Question

If a solid and hollow sphere of the same mass allowed to rolled down

Answer 1

Answer:

We need to assume that each object has uniform density and that they all roll without slipping. Perhaps surprisingly size, mass, density height don’t matter. Define

mass =m

initial height =h

gravity =g

final velocity =v

radius =r

angular velocity =ω=vr

moment of inertia =I=kmr2

(where k is a measure how close to the edge the mass is, on average)

Then by conservation of energy,

mgh=12mv2+12Iω2

2gh=v2+kv2

v=2gh1+k−−−−−√

So to maximize v we need to minimize k - in other words we want the mass to be concentrated as close to the centre as possible. This is intuitive, we don’t want to ‘waste’ energy spinning up the object - we want to concentrate on maximizing its speed.