Page 1 CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 INTRODUCTION In the earlier chapters we primarily considered the motion of a single particle. (A particle is represented as a point mass. It has practically no size.) We applied the results of our study even to the motion of bodies of finite size, assuming that motion of such bodies can be described in terms of the motion of a particle. Any real body which we encounter in daily life has a finite size. In dealing with the motion of extended bodies (bodies of finite size) often the idealised model of a particle is inadequate. In this chapter we shall try to go beyond this inadequacy. We shall attempt to build an understanding of the motion of extended bodies. An extended body, in the first place, is a system of particles. We shall begin with the consideration of motion of the system as a whole. The centre of mass of a system of particles will be a key concept here. We shall discuss the motion of the centre of mass of a system of particles and usefulness of this concept in understanding the motion of extended bodies. A large class of problems with extended bodies can be solved by considering them to be rigid bodies. Ideally a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change. It is evident from this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in many situations the deformations are negligible. In a number of situations involving bodies such as wheels, tops, steel beams, molecules and planets on the other hand, we can ignore that they warp, bend or vibrate and treat them as rigid. 7.1.1 What kind of motion can a rigid body have? Let us try to explore this question by taking some examples of the motion of rigid bodies. Let us begin with a rectangular block sliding down an inclined plane without any sidewise 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 7.11 Kinematics of rotational motion about a fixed axis 7.12 Dynamics of rotational motion about a fixed axis 7.13 Angular momentum in case of rotation about a fixed axis 7.14 Rolling motion Summary Points to Ponder Exercises Additional exercises Page 2 CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 INTRODUCTION In the earlier chapters we primarily considered the motion of a single particle. (A particle is represented as a point mass. It has practically no size.) We applied the results of our study even to the motion of bodies of finite size, assuming that motion of such bodies can be described in terms of the motion of a particle. Any real body which we encounter in daily life has a finite size. In dealing with the motion of extended bodies (bodies of finite size) often the idealised model of a particle is inadequate. In this chapter we shall try to go beyond this inadequacy. We shall attempt to build an understanding of the motion of extended bodies. An extended body, in the first place, is a system of particles. We shall begin with the consideration of motion of the system as a whole. The centre of mass of a system of particles will be a key concept here. We shall discuss the motion of the centre of mass of a system of particles and usefulness of this concept in understanding the motion of extended bodies. A large class of problems with extended bodies can be solved by considering them to be rigid bodies. Ideally a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change. It is evident from this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in many situations the deformations are negligible. In a number of situations involving bodies such as wheels, tops, steel beams, molecules and planets on the other hand, we can ignore that they warp, bend or vibrate and treat them as rigid. 7.1.1 What kind of motion can a rigid body have? Let us try to explore this question by taking some examples of the motion of rigid bodies. Let us begin with a rectangular block sliding down an inclined plane without any sidewise 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 7.11 Kinematics of rotational motion about a fixed axis 7.12 Dynamics of rotational motion about a fixed axis 7.13 Angular momentum in case of rotation about a fixed axis 7.14 Rolling motion Summary Points to Ponder Exercises Additional exercises 142 PHYSICS movement. The block is a rigid body. Its motion down the plane is such that all the particles of the body are moving together, i.e. they have the same velocity at any instant of time. The rigid body here is in pure translational motion (Fig. 7.1). In pure translational motion at any instant of time all particles of the body have the same velocity. Consider now the rolling motion of a solid metallic or wooden cylinder down the same inclined plane (Fig. 7.2). The rigid body in this problem, namely the cylinder, shifts from the top to the bottom of the inclined plane, and thus, has translational motion. But as Fig. 7.2 shows, all its particles are not moving with the same velocity at any instant. The body therefore, is not in pure translation. Its motion is translation plus â€˜something else.â€™ In order to understand what this â€˜something elseâ€™ is, let us take a rigid body so constrained that it cannot have translational motion. The most common way to constrain a rigid body so that it does not have translational motion is to fix it along a straight line. The only possible motion of such a rigid body is rotation. The line along which the body is fixed is termed as its axis of rotation. If you look around, you will come across many examples of rotation about an axis, a ceiling fan, a potterâ€™s wheel, a giant wheel in a fair, a merry-go-round and so on (Fig 7.3(a) and (b)). (a) (b) Fig. 7.3 Rotation about a fixed axis (a) A ceiling fan (b) A potterâ€™s wheel. Let us try to understand what rotation is, what characterises rotation. You may notice that in rotation of a rigid body about a fixed Fig 7.1 Translational (sliding) motion of a block down an inclined plane. (Any point like P 1 or P 2 of the block moves with the same velocity at any instant of time.) Fig. 7.2 Rolling motion of a cylinder It is not pure translational motion. Points P 1 , P 2 , P 3 and P 4 have different velocities (shown by arrows) at any instant of time. In fact, the velocity of the point of contact P 3 is zero at any instant, if the cylinder rolls without slipping. Page 3 CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 INTRODUCTION In the earlier chapters we primarily considered the motion of a single particle. (A particle is represented as a point mass. It has practically no size.) We applied the results of our study even to the motion of bodies of finite size, assuming that motion of such bodies can be described in terms of the motion of a particle. Any real body which we encounter in daily life has a finite size. In dealing with the motion of extended bodies (bodies of finite size) often the idealised model of a particle is inadequate. In this chapter we shall try to go beyond this inadequacy. We shall attempt to build an understanding of the motion of extended bodies. An extended body, in the first place, is a system of particles. We shall begin with the consideration of motion of the system as a whole. The centre of mass of a system of particles will be a key concept here. We shall discuss the motion of the centre of mass of a system of particles and usefulness of this concept in understanding the motion of extended bodies. A large class of problems with extended bodies can be solved by considering them to be rigid bodies. Ideally a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change. It is evident from this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in many situations the deformations are negligible. In a number of situations involving bodies such as wheels, tops, steel beams, molecules and planets on the other hand, we can ignore that they warp, bend or vibrate and treat them as rigid. 7.1.1 What kind of motion can a rigid body have? Let us try to explore this question by taking some examples of the motion of rigid bodies. Let us begin with a rectangular block sliding down an inclined plane without any sidewise 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 7.11 Kinematics of rotational motion about a fixed axis 7.12 Dynamics of rotational motion about a fixed axis 7.13 Angular momentum in case of rotation about a fixed axis 7.14 Rolling motion Summary Points to Ponder Exercises Additional exercises 142 PHYSICS movement. The block is a rigid body. Its motion down the plane is such that all the particles of the body are moving together, i.e. they have the same velocity at any instant of time. The rigid body here is in pure translational motion (Fig. 7.1). In pure translational motion at any instant of time all particles of the body have the same velocity. Consider now the rolling motion of a solid metallic or wooden cylinder down the same inclined plane (Fig. 7.2). The rigid body in this problem, namely the cylinder, shifts from the top to the bottom of the inclined plane, and thus, has translational motion. But as Fig. 7.2 shows, all its particles are not moving with the same velocity at any instant. The body therefore, is not in pure translation. Its motion is translation plus â€˜something else.â€™ In order to understand what this â€˜something elseâ€™ is, let us take a rigid body so constrained that it cannot have translational motion. The most common way to constrain a rigid body so that it does not have translational motion is to fix it along a straight line. The only possible motion of such a rigid body is rotation. The line along which the body is fixed is termed as its axis of rotation. If you look around, you will come across many examples of rotation about an axis, a ceiling fan, a potterâ€™s wheel, a giant wheel in a fair, a merry-go-round and so on (Fig 7.3(a) and (b)). (a) (b) Fig. 7.3 Rotation about a fixed axis (a) A ceiling fan (b) A potterâ€™s wheel. Let us try to understand what rotation is, what characterises rotation. You may notice that in rotation of a rigid body about a fixed Fig 7.1 Translational (sliding) motion of a block down an inclined plane. (Any point like P 1 or P 2 of the block moves with the same velocity at any instant of time.) Fig. 7.2 Rolling motion of a cylinder It is not pure translational motion. Points P 1 , P 2 , P 3 and P 4 have different velocities (shown by arrows) at any instant of time. In fact, the velocity of the point of contact P 3 is zero at any instant, if the cylinder rolls without slipping. SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143 axis, every particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has its centre on the axis. Fig. 7.4 shows the rotational motion of a rigid body about a fixed axis (the z-axis of the frame of reference). Let P 1 be a particle of the rigid body, arbitrarily chosen and at a distance r 1 from fixed axis. The particle P 1 describes a circle of radius r 1 with its centre C 1 on the fixed axis. The circle lies in a plane perpendicular to the axis. The figure also shows another particle P 2 of the rigid body, P 2 is at a distance r 2 from the fixed axis. The particle P 2 moves in a circle of radius r 2 and with centre C 2 on the axis. This circle, too, lies in a plane perpendicular to the axis. Note that the circles described by P 1 and P 2 may lie in different planes; both these planes, however, are perpendicular to the fixed axis. For any particle on the axis like P 3 , r = 0. Any such particle remains stationary while the body rotates. This is expected since the axis is fixed. Fig. 7.5 (a) A spinning top (The point of contact of the top with the ground, its tip O, is fixed.) Fig. 7.5 (b) An oscillating table fan. The pivot of the fan, point O, is fixed. In some examples of rotation, however, the axis may not be fixed. A prominent example of this kind of rotation is a top spinning in place [Fig. 7.5(a)]. (We assume that the top does not slip from place to place and so does not have translational motion.) We know from experience that the axis of such a spinning top moves around the vertical through its point of contact with the ground, sweeping out a cone as shown in Fig. 7.5(a). (This movement of the axis of the top around the vertical is termed precession.) Note, the point of contact of the top with ground is fixed. The axis of rotation of the top at any instant passes through the point of contact. Another simple example of this kind of rotation is the oscillating table fan or a pedestal fan. You may have observed that the axis of Fig. 7.4 A rigid body rotation about the z-axis (Each point of the body such as P 1 or P 2 describes a circle with its centre (C 1 or C 2 ) on the axis. The radius of the circle (r 1 or r 2 ) is the perpendicular distance of the point (P 1 or P 2 ) from the axis. A point on the axis like P 3 remains stationary). Page 4 CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 INTRODUCTION In the earlier chapters we primarily considered the motion of a single particle. (A particle is represented as a point mass. It has practically no size.) We applied the results of our study even to the motion of bodies of finite size, assuming that motion of such bodies can be described in terms of the motion of a particle. Any real body which we encounter in daily life has a finite size. In dealing with the motion of extended bodies (bodies of finite size) often the idealised model of a particle is inadequate. In this chapter we shall try to go beyond this inadequacy. We shall attempt to build an understanding of the motion of extended bodies. An extended body, in the first place, is a system of particles. We shall begin with the consideration of motion of the system as a whole. The centre of mass of a system of particles will be a key concept here. We shall discuss the motion of the centre of mass of a system of particles and usefulness of this concept in understanding the motion of extended bodies. A large class of problems with extended bodies can be solved by considering them to be rigid bodies. Ideally a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change. It is evident from this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in many situations the deformations are negligible. In a number of situations involving bodies such as wheels, tops, steel beams, molecules and planets on the other hand, we can ignore that they warp, bend or vibrate and treat them as rigid. 7.1.1 What kind of motion can a rigid body have? Let us try to explore this question by taking some examples of the motion of rigid bodies. Let us begin with a rectangular block sliding down an inclined plane without any sidewise 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 7.11 Kinematics of rotational motion about a fixed axis 7.12 Dynamics of rotational motion about a fixed axis 7.13 Angular momentum in case of rotation about a fixed axis 7.14 Rolling motion Summary Points to Ponder Exercises Additional exercises 142 PHYSICS movement. The block is a rigid body. Its motion down the plane is such that all the particles of the body are moving together, i.e. they have the same velocity at any instant of time. The rigid body here is in pure translational motion (Fig. 7.1). In pure translational motion at any instant of time all particles of the body have the same velocity. Consider now the rolling motion of a solid metallic or wooden cylinder down the same inclined plane (Fig. 7.2). The rigid body in this problem, namely the cylinder, shifts from the top to the bottom of the inclined plane, and thus, has translational motion. But as Fig. 7.2 shows, all its particles are not moving with the same velocity at any instant. The body therefore, is not in pure translation. Its motion is translation plus â€˜something else.â€™ In order to understand what this â€˜something elseâ€™ is, let us take a rigid body so constrained that it cannot have translational motion. The most common way to constrain a rigid body so that it does not have translational motion is to fix it along a straight line. The only possible motion of such a rigid body is rotation. The line along which the body is fixed is termed as its axis of rotation. If you look around, you will come across many examples of rotation about an axis, a ceiling fan, a potterâ€™s wheel, a giant wheel in a fair, a merry-go-round and so on (Fig 7.3(a) and (b)). (a) (b) Fig. 7.3 Rotation about a fixed axis (a) A ceiling fan (b) A potterâ€™s wheel. Let us try to understand what rotation is, what characterises rotation. You may notice that in rotation of a rigid body about a fixed Fig 7.1 Translational (sliding) motion of a block down an inclined plane. (Any point like P 1 or P 2 of the block moves with the same velocity at any instant of time.) Fig. 7.2 Rolling motion of a cylinder It is not pure translational motion. Points P 1 , P 2 , P 3 and P 4 have different velocities (shown by arrows) at any instant of time. In fact, the velocity of the point of contact P 3 is zero at any instant, if the cylinder rolls without slipping. SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143 axis, every particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has its centre on the axis. Fig. 7.4 shows the rotational motion of a rigid body about a fixed axis (the z-axis of the frame of reference). Let P 1 be a particle of the rigid body, arbitrarily chosen and at a distance r 1 from fixed axis. The particle P 1 describes a circle of radius r 1 with its centre C 1 on the fixed axis. The circle lies in a plane perpendicular to the axis. The figure also shows another particle P 2 of the rigid body, P 2 is at a distance r 2 from the fixed axis. The particle P 2 moves in a circle of radius r 2 and with centre C 2 on the axis. This circle, too, lies in a plane perpendicular to the axis. Note that the circles described by P 1 and P 2 may lie in different planes; both these planes, however, are perpendicular to the fixed axis. For any particle on the axis like P 3 , r = 0. Any such particle remains stationary while the body rotates. This is expected since the axis is fixed. Fig. 7.5 (a) A spinning top (The point of contact of the top with the ground, its tip O, is fixed.) Fig. 7.5 (b) An oscillating table fan. The pivot of the fan, point O, is fixed. In some examples of rotation, however, the axis may not be fixed. A prominent example of this kind of rotation is a top spinning in place [Fig. 7.5(a)]. (We assume that the top does not slip from place to place and so does not have translational motion.) We know from experience that the axis of such a spinning top moves around the vertical through its point of contact with the ground, sweeping out a cone as shown in Fig. 7.5(a). (This movement of the axis of the top around the vertical is termed precession.) Note, the point of contact of the top with ground is fixed. The axis of rotation of the top at any instant passes through the point of contact. Another simple example of this kind of rotation is the oscillating table fan or a pedestal fan. You may have observed that the axis of Fig. 7.4 A rigid body rotation about the z-axis (Each point of the body such as P 1 or P 2 describes a circle with its centre (C 1 or C 2 ) on the axis. The radius of the circle (r 1 or r 2 ) is the perpendicular distance of the point (P 1 or P 2 ) from the axis. A point on the axis like P 3 remains stationary). 144 PHYSICS rotation of such a fan has an oscillating (sidewise) movement in a horizontal plane about the vertical through the point at which the axis is pivoted (point O in Fig. 7.5(b)). While the fan rotates and its axis moves sidewise, this point is fixed. Thus, in more general cases of rotation, such as the rotation of a top or a pedestal fan, one point and not one line, of the rigid body is fixed. In this case the axis is not fixed, though it always passes through the fixed point. In our study, however, we mostly deal with the simpler and special case of rotation in which one line (i.e. the axis) is fixed. Thus, for us rotation will be about a fixed axis only unless stated otherwise. The rolling motion of a cylinder down an inclined plane is a combination of rotation about a fixed axis and translation. Thus, the â€˜something elseâ€™ in the case of rolling motion which we referred to earlier is rotational motion. You will find Fig. 7.6(a) and (b) instructive from this point of view. Both these figures show motion of the same body along identical translational trajectory. In one case, Fig. 7.6(a), the motion is a pure translation; in the other case [Fig. 7.6(b)] it is a combination of translation and rotation. (You may try to reproduce the two types of motion shown using a rigid object like a heavy book.) We now recapitulate the most important observations of the present section: The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is rotation. The rotation may be about an axis that is fixed (e.g. a ceiling fan) or moving (e.g. an oscillating table fan). We shall, in the present chapter, consider rotational motion about a fixed axis only. 7.2 CENTRE OF MASS We shall first see what the centre of mass of a system of particles is and then discuss its significance. For simplicity we shall start with a two particle system. We shall take the line joining the two particles to be the x- axis. Fig. 7.7 Let the distances of the two particles be x 1 and x 2 respectively from some origin O. Let m 1 and m 2 be respectively the masses of the two Fig. 7.6(a) Motion of a rigid body which is pure translation. Fig. 7.6(b) Motion of a rigid body which is a combination of translation and rotation. Fig 7.6 (a) and 7.6 (b) illustrate different motions of the same body. Note P is an arbitrary point of the body; O is the centre of mass of the body, which is defined in the next section. Suffice to say here that the trajectories of O are the translational trajectories Tr 1 and Tr 2 of the body. The positions O and P at three different instants of time are shown by O 1 , O 2 , and O 3 , and P 1 , P 2 and P 3 , respectively, in both Figs. 7.6 (a) and (b) . As seen from Fig. 7.6(a), at any instant the velocities of any particles like O and P of the body are the same in pure translation. Notice, in this case the orientation of OP , i.e. the angle OP makes with a fixed direction, say the horizontal, remains the same, i.e. a 1 = a 2 = a 3 . Fig. 7.6 (b) illustrates a case of combination of translation and rotation. In this case, at any instants the velocities of O and P differ. Also, a 1 , a 2 and a 3 may all be different. Page 5 CHAPTER SEVEN SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 7.1 INTRODUCTION In the earlier chapters we primarily considered the motion of a single particle. (A particle is represented as a point mass. It has practically no size.) We applied the results of our study even to the motion of bodies of finite size, assuming that motion of such bodies can be described in terms of the motion of a particle. Any real body which we encounter in daily life has a finite size. In dealing with the motion of extended bodies (bodies of finite size) often the idealised model of a particle is inadequate. In this chapter we shall try to go beyond this inadequacy. We shall attempt to build an understanding of the motion of extended bodies. An extended body, in the first place, is a system of particles. We shall begin with the consideration of motion of the system as a whole. The centre of mass of a system of particles will be a key concept here. We shall discuss the motion of the centre of mass of a system of particles and usefulness of this concept in understanding the motion of extended bodies. A large class of problems with extended bodies can be solved by considering them to be rigid bodies. Ideally a rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change. It is evident from this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in many situations the deformations are negligible. In a number of situations involving bodies such as wheels, tops, steel beams, molecules and planets on the other hand, we can ignore that they warp, bend or vibrate and treat them as rigid. 7.1.1 What kind of motion can a rigid body have? Let us try to explore this question by taking some examples of the motion of rigid bodies. Let us begin with a rectangular block sliding down an inclined plane without any sidewise 7.1 Introduction 7.2 Centre of mass 7.3 Motion of centre of mass 7.4 Linear momentum of a system of particles 7.5 Vector product of two vectors 7.6 Angular velocity and its relation with linear velocity 7.7 Torque and angular momentum 7.8 Equilibrium of a rigid body 7.9 Moment of inertia 7.10 Theorems of perpendicular and parallel axes 7.11 Kinematics of rotational motion about a fixed axis 7.12 Dynamics of rotational motion about a fixed axis 7.13 Angular momentum in case of rotation about a fixed axis 7.14 Rolling motion Summary Points to Ponder Exercises Additional exercises 142 PHYSICS movement. The block is a rigid body. Its motion down the plane is such that all the particles of the body are moving together, i.e. they have the same velocity at any instant of time. The rigid body here is in pure translational motion (Fig. 7.1). In pure translational motion at any instant of time all particles of the body have the same velocity. Consider now the rolling motion of a solid metallic or wooden cylinder down the same inclined plane (Fig. 7.2). The rigid body in this problem, namely the cylinder, shifts from the top to the bottom of the inclined plane, and thus, has translational motion. But as Fig. 7.2 shows, all its particles are not moving with the same velocity at any instant. The body therefore, is not in pure translation. Its motion is translation plus â€˜something else.â€™ In order to understand what this â€˜something elseâ€™ is, let us take a rigid body so constrained that it cannot have translational motion. The most common way to constrain a rigid body so that it does not have translational motion is to fix it along a straight line. The only possible motion of such a rigid body is rotation. The line along which the body is fixed is termed as its axis of rotation. If you look around, you will come across many examples of rotation about an axis, a ceiling fan, a potterâ€™s wheel, a giant wheel in a fair, a merry-go-round and so on (Fig 7.3(a) and (b)). (a) (b) Fig. 7.3 Rotation about a fixed axis (a) A ceiling fan (b) A potterâ€™s wheel. Let us try to understand what rotation is, what characterises rotation. You may notice that in rotation of a rigid body about a fixed Fig 7.1 Translational (sliding) motion of a block down an inclined plane. (Any point like P 1 or P 2 of the block moves with the same velocity at any instant of time.) Fig. 7.2 Rolling motion of a cylinder It is not pure translational motion. Points P 1 , P 2 , P 3 and P 4 have different velocities (shown by arrows) at any instant of time. In fact, the velocity of the point of contact P 3 is zero at any instant, if the cylinder rolls without slipping. SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143 axis, every particle of the body moves in a circle, which lies in a plane perpendicular to the axis and has its centre on the axis. Fig. 7.4 shows the rotational motion of a rigid body about a fixed axis (the z-axis of the frame of reference). Let P 1 be a particle of the rigid body, arbitrarily chosen and at a distance r 1 from fixed axis. The particle P 1 describes a circle of radius r 1 with its centre C 1 on the fixed axis. The circle lies in a plane perpendicular to the axis. The figure also shows another particle P 2 of the rigid body, P 2 is at a distance r 2 from the fixed axis. The particle P 2 moves in a circle of radius r 2 and with centre C 2 on the axis. This circle, too, lies in a plane perpendicular to the axis. Note that the circles described by P 1 and P 2 may lie in different planes; both these planes, however, are perpendicular to the fixed axis. For any particle on the axis like P 3 , r = 0. Any such particle remains stationary while the body rotates. This is expected since the axis is fixed. Fig. 7.5 (a) A spinning top (The point of contact of the top with the ground, its tip O, is fixed.) Fig. 7.5 (b) An oscillating table fan. The pivot of the fan, point O, is fixed. In some examples of rotation, however, the axis may not be fixed. A prominent example of this kind of rotation is a top spinning in place [Fig. 7.5(a)]. (We assume that the top does not slip from place to place and so does not have translational motion.) We know from experience that the axis of such a spinning top moves around the vertical through its point of contact with the ground, sweeping out a cone as shown in Fig. 7.5(a). (This movement of the axis of the top around the vertical is termed precession.) Note, the point of contact of the top with ground is fixed. The axis of rotation of the top at any instant passes through the point of contact. Another simple example of this kind of rotation is the oscillating table fan or a pedestal fan. You may have observed that the axis of Fig. 7.4 A rigid body rotation about the z-axis (Each point of the body such as P 1 or P 2 describes a circle with its centre (C 1 or C 2 ) on the axis. The radius of the circle (r 1 or r 2 ) is the perpendicular distance of the point (P 1 or P 2 ) from the axis. A point on the axis like P 3 remains stationary). 144 PHYSICS rotation of such a fan has an oscillating (sidewise) movement in a horizontal plane about the vertical through the point at which the axis is pivoted (point O in Fig. 7.5(b)). While the fan rotates and its axis moves sidewise, this point is fixed. Thus, in more general cases of rotation, such as the rotation of a top or a pedestal fan, one point and not one line, of the rigid body is fixed. In this case the axis is not fixed, though it always passes through the fixed point. In our study, however, we mostly deal with the simpler and special case of rotation in which one line (i.e. the axis) is fixed. Thus, for us rotation will be about a fixed axis only unless stated otherwise. The rolling motion of a cylinder down an inclined plane is a combination of rotation about a fixed axis and translation. Thus, the â€˜something elseâ€™ in the case of rolling motion which we referred to earlier is rotational motion. You will find Fig. 7.6(a) and (b) instructive from this point of view. Both these figures show motion of the same body along identical translational trajectory. In one case, Fig. 7.6(a), the motion is a pure translation; in the other case [Fig. 7.6(b)] it is a combination of translation and rotation. (You may try to reproduce the two types of motion shown using a rigid object like a heavy book.) We now recapitulate the most important observations of the present section: The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is rotation. The rotation may be about an axis that is fixed (e.g. a ceiling fan) or moving (e.g. an oscillating table fan). We shall, in the present chapter, consider rotational motion about a fixed axis only. 7.2 CENTRE OF MASS We shall first see what the centre of mass of a system of particles is and then discuss its significance. For simplicity we shall start with a two particle system. We shall take the line joining the two particles to be the x- axis. Fig. 7.7 Let the distances of the two particles be x 1 and x 2 respectively from some origin O. Let m 1 and m 2 be respectively the masses of the two Fig. 7.6(a) Motion of a rigid body which is pure translation. Fig. 7.6(b) Motion of a rigid body which is a combination of translation and rotation. Fig 7.6 (a) and 7.6 (b) illustrate different motions of the same body. Note P is an arbitrary point of the body; O is the centre of mass of the body, which is defined in the next section. Suffice to say here that the trajectories of O are the translational trajectories Tr 1 and Tr 2 of the body. The positions O and P at three different instants of time are shown by O 1 , O 2 , and O 3 , and P 1 , P 2 and P 3 , respectively, in both Figs. 7.6 (a) and (b) . As seen from Fig. 7.6(a), at any instant the velocities of any particles like O and P of the body are the same in pure translation. Notice, in this case the orientation of OP , i.e. the angle OP makes with a fixed direction, say the horizontal, remains the same, i.e. a 1 = a 2 = a 3 . Fig. 7.6 (b) illustrates a case of combination of translation and rotation. In this case, at any instants the velocities of O and P differ. Also, a 1 , a 2 and a 3 may all be different. SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 145 particles. The centre of mass of the system is that point C which is at a distance X from O, where X is given by 1 1 2 2 1 2 m x m x X m m + = + (7.1) In Eq. (7.1), X can be regarded as the mass- weighted mean of x 1 and x 2 . If the two particles have the same mass m 1 = m 2 = m , then 1 2 1 2 2 2 mx mx x x X m + + = = Thus, for two particles of equal mass the centre of mass lies exactly midway between them. If we have n particles of masses m 1 , m 2 , ...m n respectively, along a straight line taken as the x- axis, then by definition the position of the centre of the mass of the system of particles is given by 1 1 2 2 1 2 .... .... i i n n n i m x m x m x m x X m m m m + + + = = + + + ? ? (7.2) where x 1 , x 2 ,...x n are the distances of the particles from the origin; X is also measured from the same origin. The symbol ? (the Greek letter sigma) denotes summation, in this case over n particles. The sum i m M = ? is the total mass of the system. Suppose that we have three particles, not lying in a straight line. We may define x and y- axes in the plane in which the particles lie and represent the positions of the three particles by coordinates (x 1 ,y 1 ), (x 2 ,y 2 ) and (x 3 ,y 3 ) respectively. Let the masses of the three particles be m 1 , m 2 and m 3 respectively. The centre of mass C of the system of the three particles is defined and located by the coordinates (X, Y) given by 1 1 2 2 3 3 1 2 3 m x m x m x X m m m + + = + + (7.3a) 1 1 2 2 3 3 1 2 3 m y m y m y Y m m m + + = + + (7.3b) For the particles of equal mass m = m 1 = m 2 = m 3 , 1 2 3 1 2 3 ( ) 3 3 m x x x x x x X m + + + + = = 1 2 3 1 2 3 ( ) 3 3 m y y y y y y Y m + + + + = = Thus, for three particles of equal mass, the centre of mass coincides with the centroid of the triangle formed by the particles. Results of Eqs. (7.3a) and (7.3b) are generalised easily to a system of n particles, not necessarily lying in a plane, but distributed in space. The centre of mass of such a system is at (X, Y, Z ), where i i m x X M = ? (7.4a) i i m y Y M = ? (7.4b) and i i m z Z M = ? (7.4c) Here M = i m ? is the total mass of the system. The index i runs from 1 to n; m i is the mass of the i th particle and the position of the i th particle is given by (x i , y i , z i ). Eqs. (7.4a), (7.4b) and (7.4c) can be combined into one equation using the notation of position vectors. Let i r be the position vector of the i th particle and R be the position vector of the centre of mass: i i i i x y z = + + r i j k and X Y Z = + + R i j k Then i i m M = ? r R (7.4d) The sum on the right hand side is a vector sum. Note the economy of expressions we achieve by use of vectors. If the origin of the frame of reference (the coordinate system) is chosen to be the centre of mass then 0 i i m = ? r for the given system of particles. A rigid body, such as a metre stick or a flywheel, is a system of closely packed particles; Eqs. (7.4a), (7.4b), (7.4c) and (7.4d) are therefore, applicable to a rigid body. The number of particles (atoms or molecules) in such a body is so large that it is impossible to carry out the summations over individual particles in these equations. Since the spacing of the particles isRead More

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