Three objects of uniform density- a solid sphere, a solid cylinder, and a hollow cylinder- are placed at?

momen inertia of (each object is solid)

I_sphere = (2/5)mr²

I_cylinder = (1/2)mr²

I_thin_cylinder = mr²

torque = I α

(friction) r = I (a/r)

friction = Ia/r²

second Newton's law of linear motion,

ΣF = m a

m g sin θ - friction = m a

m g sin θ - Ia/r² = m a

m g sin θ = (m + I/r²)a

a = m g sin θ/(m + I/r²)

a ∼ 1/I

acceleration inversely proportional with respect to the moment of inertia,

I_sphere < I_cylinder < I_thin_cylinder

The sphere reaches the bottom for the first time and thin_cylinder for the last one.

'Try this at home and note that the result is independent of the masses and the radii of the objects.'

moment of inertia I = kmr²

hence k is a real constant.

a = m g sin θ/(m + I/r²)

a = m g sin θ/(m + kmr²/r²)

a = g sin θ/(1 + k)

acceleration a is independent of the masses and the radii of the objects.

1st = solid sphere

2nd = solid cylinder

3rd = hollow cylinder