Question

1 A hollow sphere of mass M and expression for the speed the sphere ha of mass M and radius R is suspended by a string as shown. Derive an e speed the sphere has when it falls a distance h. How long does this take? ergy is in rotation? Clear good figures, explain the math in the derivation and box your answers. MR

Answer 1

a] Let the tension be T, and linear acceleration be a,

Force equation on block : Mg - T = Ma where m is mass of block

Toque equation on cylinder: TR = I*alpha where I is moment of inertia which is 2MR^2/3

or TR = 2MR^2/3 *a/R because angular acceleration alpha is a/R in pure rolling

or T = 2Ma/3

adding this to force equation, Mg = 2Ma/3 + Ma or g = 2a/3 + a

or a = 3g/5

using third equation of motion, speed v^2 = 0^2 + 2as

v = sqrt(2as) = sqrt(2*3g/5 *h)

v = sqrt(6gh/5) answer

time taken t = v/a = sqrt(6gh/5)/(3g/5) = sqrt(6gh*5/9g^2)

t = sqrt(10h/3g) answer

   fraction of KE in rotation = (0.5iw^2)/(0.5iw^2+0.5Mv^2)

= (0.5*2MR^2/3 w^2)/(0.5*2MR^2/3 w^2+0.5Mv^2)

= (M/3 v^2)/(M/3 v^2+0.5Mv^2) after substituting v=wR

= (1/3)/(1/3+1/2) = 0.4 ANSWER