A solid uniform marble and a block of ice, each with the same mass,start from re

Question

A solid uniform marble and a block of ice, each with the same mass,start from rest at the same height H above thebottom of a hill and move down it. The marble rolls withoutslipping, and the ice slides without friction. (a) Find the speedof each of these objects when they reach the bottom of the hill.(b) Which object is moving faster at the bottom, the block of iceor the marble? (c) Which object has more kinectic energy at thebottom, the block of ice or the marble? Pls try to give theexplanation..thanks

Explanation / Answer

Using Vcm = R * w for pure rolling motion, K = (1/2)Icm * (Vcm/R) + (1/2)MVcm^2 or K = (1/2) * [(Icm/R^2) + M] * Vcm^2 Conservation of mechanical energy gives Kf + Uf = Ki + Ui or (1/2) * [(Icm/R^2) + M] * Vcm^2 + 0 = 0 + MgH or Vcm = [2gH/1 + (Icm/MR^2)]^(1/2) ------------(1) Here,"cm" denotes center of mass. (a)for marble (assuming it to be a solid sphere) substitutingIcm = (2/5)m * R^2 in (1) we get the speed at bottom. for block we take Icm = 0 and substitute in (1) (b)Substituting the values of Icm for both sphere and block weknow which object is moving faster. (c)The object which has greater velocity will have greaterkinetic energy at the bottom because kinetic energy is (1/2)m * v^2. or (1/2) * [(Icm/R^2) + M] * Vcm^2 + 0 = 0 + MgH or Vcm = [2gH/1 + (Icm/MR^2)]^(1/2) ------------(1) Here,"cm" denotes center of mass. (a)for marble (assuming it to be a solid sphere) substitutingIcm = (2/5)m * R^2 in (1) we get the speed at bottom. for block we take Icm = 0 and substitute in (1) (b)Substituting the values of Icm for both sphere and block weknow which object is moving faster. (c)The object which has greater velocity will have greaterkinetic energy at the bottom because kinetic energy is (1/2)m * v^2.

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