Welcome to this lecture on aspects of western philosophy, which would focus on the philosophy of Immanuel Kant. This is module 22nd and lecture 22nd of this course aspects of western philosophy and this lecture we would see the following topics. We will see the notion of the ideas of reason, which is very central to Kantian philosophy in the sense that the previous lecture I pointed out that the entire Kantian philosophy can be understood with the three transcendental critics or the three transcendental approaches. One is the transcendental analytic, the other one is transcendental aesthetic, the first one is actually transcendental aesthetic, second one is transcendental analytic and the third one is transcendental dialectic. In one sense transcendental dialectic, we have already discussed the other two transcendental aesthetic and the transcendental analytic in the previous lectures. This lecture would rather focus on transcendental dialectic, which would in one sense try to see how these different critical approaches can be brought into one single framework and some of the limitations of this philosophy can also be visible here or rather some of the limitations of philosophical contemplation as such as Kant envisaged or Kant's conceived them can be seen when we try to understand transcendental dialectic. So, this is a figure which we have explained in the previous lecture, so I am not going to the details. So, the sensibility part is dealt with transcendental aesthetic, understanding where we discuss the 12 categories of understanding are where discussed in transcendental analytic and now we are going to see reason. So, what Kant would assume is that the human mind or human thinking faculty be divided into two aspects understanding and reason and reason of course, he keeps at a slightly higher domain. So, we are going to see that in this lecture and transcendental dialectic in that sense is a critic of understanding and reason both because it tells us about the limitation of understanding that if you try to apply the categories of understanding to certain domain then you would ultimately land into certain troubles. So, that is what transcendental dialectic wants us and also it tries to understand reason the capabilities of reason the possibilities of reason and the role that the ideas of reason play in the entire philosophical or philosophizing enterprise. So, the critic of understanding and reason their claims to provide knowledge of things in themselves. So, in the previous lecture I have made this distinction between reality as we see it as we experience it as which is given to us in our experiences which is known as phenomenal world and there is a nominal world there is a reality which is behind it which underlies that which cannot be known. So, in that sense we have seen that Kant is an agnostic. Now, this agnosticism is being explicated further with transcendental dialectic and he wants about the misuse of the a priori concept and principles these concepts these categories of understanding the 12 categories of understanding which we have discussed in the previous lecture have got certain limitations we have already seen it in the previous lecture itself. So, this lecture would rather explicate further on that aspect that you we are not supposed to apply this these categories on nomina on things in themselves it is a critic of the metaphysical use of understanding and reason and it wants about the illegitimate extension of the a priori concepts from the objects given in sense intuition to things in general. So, these a priori concept these a priori categories their application has a limitation. So, it tells us that so far we are within these limitations they are legitimate physics for example, natural sciences for example, they actually require an application of these categories in the sensible reality. So, long as we confine ourselves to that domain it is absolutely legitimate, but once we try to cross it go beyond that and apply these categories to nomina to reality with a capital R then we would ultimately land into troubles. The cognitive function of the categories are limited to the objects of sense intuition or phenomena beyond that we are not supposed to do apply them not possible to have universal and necessary or a priori knowledge of anything non perceivable. So, sensibility transcendental aesthetic understanding transcendental analytic. So, these two things combined we will give we will get knowledge about the world, but there is no a priori knowledge possible about anything that is non perceivable, because perception is important. Percepts without concepts are blind and concepts without percepts are empty. So, both of these things are necessary for having knowledge about the phenomenal world or a priori knowledge about the world. Now, what about nomina see Ken would say that nomina is unknowable it is something which lies behind its existence is never doubted you cannot just because we do not experience it we cannot know about anything about it does not mean that it does not exist it exist without that it is a precondition without that we cannot have knowledge at all. So, the Ken never doubts its existence philosophy envisages understanding the ultimate unconditioned reality that lies behind the phenomenal world see Hume also suggest something in this direction that you can never know anything with certainty about matters of fact. And these are the two domains according to Hume matters of fact and this relations nothing else knowledge is confined to these two domains and as far as the knowledge about the matters of fact is concerned there is absolute uncertainty he rejects causality all these things we have discussed in detail in previous lectures. So, I am not going to the details, but the point here is that there is a tendency and philosophy and metaphysics to know to seek to understand what is that which lies behind that underlying reality that propensity is something which we cannot deny it is there. So, to frame conceptions of God freedom immortality all these concepts in one sense are not part of the phenomenal world they are not part of this world they are transcendental concepts, but at the same time there is a tendency philosophy has this tendency to frame conceptions about these concepts these entities if at all we can call them as immortal and immortal soul a free soul God a world a cosmos which includes all these entities in this world. So, the conception about them apparently metaphysical conception about them something which the mind cannot resist framing. So, we have the three most fundamental concepts which later can would call them as ideas of reason solve the unitary and substantial solve something which is substantial something which unites everything the cosmos which is infinite world process as a unity apart from this diverse things which is transitory we see around there is a world there is a cosmos which unites everything that goes around and there is a concept which is a supreme unity of everything. So, God is a totality of existence everything is converged in God. So, these are some concepts which we have which is in one sense we can say that the result of our metaphysical propensities, but something which mind human mind cannot avoid envisaging the tendency cannot be avoided cannot be ignored that is part of human reason. So, now can talks about them on the one hand he says that it is not possible to talk about them in the sense that we can have knowledge about them trying to know them in the same way we try to know objects in the world that is impossible that is metaphysics trying to frame concepts a priori concepts about them is impossible, but at the same time it is very important that we have to think about them we have to assume their existence because they have a very important regulative role to play in our conceptualization we will see that. So, knowledge about nominah is something when we apply the categories of understanding to nominah leads to illusions when we try to apply the same category the 12 categories quality, quantity, modality and relations to these metaphysical domain say for example, God. Let us say you know even the question of existence of God itself when you try to prove God's existence how do you try there are several methods by means of which philosophers have attempted to prove God's existence theologians and philosophers and Kant would ridicule all of them he would say that all these attempts would ultimately lead to kind of illusions transcendental illusion he calls them because thought can never explore what lies behind nature as its ultimate ground it is a ground of what we see human thinking or human thought can explore only what is provided to it through sensations, but what lies behind these sensations that is something which human mind cannot access. So, whatever you talk about it in whatever way you try to explain it or understand it, it is ultimately going to end up in a kind of illusion you can never know them nominah can never become a proper object of our investigation. So, in that sense impossibility of metaphysics, metaphysics is impossible as a science because in science what happens is that as science we try to gain knowledge about a reality which is in front of us which is given to us in experience. So, we have in Kant's own language a priori synthetic a priori propositions are possible about the world. So, only when there are synthetic a priori propositions we can talk about scientific knowledge. Now, let us extend this possibility to a domain of reality to the domain of ultimate reality to the domain of nominah there you know if you try to derive or if you try to gain the so called a priori synthetic a priori knowledge it leads to illusion it is impossible because there is no percepts. Metaphysics attempts deducing a priori synthetic knowledge from the pure concepts of understanding this is to venture the concepts alone without percepts and concepts without intuitions are empty. Therefore, metaphysics is impossible metaphysics as a science is impossible in the sense that if you try to derive or if you try to develop a metaphysics in the line of physics where which is constituted of synthetic a priori propositions it is impossible it is not going to gain it is not going to yield any result any fruits at all. Now, again applying a priori conceptions to things in themselves leads to and anomies metaphysics as a possible source of objective knowledge is completely excluded from Kantian scheme of things and again questions which are legitimate when asked about the world of experience are meaningless when asked about the transcendental reality. For example, let us say causality a causes b my clapping causes the sound this is quite legitimate I can understand it as far as you know I apply these categories of relations relation of causality to the empirical world but when I try to extend this to at the non empirical world to the world of nominal reality God whether God caused this when I raise such a question it leads to nonsense because I can never understand I can never prove it I can never be sure about it. Example notions like cause and effect substance and accident are perfect and legitimate when applied to the phenomenal order and transferred to the nominal world they lead to nonsense. So, to say that why is it that it is not raining there is doubt it is not raining because God is angry they lead to a kind of blind beliefs which have no basis in experience no basis in scientific theorizing. So, you can never claim them as knowledge they are just believes they are never they can never be elevated to the status of knowledge because there is no confluence of or there is no coming together of percepts and concepts in such cases and they are called transcendental illusion principles of the understanding are imminent principles the categories of understanding are imminent cause effect all these things are imminent and mistaking imminent or subjective they are subjective principles for objective principles or transcendental principles is the result of this will ultimately result in error and illusion according to Kant. When we apply the subjective principles to things in themselves to reality to nominal reality that leads to transcendental illusion and transcendental dialectics wants us against that it tells us that this is the past this is where you know you end up in transcendental illusion and you have to come out of it you have to how do you come out of it you can come out of it only by knowing the limitations only by knowing the limits of your understanding. So, transcendental dialectic endeavors freeing us from this dogmatic illusion or the transcendental illusion by means of a critic a critic that will limit our speculative pretensions to the sphere of possible experience it tells us categorically that you can apply these categories of understanding these a priori categories to this domain not beyond that do not extend it any more beyond this what is in front of you. So, that is how it tells you and transcendental dialectic intends to free us from our transcendental illusions it also explores the roles of transcendental ideas in that sense we can say that there are two roles of reason there is a negative role as well as a positive role the negative role is to tell you that these are the limitations and there is this propensity of human mind to go beyond to conceptualize what is lying beyond that tendency might lead you to troubles, but at the same time the reason can also tell us about transcendental ideas which are inevitable for our theorizing. So, transcendental dialectics deals with both it has a negative as a and a positive role it assigns to reason. So, let us see what is a negative role as reason it leads us to transcendental illusion I have already mentioned it and it is a source of all metaphysical errors because we have a tendency to apply these categories to things in themselves we tend to apply them to all reality that is in front of us and then it leads to anti normies I will explain what anti normies is slightly later. Now, the positive role which is a necessary role of reason this is what this lecture is primarily concerned about it is the source of the necessary ideas reason is the source of the necessary ideas and principles that play an essential role in scientific theorizing. So, this is what we are trying to see now it has a positive role this metaphysical propensity the so called metaphysical propensity which has the potential to lead us into dangers has got a very positive it actually suggests the possibility of certain ideas certain transcendental ideas which are called the ideas of reason they play an essential and a very important role in scientific theorizing. It examines the higher processes of the reason to see whether it is possible to discover the ultimate nature of things in themselves and it is a source of transcendental concepts or ideas I have just mentioned we have already seen what are these basic transcendental ideas the first one is the solve the self the second one is the world or the cosmos the third one is God. So, these are the fundamental basic transcendental concepts or ideas now when we talk about imminent and transcendental principles since I have already mentioned about it the imminent principles are known as a priori concept of understanding I have already discussed this their principles which are applied within the confines of possible experiences and our subjective on the other hand transcendental principles are known as ideas of reason and the principles which transcend within the confines of possible experience and their objective. So, we are going to focus on these objective transcendental ideas of reason. I have already mentioned in the beginning that there is a way in which Kant makes a distinction between reason and understanding reason according to him is a higher function of the mind than of understanding and it is a mind's activity which is concerned solely with the enquiry as to its own operations and metaphysics is the occupation of reason with itself in that sense metaphysics is the occupation of reason with itself on the other hand understanding is it deals with objects of knowledge in experience referring particular percepts to general concepts etcetera this we have already seen the 12 categories of understanding. So, we are going to focus more on reason and what reason does and in what sense metaphysics can be reaccommodated not as a science, but as a regulative ideal. So, the transcendental dialectics in that sense has certain very positive functions the pure reason as a faculty distinct from understanding as we have already seen to determine what are the transcendental ideas of pure reason. So, what Kant would say is that reason would analyze itself and tries to understand what are the transcendental ideas of pure reason on the basis of which it operates this is also an attempt to find out the legitimate and proper function of the ideas of pure reason we have already mentioned that these ideas the metaphysical ideas say for example, the idea of a world which encompasses all these events which we see around us it is a metaphysical idea we cannot talk about such a world in the scientific sense of the term we can never have a knowledge about it as such an a priori synthetic knowledge about it because there is no percepts or corresponding to them. So, but at the same time the concept of world as a hall cosmos or the concept of self not the empirical self, but a transcendental self which is permanently there we cannot talk about it from the scientific sense of the term see for example, we can never frame synthetic a priori knowledge about the self which is transcendental synthetic a priori knowledge is possible only about the empirical self and the empirical self is an entity which changes from time to time. So, here on the one hand Kant wants us against the extension of these categories to that domain, but at the same time he says that these ideas of pure reason like self or God or cosmos they have a very important function in human thinking. So, what is the legitimate and proper function of the ideas of pure reason is something which transcendental dialectic would explore they arise in us through the very nature of our reason. So, Kant would say that that is where he says you know reason has to find it out within itself reason when reason turns itself to itself. So, it finds these transcendental ideas only their misuse is deceptive he says that they are there, but when you try to sort of extend the categories to understand them then they are problematic. So, the proper function of the ideas of pure reason are determined by the constitution of our reason. So, transcendental dialectics aims at understanding analyzing and examining the constitution of reason itself how do you do that. So, the same method which Kant applied when he tried to understand the process of understanding the constitution of understanding what he did was he examined the logical judgments. So, here also he does something very similar to that he refers to the syllogistic process, but before that let us see more about the ideas of pure reason they are inherent in the nature of reason not innate, but not derived empirically they are they are there in reason. Transcendental ideas produced by pure reason reason contains within itself the source of these ideas. So, it they are contained within reason they are the foundations of reasons construction and account of the systematic unity of experience without this systematic unity of experience there is no knowledge possible there is no understanding possible there is no conceptualization possible. So, for reason to conceptualize and derive ideas and concepts and knowledge there should be a systematic unity of experience presupposed and that is possible only with the function of reason which synthetically unite them it is this synthetic function of reason that is reflected in the construction of these transcendental ideas. The ideas are self cosmos and God I have already explained it. So, this figure would give you an idea about the ideas of reason self is as permanent substantial subject of course there is no synthetic a priori knowledge possible about it because it is permanent substantial subject and human already demonstrated that it is impossible to have sensations or perceptions about such a permanent substance we have only bundle of perceptions according to human and that is acceptable for Kant as well to some extent, but at the same time he says that self as a permanent substantial subject should exist as an idea of pure reason as a transcendental idea and then the world the cosmos again human denied it for human it is nothing but impressions there are multitude of impressions one after the other and it is the habit of the mind to frame relationships between one and the other when I say a causes b it is my problem it is my mind which is superimbosing this causation relationship on these two events of a and b which appears successively one after another it is my habit of the mind, but Kant would say that there exist such a world as a totality of causally related phenomena a totality the world as a totality a cosmos that is another transcendental idea. So, these two are the transcendental ideas and when he comes to the third one God as the unity of the conditions of objects of thought in general it is a unity. So, it is a transcendental ideal that is the only transcendental ideal the other two are transcendental ideas, but God in one sense unites even these two world and self are united in God. So, that is all encompassing all uniting concepts and these ideas of reason are human mind as a tendency go back to that it continually sinks back to these ideas of reason to understand things you know not merely arbitrary, but has some validity the decide to grasp things as a hall. So, this is something which is a very important human urge the urge of reason to grasp things as a hall and project an ideal towards which knowledge is directed. So, that is why you have this each of these transcendental idea of pure reason is a uniting force it unites the subject unites the concept of self unites all our experiences into one point the eye the world unites everything all the causally otherwise causally interrelated discrete unrelated phenomena into one hall as world and God unites everything everything that exist is united in one single concept the concept of God. So, here the ideas of reason project an ideal towards with knowledge is directed now the question is how do we justify them? The ideas of pure reason are not given through the ordinary channels of experience this has been stated without any doubt in the very beginning itself and they arise in us through the very nature of our reason can this already pointed out that reason has this propensity or the very structure of reason itself suggest that there is a tendency to seek for unity have their function determined by the constitution of our reason this uniting this architectonic function and reason tends completing the synthesis achieved by the understanding. Understanding is already achieved a kind of synthesis, but this synthesis is again incomplete this has to be completed and this can be completed only with more uniting concepts like self cosmos and God deduced from the forms of mediate inference. So, here again as I pointed out earlier that Kant turns back to logic where he examines the very structure of syllogistic reasoning or the very nature of syllogistic reasoning what happens in syllogistic reasoning the process of reason as essentially syllogistic and here what happens is that there are three forms of syllogistic procedure which is given in this diagram. So, the first one is categorical second one is hypothetical the third one is disjunctive and corresponding to this categorical you have the psychological idea of self corresponding to the hypothetical you have the cosmological idea of the world and corresponding to disjunctive syllogism you have the transcendental idea of God. So, corresponding ideas of the pure reason is given here the three forms of syllogistic procedure the three types of possible mediate inferences which the mind makes which reason employs in understanding the world or in its exercises and this corresponding to the three types of possible mediate inferences categorical hypothetical and disjunctive there are three categories of relations substance which is represented by the self or the soul cause which is represented by cosmos and community or reciprocity which is represented by God and corresponding to the three types of inference there are three types of three kinds of unconditioned unity self cosmos and God postulated or assumed by the principles of pure reason. So, you have the entire thing represented in this figure three forms of syllogistic procedure the three types of possible mediate inferences are categorical hypothetical and disjunctive and three categories of relation that correspond to them substance to categorical cause to hypothetical community to disjunctive and again the three kinds of unconditioned unity substance self cause world and community God. Now, again to derive the three kinds of unconditioned unity from the three types of syllogistic inference ascending by a chain of categorical syllogism reason seeks something which is always subject and never a predicate that is a self a subject or the self is always a subject it is never a predicate. So, reason demands that or it seeks to identify such a subject which is never a predicate but always a subject and again ascending by a chain of hypothetical syllogism if this then that that is hypothetical syllogism reason demands an unconditioned unity in which an ultimate which is an ultimate presupposition for all these conditional things to happen all these causal relationships to function and that is the cosmos and finally, ascending by a chain of disjunctive syllogism reason demands an unconditioned unity in the concept of God. So, what exactly happens in syllogistic process here again we could see that you know Kant talks about the natural propensity of the mind I have already pointed out that there is a compulsion in the mind there is a natural compulsion of the mind to expect that its knowledge should be capable of unification and systematization. So, knowledge is possible only when there is a systematization only when a unification is possible only when everything is well established and the nature of the syllogistic procedure suggests the metaphysical ideals of God of self and the world. So, these are the unconditioned unity the principles of unconditioned unity the ideas of transcendental reason. So, they are presupposed their first particular cases to the universal which accounts for them and human thought looks for some complete central all comprehensive idea. So, this notion of a complete all central and all comprehensive idea actually is part of the natural compulsion of human mind to look for completion to look for systematization to look for unification and interestingly we can see that these three ideas of reason which we have already mentioned which is actually nothing but the result of the human mind's propensity or tendency for unification and systematization. And which can be found out from the examination of the syllogistic process they corresponding to these three you have three branches of speculative metaphysics you have the thinking subject which is the object matter of psychology the world which is the object matter of cosmology and God which is the object matter of theology and can would say that as metaphysics when you approach these three ideas metaphysically when you try to see them with the expectation of deriving synthetic a priori knowledge about them that would lead to kind of metaphysical transcendental illusions they are not given in experience none of them are given in experience since they are not given in experience there are no corresponding percepts possible since no percepts possible with concepts alone you are trying to understand them which leads to illusion. Because concepts without percepts are empty they are not phenomena. So, now let us see ideas of reason and their misuse say for example the self the self as far as we conceive the self as a transcendental idea of reason what is it it is the mind seeks a common ground for all phenomena that occurs in consciousness it is a result of that propensity of the mind which seeks for a common ground for all the phenomena that occurs in consciousness. So, everything is referred to one point one focal point that point the I always a subject and never a predicate of some other subject I everything is given to that and for the possibility of experience all representation should be related to the unity of a perception. So, everything is united everything is converge to that subject. So, that there is knowledge systematic knowledge possible that is a concept of self the I think that accompanies all experiences and reason seeks to complete the synthesis of the inner life in the idea of a central self or the absolute subject of our experiences. So, this is what reason does it tries to complete the synthesis of the inner life in the idea of a central self a permanent self an absolute self which is immortal and then reason passes beyond the empirical and the conditioned ego to the unconditioned substantial subject which is transcendental. So, from empirical to transcendental this is what reason does. Now, critical rational psychology what happens is a psychology studies the empirical ego as the self that is what psychologists do the focus is on empirical self which is in the world which is very much part of the world and again empirical ego is an object in time and is reducible to successive states. The humane subject is also an empirical subject here it is an object in time and is reducible to successive states and on the other hand the transcendental ego which is an idea of pure reason is a necessary condition for experience that is the unitary conception. The empirical ego as such never provides any unity because it is it can be reducible to successive states what brings these different discrete states into one single unity that is something which you cannot see you cannot experience that is something which needs to be presupposed something which regulates our examinations that is a transcendental ego. It is not given in experience the transcendental ego can never what is given in experience is these discrete unconnected successive states of consciousness but empirically that is the empirical ego transcendental ego is never given in consciousness hence cannot apply the categories of substance and unity to that. See to conceive of a substantial self to conceive of a permanent substance self as a permanent substance you have to conceive it as a unitary substance but unity is a category how can you apply the category of unity to something which is transcendental this is where you know you end up in metaphysics does not belong to the world. So, in one sense Kantian self or subject is the limit of the world it is not the part of the world hence cannot be studied scientifically. Again let us come to the second one the cosmos the underlying system of order and law that ground all objects of knowledge everything in this world any knowledge about this world is possible with this conception with this presupposition of a cosmos which provides an underlying unity of entities in this world. The idea of a comprehensive world system the totality of causal sequences. So, in our empirical experience we do not see this totality we see only instances of such causal sequences but a totality of the causal sequence is something which you which is never given to experience but it is presupposed. Understandings in the sizes the manifold of sense intuition according to causal relation and reason here again tries to complete it tends to complete the synthesis by reaching an unconditioned unity conceived as the totality of causal sequences. So, here reason postulates an unlimited ultimate presupposition of the totality of the causal sequences of phenomena that is the cosmos. An ultimate presupposition of the totality of the causal sequences of phenomena which is never empirically given in experience but which is postulated by reason as a necessary precondition to even venture into knowledge about the phenomenal world. And now critiquing speculative cosmology this is the result of our metaphysical propensity this might lead to kind of metaphysical illusions or transcendental illusions. In speculative cosmology the idea of the world as a totality of the causal sequence or phenomena is at the center extending our knowledge of the world as a totality of phenomena through synthetic a priori propositions lead to antinomies. So, here comes the antinomies the famous antinomies of Kant when we apply forms of intuition and the categories of understanding to things that are not experienced antinomies appear speculation concerning the nature of the world lead to antinomies. When you try to understand when you try to speculate the nature of the world it might ultimately lead to different kinds of antinomies and according to Kant there are 4 types of antinomies. What is an antinomy? An antinomy constitutes of 2 contradictory propositions mutually contradictory propositions each of which can apparently be proved. So, that is the problem you have 2 propositions one would contradict the other but you can prove both which one is right that is the question. So, there are 4 antinomies where they arise when we change thoughts into things and hypothesize them we built an imaginary science on these things both their assertion and denial are the result of illusion and there are Kant talks about mathematical and dynamical antinomies I am not going to the details I will directly go to the first antinomy there is a thesis which says that the world has a beginning in time and is also limited as regard space. This is a thesis it says that the world has a beginning in time and is also limited as regard space and the antithesis is the world is infinite and has no beginning in time and is not limited to space. Apparently both these statements one would contradict the other but apparently both of them can be proved or can be disproved you cannot disprove them. So, both of them apparently looks right now let us go to the second antinomy the thesis is everything in the world consist of simple parts and the antithesis is there is nothing simple but everything is composite again one contradicts the other but which one is right that is the question. The third one there are two kinds of causality one according to the laws of nature and the other that of freedom the antithesis says there is only causality according to the laws of nature again you can neither prove them nor disprove them. The fourth one there is an absolutely necessary being belonging to the world either as its part or as its cause the antithesis there is not an absolutely necessary being existing in the world not does it exist outside the world as its cause. So, which one is right how do you prove them how do you disprove them apparently both of the one contradicts the other but you cannot say that one is right the other one is not right. So, in this context this the conceptualization of cosmos the metaphysical conceptualization of cosmos would ultimately lead to this irresolvable you cannot resolve this problem the antinomies the problems created by antinomies. And now comes the third idea of pure reason the idea of God the transcendental idea of God is the transcendental ideal I mentioned that the other two ideas are transcendental ideas they are called transcendental ideas but the idea of God is conceived as a transcendental ideal by Kant reason seeks to unconditioned unity. So, in the other two you have you know the self unifies all those appearances in human consciousness the world unifies the concept of cosmos the idea of cosmos or the idea of world unifies all those causally interconnected phenomena into one single unity. Now, the third concept the idea of God is provides you the notion of a final unity the final idea in which the thought can rest satisfied the supreme condition of the possibility of all that is thinkable. So, everything converges to this fantastic idea of God in Kant's philosophy and mind search for more unity and comprehensiveness make it more towards some higher center of unification. So, more and more it is actually moving towards higher and higher unity from self to world and now to God mind refers both the self and the world to some all comprehensive idea. So, even these two concepts are unified with the notion of God which is the ultimate unitary concept all I an idea which grounds both the self and the world is the idea of God again the idea of the sum total of all possible predicates containing a priori the data for all particular possibility. So, every possibility is contained in it. So, it is such an idea such a grand idea which unifies everything the idea of the aggregate or some total of all possible perfections the idea of the most perfect being of the most real being and the union of unlimited pure perfections in one simple being. So, this is again another propensity of the mind its propensity to see things in a unitary fashion synthesizing way and now Kant advances a critiquing of this philosophical theology by saying that reason seeks the unconditioned unity of all possible predicates which cannot be found empirically. So, philosophers and theologians have been attempting to prove the existence of God by advancing several theoretical proofs ontological proof cosmological proof and Kant ridicules all of them Kant would say that the existence of God is not something which can be empirically proved there cannot be a synthetic a priori proposition possible about God's existence after all existence is not a property it is not a predicate for Kant reason has to pass beyond the condition and hypothetize an individual being who is perfect. So, that the concept of God is actually the result of this propensity of reason this requirement this necessity or this demand of reason to pass beyond the condition and hypothetize an individual being who is perfect its existence cannot be proved because it is not a subject matter which needs to prove it is a presupposition in that way no synthetic a priori proposition about God is possible. So, let us conclude our discussion on this topic we have started with transcendental dialectic which deals with reason. Reason as such views all can be cognitions as belonging to a unified and organized system it is a unified and organized system architectonic nature of reason is highlighted moving from the particular and contingent to the universal. So, there is higher and higher unit is attained by reason reason seeks higher and higher levels of generality in order to explain the way things are and the appendix to the transcendental dialectic suggest that it deals with the regulative use of reason these ideas these so called transcendental ideas have no cognitive use if you employ them for cognitive purposes you will end up with metaphysics which is meaningless and illusory. So, they have no cognitive use they deal with the regulative use of the ideas of pure reason attempts to identify some proper immanent use for reason to establish a necessary role for reasons principle of systematic unity. So, this is what transcendental dialectic aims at doing I repeat to establish a necessary role for reasons principle of systematic unity it argues that the ideas of reason have an important theoretical function each serves as an imaginary point toward which our investigations hypothetically converge and they have a regulative function the ideas of transcendental reason or the transcendental ideas of reason have a regulative function or metaphysical propensities are grounded in the nature of human reason as pointed out earlier solve the concept of solve serves to guide our empirical investigations in psychology everything is pointed to one converging point and transcendental idea of the world or cosmos grounds investigations in physics they all represent the systematic unity we aspire in our empirical studies and God grounds the unification of these two branches of natural science empirical psychology and physics into one unified science. The idea of God in that sense I have pointed out is it enables to conceive that every connection in the world happens according to principles of systematic unity. So, that is why it is established and we can assume that all have arising from one single all encompassing being supreme and self cause. So, this is the concept and God also plays a very important role which we will explore in the next lecture when we discuss Kant's contribution in ethics or his ethical theory. So, we have discussed the notion of ideas of transcendental reason and also the problems with speculative metaphysics and how speculative metaphysics ultimately lead to transcendental illusions you will conclude our lecture at this point. Thank you.


So, this is the second session on the oscillation or you can say simple harmonic motion, okay. So can anyone summarize what we had done the last class quickly like whatever you think of whatever you remember quickly type in or you might have seen it before coming to the class quickly type in equations of SHM okay and causes of SHM all right. So when we discuss the causes of SHM what is specific thing we had discussed last class what are the things we have discussed specific things okay what else steps of problem okay what else anything else we discussed. So we had discussed a spring block system right we had seen in a spring block system how we can find out the time period of oscillation all right yes constant external force has no effect you need a force which will try to push you towards where you need a force which will try to push towards the mean position okay what is a mean position what it is mean position is what where yeah position of equilibrium position of equilibrium when that force is 0 okay that is the mean position and when we write a is equal to omega square x what is x acceleration when we write is equal to minus omega square x this is the SHM this thing equation so what is x x is the location with respect to the mean position okay with respect to the mean position it may not be extension in the spring it may not be elongation in the spring okay it can be elongation or extension also when spring is horizontal okay otherwise spring will be already extended when you put a spring vertically like that there will be extension in the spring already so add x equal to 0 itself a spring has extension okay so keep these basic things in your mind and then only you should proceed okay so I am assuming like you guys have attempted the assignments also and last statement the last statement what I said was let me write it because it is important what I said was when you write a is equal to minus omega square x and when you write v is equal to omega root over a square minus x square the x is what x is position with respect to the mean position okay so your mean position is x equal to 0 alright that is what I said okay so now you know that just like any other mechanics chapter this chapter is also full of numericals okay so it's not that we'll just discuss a theory and just run through the chapter okay so there are still many kinds of questions that we can solve alright of course I cannot take up all the questions otherwise for entire year I'll be doing only one chapter okay so let us start the session with couple of numericals based on whatever we had learned the last session itself so that it get refreshed and try doing this Pranav got something others what is the amplitude everyone amplitude is how much amplitude is 5 meters okay what is the time period formula how you write time period what is the formula for time period looking at this can you tell anything you can find out omega right 2 pi by omega is a time period omega is what omega is 5 right a sin omega t plus 5 omega is 5 so this will be equal to 2 2 seconds alright and then we need to find maximum speed the formula for the speed is written above so using that formula how you find the maximum speed this one what is the maximum speed in SHM V max is omega into a right omega is pi is 5 so 5 pi meter per second alright and also to find velocity at certain time how you find out velocity as a function of time x as a function of time is given you can differentiate it to get V as a function of time it will be 5 pi cos of pi t plus pi by 3 and you know you could have differentiated this and then looked at that and find out when this velocity will be maximum to answer the V max like that also you can do it maximum value of cost function is 1 so V will be maximum which will be equal to 5 pi from here also you can get it now when you put t equal to 0 here t equal to 1 here it will be 5 pi cos of pi plus pi by 3 what is cos of pi plus theta you know that minus of cost right so minus of 5 pi cos of pi by 3 alright now cos of pi by 3 is half so the answer is minus of 5 pi by 2 negative velocity means what what does it mean negative velocity which direction it is going left or right it is going to the left hand side okay and can you tell me the location where it is at t equal to 1 second can you find out the location where it is how you find the location at t equal to 1 second everyone how you find it put t equal to 1 second here put t equal to 1 second there that is a function of x with time right so when you put t equal to 1 second over there you get x equal to 5 of sine pi plus pi by 3 what is sine of pi plus theta you remember that sine of pi plus theta is what minus sign you know that all that right so minus of 5 root 3 by 2 so it will be here somewhere it will be here somewhere and going this way negative side like that okay this is how you solve this question anyone has any doubt you can type in do this okay Prano got something okay do you remember for a splint block system what is a formula for time period do you remember that time period is 2 pi root of m by k okay this is a time period so from here this is 3.14 is pi only so pi is equal to 2 pi root over m by k why we are doing all this because I don't know what is k I'm trying to find k why I'm trying to find k because the answer is k into maximum extension which is amplitude okay that's why so pi and pi got cancelled and 1 by 4 is equal to m by k so k is equal to 4 m m is 5 so k is 20 Newton per meter so force is 20 into 0.1 that is 2 Newton okay so you don't need to again and again find out what would be the time period as a function of m and k I mean just like you remember the time period of the pendulum formula with practice you will remember the time period of the spring block system as well okay don't worry much about it in terms of time period t you have to find out how much time it will take okay Aditya got it Gayatri got something others see you can assume a simple form a simple simple harmonic motion equation that is a sin omega t okay omega is 2 pi by capital T this into small t now I want to find out how much time it takes to go from 0 to a by 2 distance so x equals to a by 2 okay so sine of 2 pi by t into small t is equal to half so 2 pi by capital T into small t will be equal to pi by 6 okay because sine of pi by 6 becomes half first when it starts going like this so t is equal to capital T by 12 directly you can get it okay and here you can get the multiple answers because it will not just go there once it will reach there then come back to the same position go there again go back to the same position and then again go there so when you get the solution of this equation you will get multiple answers okay but I am interested in only the first answer of sine of what angle becomes half sine of pi by 6 is also half sine of 5 pi by 6 is also half sine of 2 pi plus pi by 6 is also half multiple angles for which it will be half do this can anyone tell me what is the amplitude of the oscillation for this one in terms of k m and all that what is the amplitude so see the mean position at the mean position at the mean position spring is extended by mg by k amount okay and about this it oscillates it goes up and down from this position so when it goes down it further stretches but when it goes up the stretching comes down and if at the highest point it becomes unstretched it means that amplitude of the oscillation is exactly equal to the initial extension so that as highest highest point spring becomes the unstretched position all of you understand this right everyone now what is omega omega is root over k by m okay and time period is given to us 10 times what is this 10 times per second is what time period or what it is frequency time period is 1 by 10 1 by 10 is 2 pi root over m by k okay got it so this is time period and we need to find what maximum speed of the block maximum speed is omega times a okay omega is root over k by m root over k by m and a is m by k times g so we are getting it as root over m by k times g all right so from this equation root over m by k root over m by k is 1 by 20 pi from this equation okay so when you substitute it you will get g which is 10 into 1 by 20 pi so 1 by 2 pi meter per second is the answer for the a part all of you understand this type in everybody got this type in do the next part could you explain again see whatever is given to us we are reusing it that's all time period is given to us 1 by 10 whose formula is 2 pi under root m by k so from there I'll get the value of root over m by k root over m by k is 1 by 20 pi from equation number one which we have and maximum velocity is omega into a a is m by m g by k a because it becomes unstretched top most position so all that combined will yield this result b part anyone got the b part different answer check for the silly error of yours and mine also do the b part everyone yes for now that's fine that's fine instead of writing 10 I could have written pi square over here answer is same don't worry pi square is roughly 10 only how will you do the b part everyone can you tell me what is the amplitude of the of the spring oscillation how much is the amplitude anybody got the amplitude see from equation number one again you know I'm doing the same thing root over m by k is equal to 1 by 20 pi okay so if I square it I'll get m by k is equal to 1 by 400 into pi square and then mg by k g is pi square is 1 by 400 meters so this is the amplitude okay how much it is how much it is in centimeters 1 by 4 centimeters that is 0.25 centimeters is a okay 0.25 centimeter is a now you have to use v is equal to omega root over a square minus x square formula can anyone tell me what is the value of x for the b part if amplitude is this what's the value of x will be will it be above the mean position or below the mean position everyone will it be above the mean position or below the mean position at mean position at mean position spring is already extended by 0.25 centimeter which is the amplitude also now when spring is stretched by only 0.2 it is above the mean position because at mean position itself it was extended by 0.25 so it is how much above from the mean position 0.05 centimeter okay above the mean position so you need to calculate x from the mean position so x is 0.05 don't ever put x blindly from here as 0.2 x is what distance from the mean position it is initial stretch at mean position was 0.25 final stretch is 0.2 for the spring so it is 0.05 distance away from the mean position so now you can do it yourself right omega v no a v no and x v no just take care of the meters and centimeter s i units all of you understood this question this was a good question everyone clear right no you guys are ready for these or not let us see do this all of you what is the time period of oscillation everyone 2 pi under root small m by k others okay Hariran got something others Prano got something you don't need to redo it you already know for spring block system what is the time period you need to fit in the formula for this situation okay Shreyas also got it so both the blocks are performing the shm together so 2 pi under root of m plus capital M by k this is the time period right now total mass of m plus m is performing shm there is no slipping which we are we are assuming that there is no slipping both of them are doing the shm together okay now b part this is a yeah no slipping right now no slipping so but isn't the friction force also changing yeah let friction force change it is an internal force if I tell you a single block also like this single block then also I can say that the single block can be assumed to be made up of two parts one above part one below part okay how do you know why do you care what is the force between above part of the mass and below part of the mass as long as it is internal so if they behave like a single mass it is as good as this which block has only friction force in horizontal direction which block has only friction in the horizontal direction smaller one so draw the free by diagram of the smaller one okay look at the forces on it and write down force is equal to mass m acceleration you will get the answer it is straightforward do it let me know once it done okay here I have got something let's say it is at the right hand side of the mean position so what will be the acceleration of this block how much will be the acceleration at a distance x away from the mean position everyone acceleration will be omega square x okay and there is only one friction that one force that can provide that acceleration and that is friction okay rest all of the forces are vertical in nature vertically they are balanced normal reaction and gravity okay so the friction force should be equal to m omega square x omega is what what is omega omega is root over ky m plus capital M this is omega so when you substitute it you will get m divided by m plus capital M into k times x this is the force of friction okay now do the c part throughout we are assuming no slipping okay why do you think there is a maximum force of friction why what is the reason why there is a limit because friction force has a limit it cannot be beyond a certain value right how much maximum friction will be maximum value of friction is what mu time normal reaction is the maximum friction right normal reaction is mg so mu mg okay this force should be greater than or equal to largest value of mass time acceleration which is omega square into amplitude all of you understand this whatever is the largest value that can happen m into a friction force should be sufficient to provide that acceleration this is clear to everyone right so a and a gone so amplitude should be less than or equal to mu g divided by omega square mu g by omega square is mu m plus m by k this is the maximum value of the amplitude all of you clear right any doubts quickly type in see there are such there are questions in which two masses together are doing simple harmonic motion we have only catered till now situations in which only one mass is doing shm okay so i'm not sure whether you guys have done enough practice on the shm or not so probably next class we can take up some questions in which two two masses together are doing shm all right okay so maybe end of the today's class itself if we get time we can take such questions how does that happen a very simple example is okay i'll put the question here and we'll come back and solve it later on this is the example you can see both the masses are doing shm together okay this is doing shm this is also doing shm there is no fixed wall as such over here okay so we will take this at the end right now i'll leave the space to go to the next okay write down the angular shm that question was in your assignment that's good so you guys have little bit struggled through now when i tell you you will learn quickly okay so angular shm can anyone give me one example of angular shm whatever comes in your mind pendulum pendulum okay so in case of angular shm what happens is that the oscillation with respect to angle happens it means a very simple thing that in case of linear shm we track what we track the position with respect to mean point okay in case of angular shm we track angle with respect to mean position okay so like you guys said pendulum okay a straight forward example is a simple pendulum okay so at mean position it will be here like this mass m length l at mean position when you displace it slightly it goes this way and then it goes that way okay so it swings with respect to its initial position and this angle changes angle with respect to its mean position all right now can you tell me mean position with respect to the linear shm is when net force is zero okay so i'll write down mean position for linear shm is when net force is zero what is the mean position for angular shm what do you think net torque is zero okay net torque should be zero fine so from this position where the net torque is zero you need to then find out how it is getting deviated now things are little bit similar in fact they are exactly the same compared to the linear shm mathematically they are exactly the same what i'm trying to say is that if if a is equal to minus omega square x is for the linear shm then alpha is equal to minus omega square theta is for the angular shm mathematically both are same because a is what d square x by dt square double derivative of x and alpha is double derivative of theta now over here i can say that what else you have v is equal to omega under root a square minus x square okay here what we can write everyone what we can write here if this is true for the linear shm v is equal to omega root over a square minus x square what we can write here instead of v what it will be over here v is d theta by dt don't say v corresponds to omega because omega is angular frequency already taken up don't write omega here d theta by dt is omega root over amplitude is maximum angle minus theta square yes this is angular velocity this is angular velocity this is angular frequency omega is angular frequency d theta by dt is angular velocity okay then you have x is equal to a sign of omega t plus phi and over here what should i write everyone what should i write here right theta is equal to theta not sign of omega t plus phi okay rest all things you can differentiate it and get it from here onwards all of you are able to understand it kindly type in quickly there is this correlation between linear and angular shm how is d theta by dt different from omega what is omega omega is constant of proportionality between a and x or alpha and theta how is omega d theta by dt why do you think it is that just because we are used to seeing omega as angular velocity we tend to think that omega should be equal to d theta by dt okay what is the time period time period is same in both the cases time period is 2 by by omega only that doesn't change because do you think for mathematics these two equations they are different or same for mathematics mathematics will look at these two equation and will treat in exact same way or not these two instead of a i have written alpha instead of x i have written theta so mathematics doesn't care whether it is alpha or a or theta or x just like a is d square x by dt square similar way alpha is d square theta by dt square okay previous slide somebody has to see that's all i move forward okay fine so now the question comes how do you find out the time period of oscillation for the angular shm okay all the steps are exactly the same okay all the steps are exactly the same i will not going i'm not going to write down the same steps again just that instead of displacing by a small distance x from its mean position what do you have to do can you tell me previously when we wrote steps of solving problems for linear shm we displaced it by a small and a small distance x now in this case what will happen displaced by small angle theta okay here what you will do is that displace by small angle theta then what i have to see in earlier case i had seen that there is a force a restoring force that get developed that again tries to bring the mass towards the mean position here what will get developed correct here a restoring torque restoring torque gets developed so what you have to do is write down torque is equal to i alpha equation all right then get the alpha is equal to some constant times theta and alpha should be in opposite direction of theta okay and then you can compare it with alpha is equal to minus omega square theta you'll get omega straightforward thing okay now if theta is very small if theta is very small we are going to use this approximation again and again that sin theta is roughly equals to tan theta and is roughly equals to theta itself remember these approximation anyone has any doubts sin theta if there is a right angle triangle just taking an analogy here is perpendicular upon hyperton is this is p by b so when angle is very small hyperton is and the base becomes almost equal if the angle is very small this hyperton is and this base becomes equal almost so that is what it is and moreover it looks like an arc length which is perpendicular divided by base so that is the definition of theta itself okay so that is why is equal to theta also okay so we are going to use this approximation multiple times okay let's now again go back to the simple pendulum and see how to get the time period is this chapter done in your school by any chance not in dps not in ypr no not in hsr also okay so what i'm going to do is i'm going to i'm already displacing the mass m by a small angle theta this is the length l okay now i want you to get the answer yourself try doing it okay aditya got something there'll be mg force acting on it and there'll be tension force acting on this t so how much is the torque about this point oh everyone what is a torque of mg torque due to mg is what mg into l sin theta l sin theta not cos theta this is the perpendicular distance right this is the line of force i drop a perpendicular on it mg into l sin theta is the torque okay this is the torque not torque is equal to i alpha what is i moment of inertia about this point everyone how much it is for this mg what is the moment of inertia for mass m it's a point mass ml square so alpha is equal to mg l divided by ml square into sin theta okay m is gone one of this is gone now theta is very small so sin theta is theta only now is theta and alpha they are in opposite direction or in the same direction theta and alpha theta goes this way alpha tries to bring it back okay alpha is in this direction theta is in that direction clockwise anti clockwise so minus so now when you compare it with alpha is equal to minus omega square theta you'll get omega is equal to root or g by l so time period is 2 pi under root l by g this is the time period okay now why it is called simple pendulum it is called simple pendulum because it is a simplest kind of pendulum can you think of any other kind of pendulum okay think of any other kind of pendulum correct there can be there can be a rigid body okay like this pen itself it can have an shm this bottle itself if I displaced a bit it can perform some oscillation like this okay but then this is not a simple pendulum this is a this is a different kind of pendulum and the name for this is physical pendulum so write down physical pendulum can both angular and linear shm what do you mean didn't get okay physical pendulum let's say we'll take an example to discuss that okay let's say you have a rod like this okay this rod is hinged over here total mass of the rod is m length is l this distance the distance of the topmost part from the hinge is a okay I want you to find out the time period of oscillation for this rod at the mean position it is vertically straight like this isn't it net torque is zero right now then what you do you displace it both angular and linear shm yes they can happen together it's can be like this a rod then there is a spring mass like this so when rod is oscillating this block also oscillates but then you guys have to do a lot of problem practice your own the basic level okay so Pranav got something others anybody else got anything nobody else okay so at this position gravity force will act from where gravity will act from center of mass so this is mg can you tell me what is a torque due to mg about this position where it is hand what is this perpendicular distance this perpendicular distance is how much this is theta this angle is theta okay so this angle is also theta this angle is theta okay from here to there what is the distance this distance is what l by 2 minus a see I intensely made it little geometrically like this so that it should not be a straightforward thing right so this is l by 2 minus a so the torque is mg l by 2 minus a times sin theta all of you understand this type in this is a torque we are trying to use torque is equal to i alpha okay now can you tell me what is moment of inertia i which i should be using there if it is icm it is not ml square by 3 plus ma square if it was icm plus m into distance from center of mass square icm is what ml square by 12 plus m into distance from center of mass square that is square okay now I think let's call this as this bracket the white bracket let's say c so alpha is equal to minus c times theta is very small this okay so omega is root over c where c is this okay and time period is 2 pi root c is it clear to everyone all of you anyone has any doubt quickly type in let's proceed they are now you guys have learned almost everything about the problem solving it's all about practice now okay so we'll do one more question because you guys will see these kind of things quite often so hence you should be so I think the arrangement you can make of this is k1 spring k2 spring constant mass m length l this rod can freely rotate about one of its ends which is topmost point it can rotate so you need to find out the time period of its oscillation yes k2 is a midpoint sorry anybody close to the answer should I wait or solve okay one or two minutes take all right so nobody got it so let us say the extension on the bottom most spring is x1 so it'll be k1 x1 force over here the compression on this is spring this spring will be compressed the bottom spring will be extended but the good thing is that both will create the force in the same direction this will be k2 x2 okay and that forces gravity mg okay now the torque about point o is mg into l by 2 sin theta plus k2 x2 into l by 2 plus k1 x1 into l do you all understand this torque type in quickly angle is very small so the force into perpendicular sense remains that much only all of you type in everyone is it clear yeah yeah this is l now how how is x1 and x2 they are related is there a relation everyone type in is the relation between x1 and x2 okay can I write x1 x2 in terms of theta can I write x1 and x2 in terms of theta yes directional spring force will change and it'll always try to bring the rod in its main position that's how the force should be which is helping you to do the shm okay so x1 all of you agree is l into theta if theta is a small and x2 is equal to l by 2 into theta and sin of theta is roughly equal to theta okay so we can write down torque about o to be equal to mg into l by 2 theta plus k2 l square by 4 into theta plus k1 l square into theta okay this is the torque this is equal to i alpha that is ml square by 3 into alpha so alpha comes out to be um mg l by 2 okay we can cancel out one of the l l is gone how it is l square by 4 okay x2 is l by 2 theta substitute here one of the l is gone so alpha is mg by 2 plus k2 l by 4 plus k1 l see I just made up a question so that's why the values are not looking so good but there are many questions like this when you practice your own that divided by ml by 3 this into theta now the problem is done this bracket term this term is your omega square and time period is 2 by by omega okay so like this you solve this particular question okay so let's move forward there are many questions you know so let me just move forward and do a little bit of more theory then we'll again come back to the numericals okay so write down the energy simple harmonic motion so basically our main interest is spring block system it's a relatively small topic because there is nothing new as such whatever you know about the springs energy and the blocks energy same thing it's a repetitive stuff but slight bit of specific details are there so if a block is doing shm with a spring or spring constant k okay then the value of omega is root over k by m okay the amplitude is let's say amplitude is a so can you tell me maximum potential energy how much it will be and maximum kinetic energy how much it will be everyone maximum potential energy and maximum kinetic energy how much it will be maximum kinetic energy maximum sorry potential energy is half k into x square what is the maximum amount of x a only so half k a square maximum kinetic energy is what half m into v max square v max is what omega into a whole square omega is root over k by m so you will get half k a square here also okay got it now in terms of x at a particular moment in terms of x the potential energy is half k x square and we know that in an shm v is equal to root over a square minus x square so kinetic energy is half m into v square that is omega square a square minus x square now omega is root over k by m so this can be written as half k times a square minus x square so can you see that the sum of kinetic energy and potential energy is a constant can you see that it doesn't depend on x at whatever location you go to when you add potential energy and kinetic energy it is equal to half k a square only all the time okay so why don't you do one thing draw a plot let's say this is the this is the zero energy zero energy and there's a maximum energy which is half k a square okay can you plot the potential energy and kinetic energy on it this center line is x equal to zero this is x equal to zero this is not zero left hand side is x equal to minus a right hand side is x equal to plus a i want you to plot the potential energy on it everyone it doesn't matter from where it starts you have to plot this on the graph let me know one step okay done so potential energy will be maximum at x equal to minus a and x equal to plus a and it is a parabola like this so i hope all of you have got this okay this is from your n crt whatever i'm discussing okay now i want you to plot the kinetic energy on the same graph which are different color can you plot the kinetic energy this is your potential energy you plot the kinetic energy energy is this let me know one step look at the critical points at x equal to zero what will be the kinetic energy at x equal to plus minus a what it will be and then join it with a curve what will be the kinetic energy at x equal to minus a and x equal to plus a look at the formula zero and at x equal to zero it will be half k a square it's a downward opening curve right to plot it it'll be like this this is your kinetic energy i hope it is clear to everyone nobody has any doubt so if the shm is a system which is non-desipative non-desipative as in in which friction is absent and there is no let's say drag force or anything like that the mechanical energy will be constant okay so there are ways to solve a problem by using the mechanical energy conservation you can get the value of omega okay so if you want i can do one question where you can do the conservation of energy and get the answer of finding time period and things like that i'll take a simple question only let's say you have a spring block system mass m spring constant k okay then there is no friction anywhere whatsoever so at any time at any time kinetic energy plus potential energy will be a constant right kinetic energy is half m v square potential energy is half k x square this is a constant if it is a constant what will be the derivative of this if i differentiate it if it is a constant what should it become z will become equal to zero it's a constant so when you differentiate it you'll get m v dv by dt plus k x dx by dt is equal to zero now dx by dt is your velocity also goes away okay from here you'll get dv by dt is equal to minus of k by m times x d by it is acceleration so a is equal to minus k by m times x so you got the shm equation by using the conservation of energy also okay this kind of method is very useful and easy but it requires a lot of practice okay so i'll send you a worksheet in which you can practice something like this but right now you can focus on the force and the torque equation and if you have time right now after the class only then you can do the same question like this all you have to do at any point in time write down kinetic energy plus potential energy it should be a constant and you will get the answer okay in case of angular shm also you can write down the kinetic energy half i omega square plus mg h and differentiate it okay when you differentiate mg h you get velocity of center of mass which is equal to omega into l by 2 so there are some certain things that you need to take care of but then you'll get the answer okay yeah so let's take a question i have now i think i am um more or less done with the basics of the shm i can take few questions and then there is a third part of the chapter that i can discuss do this your concepts of rigid body motion will also be used in shm okay pranav got something others okay so moment of inertia is not coming in the equation i is not coming are you sure pi t i what is that 2 pi r t i root i by k okay root i by k okay so all of you try it right so now let us move this down by a distance of x who is doing shm the pulley is doing shm okay so you just look at the movement of the pulley that's all so movement of the pulley is such that it goes down by a distance of x from its mean position it goes down by a distance of x from its mean position how much spring will be extended extension the spring will be how much this length will be extended by x even that will be extended by x isn't it if the pulley has to go down by x it increases excellent this side excellent that side from where it comes from the spring a spring get extended by 2x are you getting it everyone quickly is it clear to everyone why 2x because if cylinder has to go down by x the length of this portion has to change by 2x okay yeah let it rotate pulley is rotating so what but the length has to change you know this length and this length both are x from where it will come from from the spring only 2x length will come then only cylinder will go x down now draw the freeway diagram this is 2kx we don't know what is the tension over here this is mg okay fine so the there was an initial extension on the also on the spring let us do that as well so that it had a complete picture when initially it was extended kx not this side is t this is your mg everything was at rest so pulley was not rotating so this tension was also equal to kx not only so initially kx not plus kx not is equal to mg now I've displaced it by a distance of x so the spring force has become k times 2x plus x not tension has changed from the earlier tension which was kx not plus let's say delta t okay so the net torque about the center is what k times 2x plus x not times r minus k times x not plus delta t times r this is equal to i alpha do you understand this equation all of you type in kx not r minus delta t r is equal to 0 okay minus sorry kx not r and kx not r will go away so you'll get k times 2x times r minus delta t times r is equal to i alpha okay now we also know that net force is equal to mass acceleration okay earlier forces were all balanced as in kx not kx not and mg they were balanced so if you write kx not plus delta t plus k times 2x plus x not this is the upward force minus mg is equal to m into x version of center of mass so kx not kx not and mg they get cancelled over because 2 kx not is equal to mg so we have delta t plus k times 2x is equal to m into a sin center of mass now what to do everyone these are the two equations now what to do i have to write alpha i have to write alpha in terms of x version of center of mass a center of mass divided by r okay pure rolling condition so substitute it over there using these two equations remove delta t remove delta t using these two equations then write down x version of center of mass to be equal to something into x and this something will be equal to omega square so like this you solve this particular question all of you understand this just go through it once there is no hurry you can spend one minute you don't need to hurry up it can be in terms of r see r will be there r is a constant it should be in terms of x so that you can compare a is equal to minus omega square x this equation when you compare then only you'll get omega right inside this bracket there can be r there can be m there can be i first equation again see first equation is torque about center of mass is equal to i alpha earlier kx not kx not and mg were acting okay so 2kx not was equal to mg at the mean position everything was at rest then i have displaced the mass by distance of x when i'm displacing by distance of x the left hand side the string length increases by x the right hand side also the length increases by x the total increase in this overall length of the string is 2x from where this 2x comes from the spring only so spring generates 2x into k force because the block gets pulled down by x so k into 2x the new extension plus the earlier extension here tension also changes when you leave it let's say tension earlier it was kx not now it is kx not plus extra tension then i'm writing torque equation torque due to this force is that force into the radius perpendicular distance torque due to this force is force into radius perpendicular distance this is equal to i alpha and this is what the equation is clear any other doubt normal reaction where is a normal reaction there is no normal reaction from where you got it any other doubt everyone okay so now we have seen many spring and block systems let's take a situation in which there is no spring block but still it is an SHM only have you used have you done this question in which there's a beaker and water have you done this no right so let's do this you have a beaker full of a liquid this is from your ncid textbook the last question of your unsolved x problem don't look at the final answer do it yourself this has density of row okay the beaker is full of a density of liquid row and there is a cube that is floating on it there is this cube whose edge length is a and it is floating over it it has a density of sigma sigma clearly is lesser than row that is why it is floating okay you need to find out time period of oscillation if what I do is that if I push it slightly down and release it if I push it slightly and release it then with what time period will oscillate all of you do this what length is it submerged to start the oscillation doesn't matter it doesn't matter you can find out initial submerged length also if you want but it doesn't matter okay Pranav got something Hariharan others when you push it down further what exactly happens what are the changes that will happen everyone when you push it down by a distance of x what is the change that will happen all of you type in what do you think buoyant force increases very good buoyant force increases so earlier we had earlier the buoyant force balancing the gravity force earlier now extra buoyant force is created how much extra buoyant force is getting created everyone if I push it down by a distance of x what is the extra buoyant force that gets created so the net force is the earlier buoyant force plus extra buoyant force minus mg okay the earlier buoyant force was equal to mg so the net force is the extra buoyant force only which is rho dv what is dv Hariharan what is that don't include another variable already we have so many variables how much volume gets submerged further a square into x so extra displaced liquid weight is rho a square x this is mass into g okay this is the net force this is equal to mass and acceleration sigma a cube is the mass times acceleration let's say capital a is acceleration small a we have already used actually so capital a is rho g by sigma a times x and clearly a and x are in opposite direction so minus sign this bracket term is your omega square all of you got this everyone any doubts this is the time period of oscillation clear to everyone type in can you explain the last two steps last two steps though nothing it is you you want to see this one this is the equation of shm a is equal to minus omega square x if acceleration is minus omega square x then it is shm so I am comparing whatever I have got with a standard shm equation and when I compare I get omega as under root of this bracket term and if you get omega time period is 2 pi by omega okay any doubt anyone has clear to everyone so we'll meet after the break now okay fine so let's take a break now we will meet at 6 22 p.m


Hello everyone, welcome to our Priam vehicles of the week. I'm Gabrielle and I'm over here at our sports HSE. Now the exterior color is in a Fuji white. So it is for alloy wheels. There is navigated heated weather seating. Now that interior is made of Corolla hatchback 2021 Kia 4K LXS. Locally owned, see if owner, you'll have a back of can. Now that interior color is in a black. Feel free to let us know.


The pass mark is 50. When you get 48, have you failed or have you passed? Failed or passed? The idea failed or passed? The person who gets 48 doesn't know some things. The problem is not, I'm not saying we don't know anything. We know some things, but what we know is at the minimum threshold, it cannot carry you far.