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Chapter 1: Q14DQ (page 294)

A hollow spherical shell of radius R that is rotating about an axis through its center has rotational kinetic energy K. If you want to modify this sphere so that it has three times as much kinetic energy at the same angular speed while keeping the same mass, what should be its radius in terms of R?

Short Answer

Expert verified

The required radius of the sphere is 3R.

Step by step solution

01

Concept/Significance of Moment of inertia and rotational kinetic energy

The moment of inertia of the hollow spherical shell with mass M and radius R about an axis through its center is given by,

I=23MR2

The rotational kinetic energy of a hollow spherical shell radius R rotating about an axis through its center is given by,

K=12l蝇2=12(23MR2)蝇2

Here,蝇 is angular speed.

02

Derive the expression of radius in terms of R

If the mass (M) and angular speed蝇are the same then rotational kinetic energy will have following relation.

K鈭�R2K1K2=R12R22..........1

It is given that R = R andK1=K. LetR2be the radius of the sphere so that the kinetic energyK2=3K.

Substitute all the values in equation (1).

K3K=R2R2213=R2R22R22=3R2R2=3R

Therefore, the required radius of the sphere is 3R.

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