1. Physics

2. Session Rotational Mechanics - 6

3. Session Objectives

4. Session Objective 1.Angular momentum of a body in rolling 2.Friction in rolling 3.Motion of rigid bodies on frictionless and frictional inclined/horizontal plane 4.Problem Solving

5. Angular Momentum in Rolling The angular momentum about point of contact (P) is given by L P = mv cm r + I c w… (i) V cm P

6. Friction in Forward Slipping V CM V CM > r

7. Friction in Backward Slipping V CM V CM < r

8. Friction in Rolling on an Inclined Plan For rotational motion, f · R = I For translational motion, mg sin – f = ma For rolling, a = R R f

9. Class Test

10. Class Exercise - 1 A heavy disc is rolling down without slipping down an inclined plane, which is rough. Let V 1 be its velocity at the bottom. Let V 2 be the velocity of the same ball if the plane were smooth. Then (a) V 1 < V 2 (b) V 1 = V 2 (c) V 1 > V 2 (d) Cannot be predicted

11. Solution V 2 > V 1 Applying conservation of energy in the case of pure rolling, Applying conservation of energy in the case of sliding Hence, answer is (a).

12. Class Exercise - 2 A hollow cylinder rolls down a rough incline. Its velocity at the bottom is affected by (a) radius(b) length (c) density(d) change in height

13. Solution a, c, d Velocity will be affected by: a.Radius — It changes the M.I. b.Density — It changes the mass distribution and, hence, the M.I. c.Change in height — It changes the total energy.

14. Class Exercise - 3 A uniform circular disc of radius r placed on a rough horizontal plane has initial velocity V 0 and an angular velocity w 0 as shown. The disc comes to rest after some time. Then (a)friction acts in the backward direction (b) the point of contact is at rest (d) V 0 = 2r  0 in magnitude VoVo

15. Solution As the point of contact moves forward, the friction will act in the backward direction. Hence, answer is (c).

16. Class Exercise - 4 A cylinder of mass m rolls down a rough inclined plane without slipping. Then which is/are correct? (a) Acceleration is lower than that on a smooth plane (b) Energy is lower than that on a smooth plane (c) KE is related to the coefficient of friction (  ) (d) The friction force may not be N, where N is the normal force

17. Solution Acceleration will be lower than that on a smooth plane, as in pure rolling Hence, answer is (a).

18. Class Exercise - 5 A solid sphere of mass M and radius R rolls down an inclined plane of inclination . The acceleration of the sphere down the plane is

19. Solution Hence, answer is (c).

20. Class Exercise - 6 Which of the following statements is not true? (a)The instantaneous speed of the point of contact of disc rolling perfectly on a rough horizontal plane is zero (b) The frictional force on the same body is zero (c) The instantaneous acceleration of the point of contact of the same body is zero (d) Work done against frictional force in the case of same body is not zero

21. Solution b, d The answers are definition of rolling.

22. Class Exercise - 7 Body A is placed on a rough horizontal surface with a horizontal speed of V 0 and a spin of  0 as shown. Then (a) it moves faster and its rotation slows down (b) it moves slower and its rotation slows down (c) there is no friction at P (d) it stops VoVo P A

23. Solution As tends to produce a forward motion at the point of contact P, friction is in backward direction. So both V 0 and reduce. Hence, answer is (b). Friction Translation due to rolling in forward direction VoVo P

24. Class Exercise - 8 A ball is released on a rough horizontal plane with a speed v 0 so that it slides on the surface without rolling. To what value should the speed decrease so that it rolls without sliding? VoVo P

25. Solution The moment the ball touches the surface with speed v 0 to right, frictional force f acts to the left. Torque due to f produces clockwise rotation (so increases from initial zero value) while f(= mmg) produces a reduction of v 0. Let the pure rolling motion start at linear velocity v. ( v = r ) VoVo P r f

26. Solution contd.. Translation: Acceleration Rotation:

27. Solution contd..

28. Class Exercise - 9 A solid sphere (radius R, mass M) is given a clockwise angular velocity  0 and lowered to a flat horizontal surface with coefficient of friction  with the sphere. The sphere first slips on the surface. After time t the motion changes to pure rolling without slipping. What is the value of t and the value of velocity of centre of mass at t?

29. Solution When rolling starts, let v and be the linear and angular velocities respectively. F = Mg is the friction force (in the forward direction), which produces an acceleration a = g. v f

30. Solution The torque is fR = Mg R and

31. Class Exercise - 10 A disc of radius b and mass m rolls down an inclined plane of vertical height h. The translational speed when it reaches the bottom of the plane will be

32. Solution Applying the conservation of energy, Hence, answer is (a).

33. Thank you