As an ice skater begins a spin, his angular velocity is 3.75 rad/s. After pulling in his arms, his angular speed increases to 6.45 rad/s. Find the ratio of the skaters final moment of inertia to his initial moment of inertia.

The ratio of the skaters final moment of inertia to his initial moment of inertia is 0.338

Explanation:

Moment of inertia = mr^2 = m (v/w) ^2 = mv^2/w^2

Initial angular velocity (w) = 3.75 rad/s

Initial moment of inertia = mv^2/3.75^2 = 0.071mv^2

Final angular velocity (w) = 6.45 rad/s

Final moment of inertia = mv^2/6.45^2 = 0.024

Ratio = 0.024/0.071 = 0.338

1 : 1.72.

Explanation:

Moment of inertia {/displaystyle I}I is defined as the ratio of the net angular momentum, L of a system to its angular velocity, w around a principal axis.

I = L/w

Given:

w1 = 3.75 rad/s

w2 = 6.45 rad/s

Therefore,

I1 = L1/3.75

= 0.27L1

I2 = L1/6.45

= 0.155L1

At constant momentum, L ratio of the initial moment of inertia to the final moment of inertia =

3.75 I1 : 6.45 I2

= 1 : 1.72.

1.74