 Let us move on with the new topic okay which is not a completely new topic it is essentially whatever we have learnt in kinematics now we are carrying over and topic is kinetics okay again the main text main resources everything remains the same. So what I am going to discuss is I am going to discuss Newton's second law of motion linear momentum of a particle just a side topic on system of units okay we do not need to bother with this equations of motion and what is this concept of dynamic equilibrium or how do we relate the forces acting on a body with the corresponding accelerations towards the end okay we will also discuss what is the angular momentum of a particle conservation of angular momentum and how to write kinetics equation in radial and transverse components or in normal and tangential components okay we will do that. Now there was a there have been many questions okay that what is the difference between kinetics and kinematics so kinematics is something for example which means that you are just looking at a trajectory of a particle or a rigid body you see how it moves but as we all know for example that if you take a discuss okay an iron discuss which is thrown by a discuss thrower and you take a flying disk a flying saucer made of plastic both can have very similar trajectories but we clearly know that to give the same trajectory to a normal flying disk that we use for playing and the same trajectory for example thrown in for the by this Olympic thrower okay it is not you need to apply different forces in order to maintain the same trajectory. So the topic is that that given that you have a particular trajectory that you want to maintain for a particle or for a rigid body then what should be the forces okay that should be applied to that body to keep it in motion is for example if you want to design a wheelchair okay if for this for a motion round down the ramp then what we need to find out is what are the forces acting on the various components of that so kinematics can only help us so much it can tell us that what is the acceleration of every point what is the acceleration of this person and so on okay but from those accelerations we want to find out what are the reactions that are exerted at the joints okay and so on then we need to really go into Newton's laws of motions and use okay them to find out what are the internal forces based on the kinematics that we have done so far okay so that is the purpose behind the kinematics of particles kinetics of particles kinematics only give you a visual picture but from that visual picture what are the underlying forces that come only from kinetics or application of Newton's laws through the to the accelerations that we see from kinematics. Newton's second law strictly speaking works on a particle now particle is an abstract concept but what we had seen so far is that particle does not mean strictly a point it only means that the overall dimensions are small when compared to the overall length scale of the problem okay that for example if it is a small ball okay so the overall if the ball is very small compared to the overall distance it travels and so on maybe we can neglect it and the rotation of the objects is neglected okay the demonstration I had shown with Mathematica okay when you neglect the rotations of the objects then that is when it becomes extremely it becomes extremely and rigorously clear that when the rotations are not there okay or we are neglecting then essentially a whole three dimensional body is a particle. Now what is the Newton's law of motion Newton's law of motion say that for a particle if f is the or the bunch of forces acting on the particle then sum of all the forces is nothing but the mass of the particles times the acceleration of the particle. Now that is where for example why we were completely obsessed with getting the acceleration of various particles that when we apply all manner of forces to the particle then the forces result in acceleration you can even pose a reverse problem that if the acceleration of a particle is given okay that the particle is moving for example if you create a channel that a particle should move through a channel or you create a road surface and you say that a car has to go along that track then the question we ask ourselves is that what should be the forces that should be generated in order to keep that particle on the track. So you can pose the question in anyway in one way is that given the forces and the mass of the particle what is the acceleration the reverse way is that given that the particle is following this trajectory what are the forces acting on it and we will see when we solve a variety of problems that there are a number of problems in real life where both of the situations can come into picture okay and where we know the trajectory we need to find out what are the forces that would be applied on the other hand we need we know what are the forces and we need to find out that what are the corresponding acceleration that the particle undergoes okay. Now this one thing okay I did not emphasize so much okay during kinematics because it does not really you do not really have to worry about that thing when we do kinematics but this law strictly speaking works okay not strictly speaking it only works in a Newtonian or an inertial frame of reference that if we are moving if you are in a stationary frame or we are moving in a frame which moves with a constant velocity only then can you apply this laws of motion f is equal to m. If for example a frame is accelerating what do you mean by accelerating frame that if it is moving in a straight line it is velocity is constantly changing or if it is moving in a curved direction then the direction is constantly changing or for example in many mechanical components what do we have we have many gears okay which are many gears connected to each other many components which rotate about which rotate with respect to each other okay and then if we fix a coordinate frame or if we fix a frame of reference with relative to this rotating members then those are not inertial frames of references. Similarly strictly speaking our earth is not an inertial frame of reference why because our earth rotates around its own axis so clearly it is not an inertial frame of reference and it we see that for example the fact that the earth is not an inertial frame of reference is reflected in what is called as a Coriolis force which can be observed on a pendulum for example that is oscillated and it also manifests itself in various different ways for example in the in the cloud in the atmospheric patterns in the patterns of clouds for example in a moving frame okay that the acceleration of the earth the rotation of the earth about its own axis it reflects itself in for example effects that for example how do clouds move how does atmosphere how does the pressure differences happen and so on okay so that is beyond the topic of this course okay but all these different concepts are affected due to the rotation of the earth which strictly speaking is not an inertial frame of reference but for all day to day problems okay rather than this climate climatology problems okay the simple problems about what are the forces acting on a car and so on we can do some very simple calculations and figure out that the Coriolis forces or the centripetal forces that are acting okay are very small and for all practical purposes we can neglect them okay so as far as this course is concerned we are not going to bother about the so-called non-inertial frames of references we are going to worry only about the inertial frame of references and this Newton's law okay will be used only in inertial frames of references where the frame of reference has no acceleration okay it moves at a speed at a velocity which is constant. Now there is another question which a lot of us keep asking that if you take a rigid body okay a full rigid body why is it fair to say okay why is it fair to say that for that rigid body the mass is concentrated at the center of mass okay that there is a center of mass the mass is concentrated at the center of mass and force is nothing but mass times acceleration of the center of mass what makes that center of mass so special so what I will do is that I would definitely deal with this when we go to rotations of rigid bodies but before that let me briefly explain that where does this concept come about. Now a rigid body okay can be thought of to be a collection of a bunch of particles okay one rigid body you take each rigid body had lot of small elements and the small elements can be thought of as a group of particles okay now these particles I have drawn to be spheres just for simplicity I can take them as cubes or whatever direction whatever thing they want whatever dimensions or whatever shapes they want to have but let us say that these are a collection of particles which I am representing my rigid body as now these particles in a rigid body what we have seen that a body is rigid when for example any two points you join them and whatever motion these undergo these two points for example the distance between them does not change this is a b this is a prime b prime and the body is rigid means that the distance a b should be equal to distance a prime b prime or in other words the distance between any of these particles should not change now how does that happen that happens because all the particles apply equal and opposite forces on each other okay this particle apply equal and opposite force on each other why by Newton's third law they can apply a pushing force or a pulling force okay so I do not need to draw all of them but you can imagine that all these particles can exert various amount of forces on each other those forces okay exerted by ith particle on jth particle let me call as f bar ij okay now on this set of particles you also have some external forces acting let me call them as f1 f2 these are the external forces these are the internal forces which happen in between additionally this can also have forces like this f1 f2 which in this discrete picture I can show to happen like this and so on fk now you can apply Newton's law to each and every particle okay clearly you can apply Newton's law on each and every particle what does that say for any ith particle okay sum of all the forces is equal to mass of the ith particle times the acceleration of the ith particle now we write down this equation for all particles and then sum over all the forces okay what will we see that sum of all internal forces vectorial sum both internal because all the forces will be taken care of internal forces and external forces because we are adding the influences of all the forces on all the particles plus forces external but what will this be equal to this will be equal to nothing but sum of masses and accelerations of each and every particle but note one thing that internal forces they come in pairs if this is positive it has to be negative equal and opposite so what will happen is that when we sum over all the particles these internal forces they cancel off and their sum becomes 0 so what we are left with is only the external forces and so what do we have that f bar external sum over all the external forces is nothing but sigma mi times acceleration of ith particle but this can also be written as 1 by sigma mi okay into sigma mi into sigma mi ai but what is this quantity this quantity is nothing but the acceleration of the center of mass and this quantity is nothing but the total mass of all these particles which is nothing but the total mass of this rigid body and as a result what do we see that for a rigid body sum of all the external forces should be nothing but equal to the mass of the rigid body times the acceleration of its center of mass so this is a simple idea that why even the Newton's law is strictly speaking applicable only for a particle we can view a rigid body that we see in any real world okay a car or a truck or a plane or a wheelchair or a chair or anything that you can think of you can think that as a combination of particles and then for those sum of all the forces is nothing but a total mass times the acceleration of the center of mass so that is a simple understanding that even though Newton's law is strictly speaking defined only for a particle the sum you can also define that equivalently for a bunch of particles which is nothing but a representation for a rigid body and the particles cannot change the distances with respect to each other because why they are representing a rigid body even in that case we can say that sum of all the forces is nothing but total mass times the acceleration of the center of mass now coming back to this now this form is true for a constant mass system there are some system like jet planes for example where the mass is being lost so those we are not going to worry about them right now if you have any worries or concerns about that we can discuss them okay offline but not in these lectures now we have to be first define this quantity it is called as the linear momentum of a particle now what is the linear momentum of a particle the linear momentum of a particle okay we define it in a in a reverse way that we saw from Newton's laws that sum of all the forces okay acting on the particle or for a completely rigid body okay sum of all the external forces acting on the rigid body is nothing but mass times acceleration of the particle or in case of rigid bodies it is the acceleration of its center of mass so sum of all the forces is equal to m times dv by dt if the mass is constant okay for example we do not have a variable mass systems means for example a truck is going we see for example a tanker on many of our roads it keeps on like leaking water okay it makes the road all wet so those are not the constant mass systems we are looking about systems where for example the mass does not change so I can bring that mass inside so what do we have that d by dt mv is the force and this particular quantity mv we call that as a linear momentum of a particle or a linear momentum of a body so this l bar is one of the typical notations that is used for the momentum which is nothing but mass of the particle or the mass of the body times the velocity of the particle or the velocity of the center of mass whatever way but for the time being let us say that it is a particle even if it is not a particle it is a rigid body we can think of that as a center of mass particle so the linear momentum of a particle is nothing but mass times its velocity now there is a principle called as linear momentum conservation principle the very straight forward principle okay it just comes from this what it says is that if the sum of all the forces on the particle is 0 the linear momentum of the particle remains constant in both magnitude and direction right now so what it is that a linear momentum remain constant what does that mean sum of all the forces is 0 so dl by dt is equal to 0 what does that mean that l should be independent of time or l is constant now l is constant means what that the velocity the momentum of the particle the direction and magnitude both remains constant now the principle of linear momentum can also be used in the in in the form of components okay we are going to do that we have seen that f bar resultant force okay f resultant is equal to dl bar by dt now we can break this l into components lx and ly fix a coordinate frame like this i j with an origin okay x and y so we can say it is lx i plus ly j so f raise is nothing but dl by dt and we can rewrite this as f resultant can also be written as fx resultant i plus fy resultant j okay this is the magnitude in the x direction magnitude in the y direction and then what do we see is that we see that fx r will be equal to dlx by dt and fy r is equal to dl y by dt and even if the resultant force is not 0 if the force in the x direction is 0 it means that the momentum in x direction is conserved okay and if fy r is 0 then correspondingly momentum in y direction is conserved and if both forces are 0 then complete momentum is conserved so the principle of conservation of momentum can also be written in two form in the form of components that you do not need to have wholesale momentum being conserved if the forces are in one particular direction are 0 then the corresponding component of momentum is conserved okay more examples we will see when we come tomorrow now these are simple system of units we have been using so 1 Newton is 1 kg 1 meter per second square in US or the imperial units okay one pound okay one pound force the unit of mass is given as this we do not need to really bother about this okay we will try to use this as much as possible let us forget about this now Newton's law equations of motion okay what is that it says that the force is equal to m times acceleration and as we saw just a few moments ago that we can say break down the forces into components we can also break down acceleration into components x y and z so essentially what does we have that sum of forces in the x direction is equal to m into ax some same for some of the forces in y direction and in the z direction and ax can be written as x double dot ay as y double dot and az as z double dot straight forward now there is an alternate expression of Newton's law okay we can say that we can say that the sum of all the forces minus m a m a bar okay is equal to 0 which is like the equilibrium problems that we had done we say that this minus a bar is a inertial vector so typically this kind of approach is used when are you are in inertial non inertial frames of references but it is a very cumbersome approach okay and as far as possible okay if it is if it is possible you should try to stick to the inertial frame of reference and not use this there is a principle called as the Lambert's principle okay which becomes convenient for some vibration problems okay which will be discussed on 5th December but as far as the rest of our portion for dynamics go we are not going to use this dynamic equilibrium concept we are simply going to use this concept that sum of all the external forces is equal to mass times the acceleration okay forget about this dynamic equilibrium but this can be used sometimes to our advantages another name for that is the Lambert's principle that we just think of this as a force in the opposite direction of acceleration and behave as if this entire system is in equilibrium and sum of all the forces including this inertial force is equal to 0 okay but this is not really required we can go to the proper inertial frame of reference and can write that f bar is equal to m times a bar now comes the most important part okay what are the free body diagrams and kinetic diagrams so free body diagrams we have already seen now this kinetic diagrams is one conceptual it is a conceptual construction which you may or may not use but it is been used extensively in this beer and Johnston 10th edition and personally I feel that this kinetic diagrams really kind of clarify your understanding a lot what this kinetic diagrams are are we will see in step by step so first is that the free body diagrams is what we have done in statics the kinematic diagram is a conceptual construction which is done okay in order to get a proper understanding it is not mandatory but it definitely definitely makes our conception okay of our understanding or our representation much more transparent now what that is let us go about it what we have here is that that on this inclined plane we have a mass of 15 kg load acting on it is through frictionless pulley 225 Newton okay the mass is given to us we want to find out what is the dynamics of this particle given all these forces are okay what is the acceleration what is the velocity and so on so first look at the free body diagram so what is the particle that we are what is the mass that we are interested in is this 15 kg so this is the body B which is of 15 kg that we are interested in so we will try and isolate this okay we will isolate this now what are the various forces that are acting on this okay first we draw our axis system okay we can draw Cartesian Cartesian mean x y polar means E R E theta path means E T E N depending on what the problem is we will see that there are various problems so depending on what is the geometry of the problem and what is asked in the problem we will decide whether to choose Cartesian polar or path coordinates now what for the time being let us say we have chosen this Cartesian frame of reference y and x in these directions now what add in the applied forces what is the forces 225 Newton pulling force we apply that which is the 225 Newton pulling force which will act on this second is the weight which act downwards okay the supports with this is the support which the mass has from the bottom wedge okay if there can be a possibility of friction on it we draw a direction for the friction okay now the drawing the direction of the friction let us hold on to it for some time we will solve a few problems where what direction to choose will become more clear but for the time being let us say that we know that under the application of this force the mass is trying to move up and as a result the friction acting on it is the kinetic friction and acting in the downward direction but for a free body diagram this is a normal reaction this is a friction now what we are done is that we have done with the free body diagram that there are no other forces that can act on this mass now we want to understand that under the application of all these forces what happens to this body in equilibrium we were done once we figured out that what are the forces acting on it for the body to be in equilibrium we then find out what is this unknown reaction what is the unknown force and we are done but in this case what we want to know is that that a body is not in equilibrium the body is moving that the support reaction the support friction force is not enough to maintain this body in equilibrium so we want to know that under the application of all these forces what will be the acceleration that the body get okay and then this particular thing is that this is the free body diagram and this particular thing which we are going to draw is what we call as the kinetic diagram what do we do we isolate the body of interest which is a free body draw the mass times acceleration for the particle okay these are the two possible acceleration because this is our x frame this is our y frame so these are the two particular motions or the accelerations that this particle can have it can have an acceleration in the y direction we call that as m a y is the corresponding inertial term that we come here so m times a y is the force that we have here okay is mass times acceleration upwards the other component of acceleration can be up the plane so m ax okay is the corresponding mass times acceleration for this particle and what does Newton's law tell us Newton's law tell us that this sum of forces should be equal to the sum of mass times acceleration now you may directly write that some of the all the forces mass times acceleration but if you write this then it becomes very clear that m times acceleration in this direction is sum of all forces in this direction and m times acceleration in this direction is the sum of all the forces in this direction so if you draw the free body diagram as well as the kinetic diagram then there is the chance of error becomes very low okay and so that is why in this case in equilibrium this a becomes 0 the sum of all forces equal to 0 but in dynamics okay this need not be 0 we have to find out what the particular accelerations will be and these are two equipolian system that these forces will be equivalent to these kind of mass into acceleration so we say that force in this direction is mass times acceleration in this direction sum of all the forces in the normal directions is mass times acceleration in the y direction we will draw another free body diagram and kinetic diagram so draw free body diagrams and kinetic diagrams for this block now what do we want we want to we want to we have now two bodies body B body A we want to draw the free body diagrams for both block A and free body diagram and KD or the kinetic diagram for block B okay so first let us do look at this block A we isolate this what are the forces acting on it this is one tension this is another tension acting here third tension fourth tension okay our x and y axes are chosen just for convenience what are the applied forces okay mass okay the gravity is the applied force replace the supports okay all the support reactions or the strings will apply internal forces what are they this is the tension this is another tension this tension coming down tension then this is the another part of the tension from this mass you can have a friction force acting downwards the direction will become more clear when we solve more problems normal reaction acting from here normal reaction acting from here friction force acting from here and now what will that do okay dimensions are shown the kinetic diagram this okay M of A y M of A x okay we want to figure out that what is happening to this at equilibrium this is equal to 0 when not in equilibrium okay if this mass for example remains in contact with this then the center of mass of this does not move in the y direction or the acceleration in the y direction is 0 only acceleration that remains non-zero is the acceleration in this direction so these are the kinetic constraints that a fact that this cannot go down or lose contact immediately tell us that M A y is equal to 0 and this is M A x this is the wide axis we have chosen M A y in this direction x axis M A x in this direction and what does Newton's law tell us that sum of all these forces in the x direction this inclined direction is equal to M times A x and sum of all the forces in the y direction is M times A y and because of the particular geometry of the problem this A y is 0 and we get equations and we can solve okay so with this much have a nice day I will see you tomorrow morning.