Four round objects namely a ring, a disc, a hollow sphere and a solid sphere with same radius R and made of same material start to roll down an inclined plane at the same time. The object that will reach the bottom third is

Four round objects namely a ring, a disc, a hollow sphere and a solid

1.	Four round objects namely a ring, a disc, a hollow sphere and a solid sphere with same radius R and made of same material start to roll down an inclined plane at the same time. The object that will reach the bottom third is
Answer» Solution :For all the FOUR objects namely the ring, disc, hollow sphere and SOLID sphere, the radii of gyration K are `R, sqrt((1)/(2))R, sqrt((2)/(3))R, sqrt((2)/(5))R`. With numerical values the RADIUS of gyration K are IR, 0.707R, 0.816R 0.632R respectively. The expression for time taken for rolling has the radius of gyration K in the numerator as per equation `Vsqrt((2gh)/((1+(K^(2))/(R^(2)))))` `t= sqrt((2h(1+(K^(2))/(R^(2))))/(g sin^(2) THETA))` The one with least value of radius of gyration K will take the shortest time to reach the bottom of the inclined plane. The ORDER of objects reaching the bottom is first, solid sphere, second, dise: third, hollow sphere and last, rin