Rotational Dynamics Example Example: cord of negligible mass is wrapped around a frictionless pulley of mass M and radius R. The pulley rotates about a fixed axle which passes through its center. A bucket of mass m hangs from the cord. Calculate the angular acceleration of the pulley and the linear acceleration of the bucket. The litany for force problems still works! Step 1. Draw a basic sketch. R M m

Step 3. Label each vector (done). Step 2. Draw free-body diagrams. For objects that rotate, the free-body diagram must be extended; it must show the actual points of application of forces. P (force due to axle) W=Mg T R O +  x pulley T w=mg a x bucket To avoid extraneous minus signs, make sure your a and  have the same “sense.” Step 3. Label each vector (done). Step 4. Draw axes (done). Step 5. Draw projections of forces not along axes (done).

Tx + Wx = max T,z + W,z + P,z = Iz -T + mg = ma +RT + 0 + 0 = I P (force due to axle) W=Mg T R O +  x pulley Step 6. OSE T w=mg a x bucket bucket: Fx = max pulley: z = Iz (We don’t need the sum of forces equation for the pulley in this example.) Step 7. Write out sum of forces/torques equation explicitly, then replace generic quantities with labeled quantities. bucket pulley Tx + Wx = max T,z + W,z + P,z = Iz -T + mg = ma +RT + 0 + 0 = I Step 8. Solve. You need to use the OSE a = R to connect a and .

Rotational Kinetic Energy Example If an object is rolling (with axis of rotation through its center of mass) while undergoing translational motion, then OSE: Ktotal = Ktrans + Krot = ½ MVCM2 + ½ ICMCM2. Example: a solid cylinder and a thin hollow cylinder, both of mass M and outer radius R, are released from rest at the start of an inclined plane of height H and length L. What are the speeds of each cylinder when they reach the bottom? Which cylinder gets to the bottom first?

Example: Physics 23 Problem on KE of Rotation (from Test 3 Winter 2001). A cylinder of mass M, radius R and length L can freely rotate about a light metal rod through its center. The ends of the rod are attached to a massless yoke that can be used to pull the cylinder along a surface. D P=2Mg r M  The cylinder is placed at rest on a horizontal surface. The yoke is then pulled by a worker with a constant force of magnitude P=2Mg at an angle of  with respect to the vertical. It rolls without slipping at all times.

D P=2Mg r M  Derive an expression for the linear speed of the cylinder, in terms of relevant system parameters, after it has rolled a distance D. Vi=0 Vf? 90- d For an OSE you can use Ef-Ei=[Wother]i→f or [Wnet]i→f=K. Because the only force that does work is P, in either equation, the calculation of work is the same. There is no change of height and there are no springs, so the K equation is “easier.” K = WP = P·d → K = PD cos(90-) = PD sin 

½ M Vf2 + ½ (½ Mr2) (Vf /r)2 = (2Mg)D sin  P=2Mg r M  Vi=0 Vf? d The cylinder is solid (if that is not clear, you should ask) and has a moment of inertia ½ Mr2. 90- Kf - Ki = PD sin  ½ M Vcm,f2 + ½ I f2 = PD sin  ½ M Vf2 + ½ (½ Mr2) (Vf /r)2 = (2Mg)D sin  ½ Vf2 + ¼Vf2 = 2gD sin  ¾Vf2 = 2gD sin  Vf2 = (8gD sin )/3