* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download rotational kinetic energy
Uncertainty principle wikipedia , lookup
Quantum vacuum thruster wikipedia , lookup
Specific impulse wikipedia , lookup
Atomic theory wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Sagnac effect wikipedia , lookup
Classical mechanics wikipedia , lookup
Jerk (physics) wikipedia , lookup
Routhian mechanics wikipedia , lookup
Centripetal force wikipedia , lookup
Hunting oscillation wikipedia , lookup
Center of mass wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Matter wave wikipedia , lookup
Rotational spectroscopy wikipedia , lookup
Work (physics) wikipedia , lookup
Tensor operator wikipedia , lookup
Old quantum theory wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Moment of inertia wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Equations of motion wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Accretion disk wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Classical central-force problem wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Photon polarization wikipedia , lookup
Angular momentum wikipedia , lookup
Angular momentum operator wikipedia , lookup
Physics Session Rotational Mechanics - 4 Session Objectives Session Objective 1. Angular Momentum of a Particle 2. Conservation of angular momentum 3. Angular Impulse Angular Momentum Angular momentum of a particle Particle P rotates around Point O Angular Momentum L of P v L r p (p mv) mr ( r) P(m) L is an axial vector units : Kg m2 / s L mvr sin L I ( I mr2 ) r O Angular Momentum for a System of Particles For a system of particles (i = 1 to n) L n r i 1 i pi I ( is a cons tan t. So I m i ri2 ) Angular momentum depends on the position of the axis Angular Momentum for a Rigid Body For a rigid body L I ( I : moment of inertia around the axis of rotation) Z Y r O m X Conservation of Angular Momentum L l dL d l l dt dt ext dL dt If, ext d dt ext l dL 0 then, 0 dt L cons tan t If ext 0 lii lf f Conservation of Angular Momentum If no external torque acts on a body dL 0 so the total angular momentum dt of the body (or system of bodies) remains a constant , and the vector sum of all torques add up to zero. Equilibrium of a Rigid Body Two conditions are necessary. (i) Total force acting on the body must add up to zero (equilibrium of linear motion) (ii) Total torque acting on the body must add up to zero (equilibrium of rotational motion) Comparison LINEAR Mass Momentum m P=Mv ANGULAR I L I Newton’s Second Law F=ma Work W Fdx W d Kinetic Energy 1 K mv 2 2 1 2 K I 2 Power(const ant force) P=Fv P I Class Test Class Exercise - 1 A planet revolves round the sun as shown. The KE is greatest at (a) A (b) B (c) C (d) D C B A D Solution Kinetic energy at any point: 2 2 L L K 2I 2mr 2 L is a constant for the system (as motion of sun is negligible). L is constant for planet. So K is 1 maximum for smallest r K which is A. r2 Hence, answer is (a). Class Exercise - 2 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? (a) 22 rpm (b) 11 rpm (c) 44 rpm (d) 66 rpm Solution Moment of inertia of platform: 1 2 mPr I1 2 Moment of inertia of man at edge: mmr 2 I2 Moment of inertia of man at centre: O ( r = 0) Solution contd.. L is conserved: I1 I2 1 I12 1 1 100 60 10 100 2 2 2 1100 502 2 22 rpm Hence, answer is (a). Class Exercise - 3 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 3c (a) b (c) 3c b (b) c b (d) 3b c Solution dv y Acceleration a y by 2 c dt dv y vy by 2 c dy vy 0 y v y dv y 2 by c dy 0 3 by 2 vy cy 2 3 by3 cy y 3 3c b L (= mvyr) is zero when vy = 0 Hence, the answer is (c). Class Exercise - 4 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. M (a) Mm (b) M (c) M 2m (d) M 2m M 2m M 2m M Solution I (ring) = MR2 , L = I = MR2 when two masses are attached, M.I. becomes I' MR2 mR2 mR2 (M 2m)R2 L I I' ' I M ' I' M 2m Hence, answer is (c). Class Exercise - 5 A mass M is moving with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin (a) is zero (b) remains constant (c) goes on increasing (d) goes on decreasing Solution Let the motion of P be in xy plane. y z=0 P Vx b x O Then y = b is constant. Velocity Vx is along x-axis (constant) L k L Z k xVy yVx m k mbVx (Constant) (Vy, Vz = 0) Hence, answer is (b). Class Exercise - 6 If the rotational kinetic energy and translation kinetic energy of a rolling body are same, the body is (a) disc (b) sphere (c) cylinder (d) ring Solution 1 KE (translatory) = mv2 2 2 1 2 1 v KE(rotatary) I I 2 2 R2 I always has the form kmR2, where k is a fraction or unity. 2 1 1 2v 2 KE (rotatory) kmR kmv 2 2 2 R KE (rotatory) = KE (translatory) if k = 1 or I = mR2. This is true for a ring. Hence, answer is (d). Class Exercise - 7 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) (a) 1 : 2 (b) 1 : 1 (c) 1 : 4 (d) 1: 2 Solution Linear speed v of both wheels is the same. v ARA BRB A B RB RA 2 A 2B L IAA IBB IA B 1 IB A 2 Hence, answer is (a). Class Exercise - 8 Consider the previous problem. If the mass ratio mA : mB = 1 : 4, what is the ratio of their rotational kinetic energies? (a) 4 : 1 (b) 2 : 1 (c) 1 : 2 (d) 1 : 4 Solution IA : IB = mARA2 : mBRB2 = mARA2 : 4mA(2RA)2 = 1 : 16 A = 2B 2 A 1 1 2 1 2 IA A , IBB 16IA 2 2 2 2 (KE)A 4 KEB Hence, answer is (d). 1 16 2 IA A 2 4 Class Exercise - 9 Two masses mA and mB are attached to each 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 . Then their angular momentum is (a) 1 mA mB r 2 along the x axis 2 z (b) mA mB r along the y axis 2 1 (c) mA mB r 2 along the z axis 2 2 (d) mA mB r in the x y plane MA 2r r r MB x Solution 1 1 2 IA mAr , IB mBr 2 2 2 1 2 I mA mB r 2 1 L mA mB r 2 2 As the rotation is in the xy plane, L is along the z-axis. Hence, answer is (c). Thank you