Torque and Angular Momentum

Transcription

1 CHAPTER 10 (TRQUE, 2 ND LAW FR RTATIN) CHAPTER 11 (TRQUE, ANGULAR MMENTUM) Torque and Angular Momentum 1. Torque a. Definition b. Work, Power, W-K Theorem 2. Angular Momentum a. Newton s 2 nd Law in angular form b. Systems c. Conservation of Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 1

2 PRDUCING RTATIN ABUT AN AXIS USING A FRCE pening a door Top view Hinge Door F 1 and F 2 will not produce rotation Producing rotation around an axis is much easier when: - the force is greater magnitude of - the force is applied farther from the axis distance r from axis - the direction of is closer to 90 with respect to the arm Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 2

3 RTATIN CAUSED BY TANGENTIAL CMPNENT F F F t φ P Fr causes rotation around the axis Rotation axis But actually only the tangential component F t of the force causes rotation Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 3

4 TRQUE we must define a new quantity that describes the effectiveness of an interaction at producing rotation about a specific axis. This quantity is called torque τ and is a vector whose: - magnitude is Distance from axis X or (Moment arm* of ) X - direction is related to the direction of rotation (remember ). *Also called lever arm Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 4

5 MAGNITUDE F THE TRQUE F t φ P Fr = = sin Rotation axis Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 5

6 2 WAYS T EXPRESS THE TRQUE MAGNITUDE Line of action of F t φ P Fr φ P Rotation axis Rotation axis r Moment of arm of = = ( sin ) = sin = Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 6

7 TRQUE AS A VECTR z = y To get the cross-product right, use a xyz reference obeying the righthand rule: = x P F t = sin = φ φ Direction of : (2) Screw rule Rotation direction that tends to induce (1) Right-hand rule Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 7

8 SIGN F A TRQUE = y - If the torque tends to produce a counterclockwise rotation, it is Positive - therwise, it is negative. x = = sin P F t φ φ positive direction Sign of given by direction along z axis Sign of is in sin, negative when going clockwise Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 8

9 WRK DNE BY A TRQUE (PURE RTATIN) = = cos = = cos = = = cos = lim = = If τ is constant, = = = = Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 9

10 WRK-KINETIC ENERGY THEREM (PURE RTATIN) = = = ( ) When using conservation of energy in general, rotational kinetic energy should also be considered. Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 10

11 NEWTN S SECND LAW FR PURE RTATIN = = = F t m F r = = = For more than one force, we can generalize:, = ( ) C Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 11

12 TRANSLATINAL AND RTATINAL MTIN Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 12

13 ANGULAR MMENTUM (SINGLE PARTICLE) Linear Angular = = = Angular momentum defined: = = In general, of the particle of mass m is defined about a point ( origin of ) m Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 13

14 MAGNITUDE F ANGULAR MMENTUM = = of particle A about point = y = y P p t φ φ r φ P φ x x Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 14

15 PURE RTATIN SINGLE PARTICLE Single particle in pure rotation about axis z (oriented out of page) = = = = = unit vector along z M = = Another way to write in this case: = Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 15

16 NEWTN S SECND LAW IN ANGULAR FRM FR A SINGLE PARTICLE = = + = + = = = =, = The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle. All taken with respect to point. Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 16

17 ANGULAR MMENTUM F A SYSTEM F PARTICLES The total angular momentum of the system is the (vector) sum of the angular momenta of the individual particles (here with label i): = = With time, the angular momenta of individual particles may change because of interactions between the particles or with the outside. = =, Therefore, the net external torque acting on a system of particles is equal to the time rate of change of the system s total angular momentum., = Newton s 2 nd law in angular form for a system of particles Analogy with Newton s 2 nd law in linear form: F=dp/dt Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 17

18 CNSERVATIN F ANGULAR MMENTUM If total net torque is zero If the net external torque acting on a system is zero, the angular momentum of the system is constant, no matter what changes take place within the system. = = 0 = = ( ) If net torque along axis z is zero If the net external torque acting on a system is zero along a given axis z, the angular momentum L z of the system is constant., = = 0 = Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics II 18

19 FRMULA SUMMARY Work (pure rotation): Torque: = = If τ is constant: = Power: = = = W-K Theorem for pure rotation: = = = ( ) Angular momentum: Newton s 2 nd law: = = Pure rotation: = = Bernard Gelloz - Nagoya U. G30 Fundamentals of Physics I 19