1 Work in Rotational Motion Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r d  The radial component of the force does no work because it is perpendicular to the displacement

2 Work in Rotational Motion, cont Work is also related to rotational kinetic energy: This is the same mathematical form as the work-kinetic energy theorem for translation If an object is both rotating and translating, W =  K +  K R

3 Power in Rotational Motion The rate at which work is being done in a time interval dt is the power This is analogous to P = Fv in a linear system

4

5

6

7

Angular Momentum The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum The SI units of angular momentum are (kg. m 2 )/ s

9 Torque and Angular Momentum The torque is related to the angular momentum Similar to the way force is related to linear momentum This is the rotational analog of Newton’s Second Law The torque and angular momentum must be measured about the same origin This is valid for any origin fixed in an inertial frame

10 Angular Momentum of a System of Particles The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles Differentiating with respect to time

11 Angular Momentum of a Rotating Rigid Object, cont To find the angular momentum of the entire object, add the angular momenta of all the individual particles This is analogous to the translational momentum of p = m v

12 Summary of Useful Equations

13

14

15

16

17

Conservation of Angular Momentum The total angular momentum of a system is conserved if the resultant external torque acting on the system is zero Net torque = 0 -> means that the system is isolated For a system of particles, L tot =  L n = constant

19 Conservation of Angular Momentum, cont If the mass of an isolated system undergoes redistribution, the moment of inertia changes The conservation of angular momentum requires a compensating change in the angular velocity I i  i = I f  f This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system The net torque must be zero in any case

20 An example for conservation of angular momentum

21

22

23

24

25 Conservation Law Summary For an isolated system - (1) Conservation of Energy: E i = E f (2) Conservation of Linear Momentum: (3) Conservation of Angular Momentum:

Rolling Motion The red curve shows the path moved by a point on the rim of the object This path is called a cycloid The green line shows the path of the center of mass of the object

27 Pure Rolling Motion The surfaces must exert friction forces on each other Otherwise the object would slide rather than roll In pure rolling motion, an object rolls without slipping In such a case, there is a simple relationship between its rotational and translational motions

28 Rolling Object, Other Points A point on the rim, P, rotates to various positions such as Q and P ’ At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs Fig 10.27

29 Rolling Object, Center of Mass The velocity of the center of mass is The acceleration of the center of mass is

30 Parallel-Axis Theorem For an arbitrary axis, the parallel-axis theorem often simplifies calculations The theorem states I p = I CM + MD 2 I p is about any axis parallel to the axis through the center of mass of the object I CM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis

31 Total Kinetic Energy of a Rolling Object The total kinetic energy of a rolling object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass K = 1/2 I CM  2 + 1/2 Mv CM 2

32 Total Kinetic Energy, Example Accelerated rolling motion is possible only if friction is present between the sphere and the incline The friction produces the net torque required for rotation

33

34

35

36 Exercises of Chapter 10 5, 13, 15, 23, 26, 32, 37, 43, 49, 56, 61, 72, 79