 So today we will start with kinematics of the rigid body, fine please write down and change its volumes, is water a rigid body? You can't deform it basically. A rigid body is a hypothetical object, such object do not exist but approximately we can say that any solid behaves like a rigid body. Please write down in a rigid body, a rigid body is an object which do not get deformed. Now in reality every solid if you apply some force will have a stress because of the strain will be produced and there is a deformation. But we will assume that the forces are so less, so little that the deformation will number if I stretch this television from left hand side. There is slight deformation but that is negligible. We say that the amount of force that we are applying is so less that we can ignore it. Now this is what the rigid body is whatever you have written but mathematically how can you talk about it? A rigid body is something which do not get deformed but can I say this in a mathematical way? Suppose I take two points on a rigid body, we have tension infinity but again I am telling you the way you should tell me. Tell me with respect to the two points on the rigid body, what should happen between the two points on the rigid body for it to be a rigid body? The distance between any two points on the rigid body is fixed. Distance between any two points on a rigid body is fixed. Distance between them neither increase nor decrease. So suppose you have point A and this is point B. Distance between them is suppose 20 centimeter. It will remain. If it changes then essentially rigid body is getting rigid body. Is this the distance between the points are changing? These two points. When changing, so as a whole rigid body are you getting it? Now if distance between the two points is fixed, can one point move relative to the other point? Is it possible? Can they move? Yes. How many said no? What are you doing here? I am saying that if distance between the two points is fixed, like the case in the rigid body, can they move relative to each other? Do you understand? Is it the tip is moving with any motion that distance should remain same? So what is the possible path of relative? If B has to move relative to A, it should move in a circle. So it will move in a circle like this. Keeping the distance fixed all the time. Are you getting it? Same way all the points on the fan. So please write down the rigid body every point in a circle, every point move in a circle with respect to every other point. And center is also moving in a circle with respect to tip. But if you see that because you are at rest and the center of the fan is also at rest. So you are only seeing what is happening with respect to center because you are also rigid body. All of you are relatively simpler. If it is not a rigid body, it becomes very difficult or complicated to analyze and look at the relative velocity. Now all of you please draw a rigid body. You can draw some random shape and size of a rigid body. This is a rigid body. You can perhaps this as a rigid body and suppose this rigid body rotates. Let's observe what are the points doing. I am taking 0.1, 0.2 and 0.3. So this line represents the axis of rotation. About this axis, this object is spinning. Just like fan is spinning. That's how. Now can you tell what is happening with the point number 1 when it rotates? It is moving in a circle. So this point number 1 moves in a circle like this. What is happening with point number 3? This is also describing a circle. Point number 2 also moves in a circle like this. So basically the axis is got locus of all the centers. Yes or no? All the circular motions center lie on the axis. Now tell me one thing. If this point 1 completes one full circle in 2 seconds. How much time point 3 will take to complete the full circle? 2 seconds only. If it takes less or more time what will happen? Then the twisting will happen. Twisting happens of a rigid body. So it gets deformed. Fine. So all the points should move by same angle at same time. Are you getting it? Please write down all the points on the rigid body. All the points on the rigid body should rotate by same angle at same time. If it is not a rigid body all of these are not valid. So all the points rotate by same angle at the same time. All of you understood this? Now tell me the speed of point 1 is more or point 3 is more? It becomes very difficult to analyze the motion of the rigid body in terms of distance, speed and acceleration. Because there are infinite points all those infinite points will have infinite velocity, distance, velocity and acceleration when they are analyzing a rigid body. But the property of rigid body that they will all move same angle the same time makes these angular variables handy when you analyze the rigid body. Rather than displacement you track how much angle they have moved. It will be same for all particles. Rather than velocity you are tracking what is the rate at which angle is changing. Which is angular velocity and rather than address relation. So we are tracking when we are dealing with rigid body. But finally I should know that if I know what is the angular velocity I must know what is the relation between angular velocity and the linear velocity or the actual velocity. Because ultimately kinetic energy when you write it is half MV. So let us first see what is the relation between the angular variables and what is its relation between what is the relation between the linear variable. Now tell me if this point 1 has traveled an angle of delta theta how much distance the point 1 has moved. Let us say this radius is r1 how much distance. Take a circle like this if the point 1 has gone from this point to that point this angle is delta theta. This distance is what? This is r1 delta divided by radius is angle. So r1 right? So delta theta angle correspond to r1 delta theta distance. Getting it? What is the assumption here? The assumption is that the rigid body is staying at one place and rotating. If it starts moving and rotating if it starts to move as well as rotate then this is not a angular vector. Are you getting it? Right now we are dealing with fixed axis. The axis is fixed or you can say that it is spinning. So this is angular variable and this is linear variable. This is what distance term it delta s you can say. Rate at which angle is changing is referred as and you have lost the omega. We have done this in several motions yes or no? Something similar we have done already. So this is and you differentiate this. If you differentiate distance you will get yes or no? So differentiate it r1 is constant so r1 comes out. So it is d theta by dt this is my velocity at any point in time which is r1 into omega. We have done this earlier also. And there is something called angular acceleration. So d omega by dt is alpha. So this is my velocity. So if I differentiate velocity I will get acceleration dv by dt is r1 into d omega by dt which is r1 alpha. So if I know omega at any point in time I will get the distance from the axis. If I know at any point in time I will get the linear acceleration of a point whose distance from the axis is r1. Are you getting it? Please write down in bracket this r1, r1, r2, r3 are the perpendicular distance from the axis. They are not any distances. You should find out the perpendicular distance. If I know the angular variable at any point in time if I can easily know what is the distance kind of it. What is the velocity and what is the acceleration? Yes or no? So rather than tracking these distance velocity and angular efficient I will keep a track of angle, angular velocity and alpha because it is easier to keep track of these. There are different points of the rigid body. I can't keep track of all of the incident and distance. Are you getting the point? Acceleration is dv by dt to d square s by dt square. This we have learned in kinematics. So if the efficient is constant the solution of this equation is what? It is equal to u plus v square equal to u square plus 2 as s equal to ut plus half dt square. This we have learned. This we can use only when efficient is constant. And I also have alpha equal to d of omega by dt. Rate of change of angular velocity. This is equal to d square theta by dt square. So what are the solutions of these equations if alpha is constant? Let me write down if alpha is constant. Instead of v what is there? Omega is equal to initial angular velocity plus alpha into t to omega naught square to alpha delta theta. And what delta theta equals to omega naught t plus half alpha. Things are exactly same. When you deal with it in algebraic way it is exactly same. What if if efficient was not constant then how we used to solve the problem you remember? We used to use these equations directly. So here I write these equations if a is not constant. Similarly here if alpha is not constant I will use this equation directly. So let us take couple of questions and see whether we can replicate whatever we have learned here on to this scenario. Find and it out is minus of per second square. You need to tell me how many revolutions the object makes before coming to rest. Initial one is minus pi into delta theta. What? Fifty pi. So it will stop and then again as I write down it is given as five minus t into per second square. Initial angular velocity is zero. You need to find out zero again. Minus t into t is also zero. This will be from zero to t. This will be again from zero to t. This will be again five t minus t square by two is equal to zero. So t is equal to when you equate five minus t equal to zero you are getting a time when angular acceleration becomes zero. Just because alpha has become zero does not mean omega is zero. It is like saying that if efficient is zero it need not be. What happens is when t is very small alpha is positive. It will accelerate becomes large alpha become negative. It started its some angular velocity after sometime alpha become negative so it is trying to stop it now. So like this there are several questions and you will see that the method of solving this question. These kind of questions are exactly same to the way we have done motion in 1D or motion in 2D. We have done lot of problems. The concept, the underlying concept that you should equate d omega by dt is exactly same as the way we used to do it earlier. That is all right. So just have to do couple of problem practice and everything will be similar looking to you. So coming back to the kinematics of rigid body. How do you think the rigid body can move the motion of rigid body? What kind of motion this is? This is only translational. We have discussed it. It can just move forward. This is translation plus rotation combined. And the way translation and rotation are combined can also be in a different different way. Have you seen that when the wheel is moving forward what is happening to the axis? Is the axis remain fixed? What is happening to it? The axis also moving. This is the axis like that. When wheel moves forward is axis moving forward? Right. So axis can move forward. And have you seen the top plane top? How does it rotate? It rotates like this. So it is axis moving forward. What is the axis? The axis is on the test. So I used to get punishment like this. You have not got such kind of punishment ever. So what is happening? So what is happening? It is not properly. No. It is not properly. What is happening? The center is here. Take care. Alright. So what I was saying? So you have played that top. Seems top. You have wrapped the... Your finger is there. So translation plus rotation can happen in different different... In our syllabus only one kind of translation plus rotation is there. When axis translates. Axis should not rotate. Are you getting what I am saying? So like for example the wheel when it is moving forward the axis... What it is doing? It is following the straight line path or not? When the wheel is rotating and moving forward. This kind of motion is there in our syllabus. And axis also rotates. We can summarize. Stop talking. This is pure translation. All the points are in the straight line. Change its orientation. It is pure translation. That line will go straight. Why? Because all the points have the same velocity. Same acceleration. Some pure spinning. And axis is fixed. The rigid body is not going anywhere. The axis is fixed. And we have just learned. With center where? On the axis. There can be a case of spinning translation. Moves forward. Moves up and down. This is more generate motion. It is not going in a circle. It is just doing some weird motion. For example. In this path. It is not a circle motion. It is a motion in this case. Why? Not the movement but the angle. And anyone? It is a motion. He is saying separate the spinning and translation. Which is correct in a way. But how can I separate? I mean the process of separating. How will you separate it? I am not asking how can you do it. How will you separate it? The process of? Fixed axis. Is there a fixed axis over here? No. There is no fixed axis. It is moving forward. If axis is fixed the object moves. What do you mean by that? The object is moving forward. If the object moves forward. Then the axis is fixed. Ok. So please write down. Rotation plus translation motion. We observe. We observe the motion of. We observe the motion of. All the points. We observe the motion of all the points. Relative to the center of mass. The motion of all the points. of mass and then add the center of mass motion on it. It is moving and then when you add its suppose I am going forward let's say I am going forward 5 meter per second then with what velocity this object will appear to me this one. So observer sees the word by subtracting is or not. What is the relative velocity? The velocity of object minus the velocity of observer center of mass so you got the velocity of velocity. Understood? Okay, simpler. What kind of with respect to center of mass everything is because the distance is fixed between any two points not this. You take any point with that point it will be circular motion because distance is fixed. Okay, now the reason why you come from it? The rigid body and still I get circular motion only relative to that point all the points motion yes I get the circular motion only but why center of mass? Why I am looking everything with respect to center of mass and not with respect to any other point? First is we already know the location center of mass how to find the formula and all okay easy to locate. Second or more important thing you will understand later on where you will learn about if you don't take the center of mass if you don't observe with respect to center of mass the pseudo forces act from the center of mass and it will have some torque. Okay, we will learn about all this later on to nullify the torque because of the pseudo forces we usually observe with respect to center of mass and then at center of mass is motion on it okay and whenever you are confused what to do in this chapter always follow the center of mass okay that is the thumb rule at times the explanation why the center of mass is not so straight forward but you can use it like a thumb rule that observe with respect to center of mass what is happening and then add what center of mass is doing got it there will be force pseudo force will be there but torque due to pseudo force will be 0 that is why pseudo force will be there right? pseudo force torque will be distance from the center of mass will be 0 because it is acting from the center of mass okay focus here let's take a wheel like this which is moving forward and velocity of center of mass is given as VCM okay and let's say angular velocity is omega whose angular velocity is this? which points? all the points on the rigid body all the points all the points every point has the same angular velocity on the rigid body okay so I don't need to specify which point has this angular velocity entire rigid body has omega so if one point is rotating for example you are observing with respect to center of mass at what angular velocity the other point will rotate with respect to center of mass same omega okay with respect to center of mass all the points rotate with omega only which is the angular velocity of the entire rigid body the person who is observing is suppose someone observer rotates and then observe he will observe hit line angular velocity is 0 the angular velocity will remain omega only okay see the problem is that if I get into too much of explanation it doesn't get simplified like if you have to dig deeper and deeper it becomes more and more involved okay so I will tell you just to take it on the face okay now tell me related to center of mass what is the velocity of this point and in which direction if the radius is r it is moving in a circle with respect to center it is with respect to center of mass r omega that point and this point can you draw it to the direction of magnitude r omega dot things are simple okay so we keep it simple only this is r omega that side okay and this is r omega stop talking a point over here this angle is 45 degree and the distance is r by 2 from the center what is the relative velocity of this point with respect to center if this is the radius it has to be perpendicular the velocity of this point related to the center of mass is r by 2 omega like that okay it is a circle these 4 points do it or vcm minus r omega r omega plus vcm that point vcm squared and if you do the vector addition net velocity will be this way okay fine vector chapter which is getting little bit of diabetes used here so isn't it much easier to observe first with respect to center of mass then add whatever center of mass is doing to find the total velocity if you don't do that the motion of all this point become very complicated this point as this will move forward will follow some weird path it will do something weird okay no doubts okay so let's solve this question simple one draw a circle try to draw as much as this circle center is going forward vcm is equal to r omega it is given you need to find the total velocity of this point which is at the distance of r by 2 and making an angle of 30 degrees which is