Work, Power and Energy in Rotational Motion AP Physics C Mrs. Coyle

Relating Torque to Work and Power Consider a rigid body rotating dθ during an infinitesimal time interval dt.

Work and Torque During the time dt, the object moves a distance ds= r d θ The work dW is: dW= F · ds = F · r d θ = (Fcos ɸ) r d θ The torque τ = F sin (90- ɸ) r = (Fcos ɸ) r dW = τ dθ (Analogous to dW= Fds for translational motion)

dW = τ dθ dW = τ dθ dt dt So power is P= τ ω (Analogous to P= Fv for translational motion) Power and Torque

Work –Kinetic Energy Theorem for Rotational Motion ΣW= ½ I ω 2 f - ½ I ω 2 i The net work done by external forces in rotating a symmetric rigid object about a fixed axis equals the change in the object’s rotational kinetic energy.

Ex: #42

A top has a moment of inertia of 4.00 x10 -4 kg m 2 and is initially at rest. It is free to rotate about the stationary axis AA’. A string, wrapped around a peg along the axis of the top, is pulled in such a manner as to maintain a constant tension of 5.57N. If the string does not slip while it is unwound from the peg, what is the angular speed of the top after 80.0cm of string has been pulled off the peg? Ans: 149rad/s

Ex: # 44 A cylindrical rod 24.0cm long with mass 1.20kg and radius 1.50cm has a ball of diameter 8.00cm and mass 2.00kg attached to one end. The arrangement is originally vertical and stationary with the ball on top. The system is free to pivot about the bottom end of the rod after being given a nudge. a) After the rod rotates through 90 o, what is its rotational kinetic energy?

#44 cont’d b) What is the angular speed of the rod and the ball? c) What is the linear speed of the ball? d) How does this compare to the speed if the ball had fallen freely through the same distance of 28cm? Ans: a)6.90J, b)8.73rad/s, (I tot =0.181kgm 2 ), c)2.44 m/s, d) times greater

Ex:#45

An object with a weight of 50.0N is attached to the free end of a light string wrapped around a reel off radius 0.250m and mass 3.00kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center. The suspended object is released 6.00m above the floor. a) Determine the tension in the string, the acceleration of the object and the speed with which the object hits the floor.

#45 Cont’d b) Verify your last answer by using the principle of conservation of energy to find the speed with which the object hits the floor. Ans: a)11.4N, 7.57m/s 2, 9.53 m/s, b) 9.53m/s