Question

Consider a rigid body that rotates but whose center of mass is at rest. True or false: the rotational kinetic energy of the entire object is equivalent to the sum of the translational kinetic energy of each small piece of the object.

Answer 1

Answer:

True

Explanation:

Let us consider a rigid body which contains tiny particles of mass dm each. and the angular velocity ω for each of the particle is same and teh radius of rotation is r.

the linear velocity, v = r ω

The total kinetic energy is given by

$K=\int \frac{1}{2}dm v^{2}$

$K=\int \frac{1}{2}dm r^{2}\omega^{2}$

$K=\frac{1}{2}\times \omega^{2}\int dm r^{2}$

K = 1/2 Iω^2

where I is the moment of inertia.