Question

A spinning giant star of initial radius R1 and angular speed ω1 suddenly collapses radially inward reaching a new radius R2 = 0.5R1 ; its mass remains the same. What is the relation between the kinetic energies of rotation (spin) before and after the collapse? In both cases, approximate the star as a uniform solid sphere.

Answer 1

K1/K2=4

Explanation:

The kinetic energy of a rotating sphere is given by:

`K=(I*\omega^(2) )/(2)`

The moment of inertia of a solid sphere is given by

`I=(2MR^(2) )/(5)`

The initial kinetic energy is therefore

`K_1=(2MR^(2)*\omega^(2) )/(10)`

`K_1=(MR_1^(2)*\omega^(2) )/(5)`

The final kinetic energy is given by

`K_2=(MR_2^(2)*\omega^(2) )/(5)`

Therefore the relation K1/K2 if R2 = 0.5R1

`(K_1)/(K_2) =(5M(R_1)^(2)*\omega^(2) )/(5*M(0.5R_1)^(2) \omega^(2))`

The text says nothing about the final angular velocity just the collapse of the collapse of the radius

`(K_1)/(K_2) =(1 )/((0.5)^(2) )=4`