Physics homework help

Question

A spherical object (with non-uniform density) of mass 16 kg and radius 0.14 m rolls along a horizontal surface at a constant linear speed without slipping. The moment of inertia of the object about a diameter is 0.25 M R2 . The object’s rotational kinetic energy about its own center is what fraction of the object’s total kinetic energy?

So far I have (1/2)(.25)(x)^2. Need to find angular velocity in order to get answer

Answer 1

The short answer is that you do not need to find the angular velocity.

Let

m = 16 kg be the mass of the object,

r = 0.14 m be the radius of the object,

I = 0.25 kgm² be the object's moment of inertia,

ω (unknown) be the object's angular velocity,

v (unknown) be the object's translational velocity.

Rotational energy is

Iω²/2

Translational energy is

mv²/2

The relation between angular and translational velocity is

v = ωr

The ratio to be determined is

(Iω²/2) / (Iω²/2 + mv²/2) =

Iω² / (Iω² + mv²) =

Iω² / (Iω² + mω²r²) =

I / (I + mr²) =

0.25 / (0.25 + 16×0.14²) = 0.44