A hot solid sphere is rotating about a diameter at an angular velocity w. If it cools so that its radius reduces to 1/n of its initial value, then its angular velocity becomes what????

To understand how the angular velocity of a rotating sphere changes when its radius decreases, we can apply the principle of conservation of angular momentum. Angular momentum is a measure of the rotational motion of an object and is given by the product of the moment of inertia and the angular velocity. Let's break this down step by step.

Angular Momentum and Moment of Inertia

The angular momentum (L) of a solid sphere can be expressed as:

L = I * ω

where:

• I is the moment of inertia of the sphere, and
• ω is the angular velocity.

For a solid sphere, the moment of inertia is given by:

I = (2/5) * m * r²

Here, m is the mass of the sphere and r is its radius. When the sphere cools and its radius reduces to r/n, we need to find out how this affects the angular velocity.

Change in Radius and Mass

Assuming the sphere is made of a material that maintains its density, when the radius decreases, the mass will also change. The volume of the sphere is given by:

V = (4/3) * π * r³

When the radius becomes r/n, the new volume (V') is:

V' = (4/3) * π * (r/n)³ = (4/3) * π * (r³/n³)

Thus, the new mass (m') will be:

m' = ρ * V' = ρ * (4/3) * π * (r³/n³) = m/n³

Calculating New Moment of Inertia

Now, substituting the new mass and radius into the moment of inertia formula:

I' = (2/5) * m' * (r/n)² = (2/5) * (m/n³) * (r²/n²) = (2/5) * m * r² / n⁵

Applying Conservation of Angular Momentum

According to the conservation of angular momentum, the initial angular momentum must equal the final angular momentum:

L_initial = L_final

Substituting the expressions for angular momentum:

I * ω = I' * ω'

Now, substituting the values we derived:

(2/5) * m * r² * ω = (2/5) * (m/n³) * (r²/n²) * ω'

We can cancel out common terms:

m * ω = (m/n³) * (1/n²) * ω'

After canceling m and rearranging, we find:

ω' = ω * n⁵

Final Result

Thus, when the radius of the sphere reduces to 1/n of its initial value, the new angular velocity becomes:

ω' = n⁵ * ω

This means that the angular velocity increases significantly as the radius decreases, specifically by a factor of n to the fifth power. This illustrates how changes in physical dimensions can dramatically affect rotational dynamics.