What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) A uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere; (d) a hollow cylinder with outer radius R and inner radius R/2.

Solution 21E Step 1 of 5: The fraction of the total kinetic energy that is rotational is (1/2)cm2 1 1 (1/2)Mv +(1/2)Icm2 = 1+(M/Icm)v /2 = 1+(MR /Icm) cm cm Step 2 of 5: a)For uniform solid cylinder, I = MR , 2 cm 2 so the fraction of the K.E that is rotational is 1 1 2MR = 3 1+( MR2 ) Step 3 of 5: 2 2 b)For uniform sphere, I cm = 5R , so the fraction of the K.E that is rotational is 1 = 2 1+( 5MR2) 7 2MR