i} + 3\hat{j} - 4\hat{k}$ to a final position vector $\vec{r}_f = 5\hat{i} - 2\hat{j} + \hat{k}$.
<br/><br/>To find the work done, we use the dot product of the force and displacement vectors :
<br/><br/>$$
\begin{aligned}
& W=\vec{F} \cdot\left(\vec{r}_f-\vec{r}_{\mathrm{i}}\right) \\\\
& =(5 \hat{i}+2 \hat{j}+7 \hat{k}) \cdot((5 \hat{i}-2 \hat{j}+\hat{k})-(2 \hat{i}+3 \hat{j}-4 \hat{k})) \\\\
& =(5 \hat{i}+2 \hat{j}+7 \hat{k}) \cdot(3 \hat{i}-5 \hat{j}+5 \hat{k}) \\\\
& =15-10+35 \\\\
& =40 \mathrm{~J}
\end{aligned}
$$ | integer | jee-main-2023-online-1st-february-morning-shift | 13,384 |
1ldsa408f | physics | work-power-and-energy | work | <p>Identify the correct statements from the following :</p>
<p>A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.</p>
<p>B. Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative.</p>
<p>C. Work done by friction on a body sliding down an inclined plane is positive.</p>
<p>D. Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero.</p>
<p>E. Work done by the air resistance on an oscillating pendulum is negative.</p>
<p>Choose the correct answer from the options given below :</p> | [{"identifier": "A", "content": "A and C only"}, {"identifier": "B", "content": "B and D only"}, {"identifier": "C", "content": "B, D and E only"}, {"identifier": "D", "content": "B and E only"}] | ["D"] | null | <p>When a man lifts a bucket out of a well using a rope, work is done by the man and the gravitational force. The work done by the man is positive as he has to exert an upward force to lift the bucket. The work done by the gravitational force is negative because the direction of the force is opposite to the direction of displacement.</p>
<p>Therefore, the <b>statement (A)</b> "Work done by a man in lifting a bucket out of a well by means of rope tied to the bucket is negative." is <b>incorrect</b>.</p>
<p>Therefore, the <b>statement (B)</b> "Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative." is <b>correct</b>.</p>
<p>Work is defined as the product of force and displacement in the direction of the force. When a body slides down an inclined plane, the force of friction acts against the motion of the body, opposing its descent.</p>
<p>The direction of the force of friction is opposite to the direction of the displacement of the body, which is downwards. Hence, the work done by the force of friction is negative.</p>
<p>Therefore, the <b>statement (C)</b> "Work done by friction on a body sliding down an inclined plane is positive" is <b>incorrect</b>.</p>
<p>
If the body is moving on a rough horizontal plane, there will be friction present, which will act in the opposite direction to the applied force. The force of friction will oppose the motion of the body, reducing its velocity. As a result, the net work done on the body will not be zero, as the force of friction and the applied force will not cancel each other out completely.</p>
<p>Therefore, the <b>statement (D)</b> "Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity is zero." is <b>incorrect</b>.</p>
<p><b>Statement E:</b> "Work done by the air resistance on an oscillating pendulum is negative."</p>
<p>This statement refers to the work done by the air resistance on an oscillating pendulum, which is a physical system that swings back and forth under the influence of gravity.</p>
<p>As the pendulum oscillates, it experiences air resistance, which opposes its motion and slows it down. The direction of the air resistance force is opposite to the direction of the displacement of the pendulum, which is back and forth.
</p>
<p>Hence, the work done by the air resistance force is negative, as the direction of the force and the displacement are opposite.
</p>
<p>Therefore, the <b>statement (E) </b>"Work done by the air resistance on an oscillating pendulum is negative" is <b>correct</b>.</p> | mcq | jee-main-2023-online-29th-january-evening-shift | 13,385 |
lgnz215a | physics | work-power-and-energy | work | A block of mass $10 \mathrm{~kg}$ is moving along $\mathrm{x}$-axis under the action of force $F=5 x~ N$. The work done by the force in moving the block from $x=2 m$ to $4 m$ will be __________ J. | [] | null | 30 | To calculate the work done by the force $F = 5x$ in moving the block from $x = 2m$ to $x = 4m$, we can use the formula for work done by a variable force:
<br/><br/>
$W = \int_{x_1}^{x_2} F(x) dx$
<br/><br/>
In this case, $F(x) = 5x$, $x_1 = 2m$, and $x_2 = 4m$. Now, we can substitute these values into the formula and evaluate the integral:
<br/><br/>
$W = \int_{2}^{4} 5x dx$
<br/><br/>
To evaluate the integral, we find the antiderivative of $5x$:
<br/><br/>
$\int 5x dx = \frac{5}{2}x^2 + C$
<br/><br/>
Now, we can find the work done by evaluating the antiderivative at the limits of integration:
<br/><br/>
$W = \left[\frac{5}{2}x^2\right]_{2}^{4} = \frac{5}{2}(4^2) - \frac{5}{2}(2^2)$
<br/><br/>
$W = \frac{5}{2}(16) - \frac{5}{2}(4) = 40 - 10 = 30 \mathrm{J}$
<br/><br/>
The work done by the force in moving the block from $x = 2m$ to $x = 4m$ is 30 J. | integer | jee-main-2023-online-15th-april-morning-shift | 13,386 |
1lguymsvs | physics | work-power-and-energy | work | <p>A force $$\vec{F}=(2+3 x) \hat{i}$$ acts on a particle in the $$x$$ direction where F is in newton and $$x$$ is in meter. The work done by this force during a displacement from $$x=0$$ to $$x=4 \mathrm{~m}$$, is __________ J.</p> | [] | null | 32 | <p>To find the work done by a force during a displacement, we can use the formula:</p>
<p>$$W = \int_{x_1}^{x_2} \vec{F} \cdot d\vec{x}$$</p>
<p>Here, the force is given by $$\vec{F} = (2+3x) \hat{i}$$, and we need to find the work done during a displacement from $$x = 0$$ to $$x = 4 \mathrm{~m}$$. Since the force is only in the $$x$$ direction, we can write the integral as:</p>
<p>$$W = \int_{0}^{4} (2+3x) dx$$</p>
<p>Now we can integrate the function with respect to $$x$$:</p>
<p>$$W = \int_{0}^{4} (2+3x) dx = \int_{0}^{4} 2 dx + \int_{0}^{4} 3x dx$$</p>
<p>$$W = \left[ 2x \right]_0^4 + \left[ \frac{3}{2}x^2 \right]_0^4$$</p>
<p>Now we can plug in the limits of integration:</p>
<p>$$W = (2 \cdot 4 - 2 \cdot 0) + \left(\frac{3}{2} \cdot 4^2 - \frac{3}{2} \cdot 0^2 \right)$$</p>
<p>$$W = 8 + 24$$</p>
<p>$$W = 32 \mathrm{~J}$$</p>
<p>So the work done by the force during the displacement from $$x = 0$$ to $$x = 4 \mathrm{~m}$$ is 32 Joules.</p>
| integer | jee-main-2023-online-11th-april-morning-shift | 13,388 |
1lgyflfn1 | physics | work-power-and-energy | work | <p>A closed circular tube of average radius 15 cm, whose inner walls are rough, is kept in vertical plane. A block of mass 1 kg just fit inside the tube. The speed of block is 22 m/s, when it is introduced at the top of tube. After completing five oscillations, the block stops at the bottom region of tube. The work done by the tube on the block is __________ J. (Given g = 10 m/s$$^2$$).</p>
<p><img src="data:image/png;base64,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"/></p> | [] | null | 245 | From work energy theorem
<br/><br/>$$
\begin{aligned}
&W_{\text {gravity }}+W_{\text {friction }} =\Delta(K E)=K E_f-K E_i \\\\
&W_{\text {gravity }} =m g h=1 \times 10 \times 0.3=3 \mathrm{~J} \\\\
&W_{\text {friction }} =0-\frac{1}{2} \times(22)^2-3 \\\\
& =-(242+3)=-245 \mathrm{~J}
\end{aligned}
$$
<p>The negative sign indicates that the work done by frictional force (the tube) is in the direction opposite to the displacement of the block. In conclusion, the work done by the tube on the block is <strong>245 J</strong>.</p> | integer | jee-main-2023-online-10th-april-morning-shift | 13,389 |
1lgyqdov2 | physics | work-power-and-energy | work | <p>A bullet of mass $$0.1 \mathrm{~kg}$$ moving horizontally with speed $$400 \mathrm{~ms}^{-1}$$ hits a wooden block of mass $$3.9 \mathrm{~kg}$$ kept on a horizontal rough surface. The bullet gets embedded into the block and moves $$20 \mathrm{~m}$$ before coming to rest. The coefficient of friction between the block and the surface is __________.</p>
<p>(Given $$g=10 \mathrm{~m} / \mathrm{s}^{2}$$ )</p> | [{"identifier": "A", "content": "0.65"}, {"identifier": "B", "content": "0.25"}, {"identifier": "C", "content": "0.50"}, {"identifier": "D", "content": "0.90"}] | ["B"] | null | <p>First, we will use conservation of momentum to find the velocity of the bullet-block system just after the bullet gets embedded into the block.</p>
<p>The initial momentum of the system is given by the momentum of the bullet (as the block is initially at rest), and the final momentum of the system is the combined momentum of the bullet and the block.</p>
<p>Setting initial momentum equal to final momentum:</p>
<p>$m_{\text{bullet}} \cdot v_{\text{bullet}} = (m_{\text{bullet}} + m_{\text{block}}) \cdot v_{\text{final}}$</p>
<p>Solving for ($v_{\text{final}}$):</p>
<p>$v_{\text{final}} = \frac{m_{\text{bullet}} \cdot v_{\text{bullet}}}{m_{\text{bullet}} + m_{\text{block}}}$</p>
<p>Substituting the given values:</p>
<p>$v_{\text{final}} = \frac{0.1 \, \text{kg} \cdot 400 \, \text{m/s}}{0.1 \, \text{kg} + 3.9 \, \text{kg}} = 10 \, \text{m/s}$</p>
<p>Next, we know the block comes to rest after moving 20 m due to friction. The work done by the friction force is equal to the initial kinetic energy of the block (since it comes to rest, the final kinetic energy is 0). The work done by friction is given by the friction force times the distance, and the friction force is equal to the coefficient of friction times the normal force (which is equal to the weight of the block). </p>
<p>So, setting the work done by friction equal to the initial kinetic energy of the block:</p>
<p>$\mu \cdot (m_{\text{bullet}} + m_{\text{block}}) \cdot g \cdot d = \frac{1}{2} \cdot (m_{\text{bullet}} + m_{\text{block}}) \cdot v_{\text{final}}^2$</p>
<p>Solving for ($\mu$):</p>
<p>$\mu = \frac{\frac{1}{2} \cdot (m_{\text{bullet}} + m_{\text{block}}) \cdot v_{\text{final}}^2}{(m_{\text{bullet}} + m_{\text{block}}) \cdot g \cdot d}$</p>
<p>Substituting the given values:</p>
<p>$\mu = \frac{\frac{1}{2} \cdot (0.1 \, \text{kg} + 3.9 \, \text{kg}) \cdot (10 \, \text{m/s})^2}{(0.1 \, \text{kg} + 3.9 \, \text{kg}) \cdot 10 \, \text{m/s}^2 \cdot 20 \, \text{m}} = 0.25$</p>
| mcq | jee-main-2023-online-8th-april-evening-shift | 13,390 |
luxwcu3m | physics | work-power-and-energy | work | <p>A force $$(3 x^2+2 x-5) \mathrm{N}$$ displaces a body from $$x=2 \mathrm{~m}$$ to $$x=4 \mathrm{~m}$$. Work done by this force is ________ J.</p> | [] | null | 58 | <p>To find the work done by the force when the body is displaced from $$x = 2 \, \mathrm{m}$$ to $$x = 4 \, \mathrm{m}$$, we use the formula for work done by a variable force in one dimension, which is the integral of the force with respect to displacement:</p>
<p>$$ W = \int_{x_1}^{x_2} F \, dx $$</p>
<p>Given the force $$F(x) = (3x^2 + 2x - 5) \, \mathrm{N}$$ and the limits of integration from $$x = 2 \, \mathrm{m}$$ to $$x = 4 \, \mathrm{m}$$, we can substitute these values into the equation:</p>
<p>$$ W = \int_{2}^{4} (3x^2 + 2x - 5) \, dx $$</p>
<p>Calculating the integral, we get:</p>
<p>$$ W = \left[\frac{3x^3}{3} + \frac{2x^2}{2} - 5x\right]_2^4 $$</p>
<p>This simplifies to:</p>
<p>$$ W = \left[x^3 + x^2 - 5x\right]_2^4 $$</p>
<p>Substituting the upper limit ($$x = 4$$) and then the lower limit ($$x = 2$$) into the antiderivative, and subtracting the latter from the former, we get:</p>
<p>$$ W = \left[(4)^3 + (4)^2 - 5(4)\right] - \left[(2)^3 + (2)^2 - 5(2)\right] $$</p>
<p>$$ W = (64 + 16 - 20) - (8 + 4 - 10) $$</p>
<p>$$ W = 60 - 2 $$</p>
<p>$$ W = 58 \, \mathrm{J} $$</p>
<p>Therefore, the work done by the force as the body displaces from $$x = 2 \, \mathrm{m}$$ to $$x = 4 \, \mathrm{m}$$ is $$58 \, \mathrm{J}$$.</p> | integer | jee-main-2024-online-9th-april-evening-shift | 13,393 |
luyit9wg | physics | work-power-and-energy | work | <p>A particle of mass $$m$$ moves on a straight line with its velocity increasing with distance according to the equation $$v=\alpha \sqrt{x}$$, where $$\alpha$$ is a constant. The total work done by all the forces applied on the particle during its displacement from $$x=0$$ to $$x=\mathrm{d}$$, will be :</p> | [{"identifier": "A", "content": "$$\\frac{\\mathrm{m}}{2 \\alpha^2 \\mathrm{~d}}$$\n"}, {"identifier": "B", "content": "$$\\frac{\\mathrm{md}}{2 \\alpha^2}$$\n"}, {"identifier": "C", "content": "$$\\frac{\\mathrm{m} \\alpha^2 \\mathrm{~d}}{2}$$\n"}, {"identifier": "D", "content": "$$2 \\mathrm{~m} \\alpha^2 \\mathrm{~d}$$"}] | ["C"] | null | <p>To find the total work done by all forces applied on the particle during its displacement, we can use the work-energy theorem which states that the work done by all forces on an object is equal to the change in kinetic energy of the object. So, we first need to find the initial and final kinetic energies of the particle and then calculate the work done.</p>
<p>The velocity of the particle is given by $$v = \alpha \sqrt{x}$$,
<p>and the kinetic energy $$K$$ of the particle is given by $$K = \frac{1}{2} m v^2$$. We can substitute the expression for $$v$$ into this formula to get the kinetic energy as a function of position $$x$$:</p></p>
<p>$$K(x) = \frac{1}{2} m (\alpha \sqrt{x})^2 = \frac{1}{2} m \alpha^2 x$$</p>
<p>To find the total work done from $$x = 0$$ to $$x = d$$, we need to compute the difference in kinetic energy between these two points:</p>
<p>$$W = K(d) - K(0)$$</p>
<p>At $$x = d$$,</p>
<p>$$K(d) = \frac{1}{2} m \alpha^2 d$$</p>
<p>At $$x = 0$$, since the particle starts from this position,</p>
<p>$$K(0) = \frac{1}{2} m \alpha^2 (0) = 0$$</p>
<p>So, the work done $$W$$ is simply the kinetic energy at $$x = d$$,</p>
<p>$$W = \frac{1}{2} m \alpha^2 d - 0 = \frac{1}{2} m \alpha^2 d$$</p>
<p>This matches with Option C:
<p>$$\frac{m \alpha^2 d}{2}$$.</p></p> | mcq | jee-main-2024-online-9th-april-morning-shift | 13,394 |
lv7v4rao | physics | work-power-and-energy | work | <p>A body of mass $$50 \mathrm{~kg}$$ is lifted to a height of $$20 \mathrm{~m}$$ from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :</p>
<p><img src="data:image/png;base64,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"/></p> | [{"identifier": "A", "content": "$$2: 1$$\n"}, {"identifier": "B", "content": "$$\\sqrt{3}: 2$$\n"}, {"identifier": "C", "content": "$$1: 1$$\n"}, {"identifier": "D", "content": "$$1: 2$$"}] | ["C"] | null | <p>Work done in both cases is equal to $$-m g \Delta h$$</p>
<p>$$\therefore \quad \text { Ratio }=1: 1$$</p> | mcq | jee-main-2024-online-5th-april-morning-shift | 13,395 |
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