Rotational motion

(c) Both will reach the wall simultaneously.

KE of a rolling body = KE translational + KE rotational

= (1/2)MV_cm² + (1/2)Iw² =(1/2)[(MV_cm² + I(V_cm/R)²]

= (1/2)[(MV_cm² + I(V_cm/R)²] = (V_cm² /2)(M + I/R²) = (MV_cm² /2)(1 + c)

... since I = cMR²  , for example, for ring I = MR², hence c = 1.

Now since both have the same KE,

V_cm = (2KE/((1 + c)M))^1/2

(V_cm)_ring/(V_cm)_cylinder = (1/((1+c)M)_ring^1/2/(1/((1+c)M)_cylinder^1/2

 (V_cm)_ring/(V_cm)_cylinder = = ((1.5x0.4)/(0.3x2)^1/2 =  1

Both are moving at the same speed, and will reach at the same time.

Regards,

Team,

TopperLearning.