Presentation on theme: "Torque Torque and golden rule of mechanics F1F1 F2F2 r1r1 r2r2 r1r1 r2r2 d2d2 d1d1 W 1 =W 2 F 1 d 1 = F 2 d 2 r 1 /d 1 = r 2 /d 2 F 1 r 1 = F 2 r 2 F1F1."— Presentation transcript:
Torque Torque and golden rule of mechanics F1F1 F2F2 r1r1 r2r2 r1r1 r2r2 d2d2 d1d1 W 1 =W 2 F 1 d 1 = F 2 d 2 r 1 /d 1 = r 2 /d 2 F 1 r 1 = F 2 r 2 F1F1 F2F2 r1r1 r2r2 F1F1 F 1 || r F Definition of torque
Torque and moment of inertia F r Analog of Newton’s Law for rotation Analog of mass for rotation
Example: Two blocks of masses m 1 = 15 kg and m 2 =20 kg are connected through a string that goes through the pulley of mass M = 2.00 kg and radiuss R = 25.0 cm. What is the angular acceleration of the pulley? R M m2m2 m1m1 m1gm1g m2gm2g
Example: What is the angular acceleration of a tree as it falls down? Model of a tree as uniform rod of length L. mg L The top of the tree will have an acceleration larger than g when:
Rolling += 45° Example: The rear wheel on a clown’s bicycle has twice the radius of the front wheel. Is the linear speed at the very top of the rear wheel greater than, less than, or the same as that of the front wheel? A. Twice greater thanB. Twice less thanC. Four times greater than D. Four times less thanE. The sameF. Non of these Rolling without slipping:
Example of Rolling: roller race Which of these get to the bottom of the ramp first?
1 Rolling down 2 Rolling up A. B. C. Example: Cylinder 1 is released on an incline and rolls down w/o slipping. Cylinder 2 has an initial angular speed at the bottom of the ramp and starts rolling up w/o slipping. What is the direction of the static friction force in each case? Friction must oppose relative motion. In the absence of friction: cylinder 1 would slide down (no rotation), cylinder 2 would keep rotating at the base of the ramp without going up. In both cases, relative velocity (at the point of contact) is down the incline, but for motion w/o slipping it should be zero, and f S must point up. The weight and the normal force produce zero torque about the CM. To produce “clockwise” angular acceleration and the appropriate torque direction (into the page), f S must point up in both cases. V relative fsfs fsfs
θ N mg fSfS Newton's 2nd law for translation of the CM: Newton's 2 nd law for rotation: Example: A cylinder of mass M and radius R rolls down an incline of angle θ with the horizontal. If the cylinder rolls without slipping, what is its acceleration? Rolling without slipping: compare to g sinθ, the results for an object sliding without friction
Example: A disk of radius R and mass M that mounted on a massless shaft of radius r << R and rolling down an incline with a groove. What is its acceleration? R r N Mg fSfS M m ~ 0 Very small if R >> r !