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Physics

Physics. Session. Rotational Mechanics - 4. Session Objectives. Session Objective. Angular Momentum of a Particle Conservation of angular momentum Angular Impulse. Particle P rotates around Point O. Angular Momentum of P. P(m). O. Angular Momentum. Angular momentum of a particle.

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Physics

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  1. Physics

  2. Session Rotational Mechanics - 4

  3. Session Objectives

  4. Session Objective • Angular Momentum of a Particle • Conservation of angular momentum • Angular Impulse

  5. Particle P rotates around Point O Angular Momentum of P P(m) O Angular Momentum Angular momentum of a particle

  6. Angular Momentum for a System of Particles For a system of particles (i = 1 to n) Angular momentum depends on the position of the axis

  7. Z Y r O m X Angular Momentum for a Rigid Body For a rigid body ( I : moment of inertia around the axis of rotation)

  8. Conservation of Angular Momentum

  9. If no external torque acts on a body so the total angular momentum of the body (or system of bodies) remains a constant , and the vector sum of all torques add up to zero. Conservation of Angular Momentum

  10. Equilibrium of a Rigid Body Two conditions are necessary. • Total force acting on the body • must add up to zero (equilibrium of • linear motion) • Total torque acting on the body • must add up to zero (equilibrium of • rotational motion)

  11. Comparison m I P=Mv F=ma P=Fv

  12. Class Test

  13. Class Exercise - 1 A planet revolves round the sun as shown. The KE is greatest at • A (b) B • (c) C (d) D

  14. Solution L is a constant for the system (as motion of sunis negligible). L is constant for planet.So K is maximumfor smallest r which is A. Kinetic energy at any point: Hence, answer is (a).

  15. A horizontal platform with a mass of 100 kg rotates at 10 rpm about a vertical axis passing through its centre. A man weighing 60 kg is standing on the edge. With what velocity will the platform begin to rotate if the man moves from the edge to the centre? • 22 rpm (b) 11 rpm • (c) 44 rpm (d) 66 rpm Class Exercise - 2

  16. Solution Moment of inertia of platform: Moment of inertia of man at edge: Moment of inertia of man at centre: O (r = 0)

  17. Solution contd.. L is conserved: Hence, answer is (a).

  18. A mass moves about the Y-axis with acceleration ay = (by2 – c); b and c are constants. The value of y for which the angular momentum is zero, is Class Exercise - 3

  19. Solution L (= mvyr) is zero when vy = 0 Hence, the answer is (c).

  20. A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity . Two objects each of mass m are attached gently to the ring. The ring now rotates with an angular velocity. Class Exercise - 4

  21. Solution when two masses are attached,M.I. becomes Hence, answer is (c).

  22. A mass M is moving with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin • is zero (b) remains constant • (c) goes on increasing (d) goes on decreasing Class Exercise - 5

  23. Solution Velocity Vx is along x-axis (constant) (Vy, Vz = 0) Let the motion of P be in xy plane.z = 0 Then y = b is constant. Hence, answer is (b).

  24. If the rotational kinetic energy and translation kinetic energy of a rolling body are same, the body is • disc (b) sphere • (c) cylinder (d) ring Class Exercise - 6

  25. Solution KE (translatory) = mv2 I always has the form kmR2, where k is a fraction or unity. KE (rotatory) = KE (translatory) if k = 1 or I = mR2. This is true for a ring. Hence, answer is (d).

  26. Two gear wheels, A and B, whose radii are in the ratio RA : RB = 1 : 2, are attached to each other by an endless belt. They are mounted with their axes parallel to each other. The system of two wheels is set into rotation. It is seen that both have the same angular momentum. What is the ratio of their moments of inertia? (Belts do not slip) • 1 : 2 (b) 1 : 1 • (c) 1 : 4 (d) Class Exercise - 7

  27. Solution Linear speed v of both wheels is the same. Hence, answer is (a).

  28. Consider the previous problem. If the mass ratio mA : mB = 1 : 4, what is the ratio of their rotational kinetic energies? • 4 : 1 (b) 2 : 1 • (c) 1 : 2 (d) 1 : 4 Class Exercise - 8

  29. Solution IA : IB = mARA2 : mBRB2 = mARA2 : 4mA(2RA)2 = 1 : 16 Hence, answer is (d).

  30. Two masses mA and mB are attached toeach other by a rigid, mass less rod of length 2r.They are set to rotation about the centre of the rod with an angular speed w. Then their angular momentum is Class Exercise - 9

  31. Solution As the rotation is in the xy plane, is along thez-axis. Hence, answer is (c).

  32. Thank you

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