Question 587: Centre of Mass and Rotational Motion

Question

A solid sphere of mass '$$M$$' and radius '$$R$$' is rotating about its diameter. A solid cylinder of same mass and same radius is also rotating about its geometrical axis with an angular speet twice that of the sphere. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder is

Accepted Answer

To find the ratio of the kinetic energy of rotation of the sphere to that of the cylinder, we need to use the formulas for the rotational kinetic energy and the moments of inertia for both shapes.
The kinetic energy of rotation is given by:
$$ KE = \frac{1}{2} I \omega^2 $$
where $$I$$ is the moment of inertia and $$\omega$$ is the angular speed.
For a solid sphere rotating about its diameter, the moment of inertia is:
$$ I_{	ext{sphere}} = \frac{2}{5} M R^2 $$
For a solid cylinder rotating about its geometrical axis, the moment of inertia is:
$$ I_{	ext{cylinder}} = \frac{1}{2} M R^2 $$
Given that the angular speed of the cylinder is twice that of the sphere, if we let the angular speed of the sphere be $$\omega_{	ext{sphere}} = \omega$$, then the angular speed of the cylinder will be $$\omega_{	ext{cylinder}} = 2\omega$$.
Now, we can write the rotational kinetic energies:
For the sphere:
$$ KE_{	ext{sphere}} = \frac{1}{2} I_{	ext{sphere}} \omega^2 = \frac{1}{2} \left( \frac{2}{5} M R^2 ight) \omega^2 = \frac{1}{5} M R^2 \omega^2 $$
For the cylinder:
$$ KE_{	ext{cylinder}} = \frac{1}{2} I_{	ext{cylinder}} \left( 2\omega ight)^2 = \frac{1}{2} \left( \frac{1}{2} M R^2 ight) \left( 2\omega ight)^2 = \frac{1}{2} \left( \frac{1}{2} M R^2 ight) 4 \omega^2 = M R^2 \omega^2 $$
So, the ratio of the kinetic energy of rotation of the sphere to that of the cylinder is:
$$ 	ext{Ratio} = \frac{KE_{	ext{sphere}}}{KE_{	ext{cylinder}}} = \frac{\frac{1}{5} M R^2 \omega^2}{M R^2 \omega^2} = \frac{1}{5} $$
Therefore, the correct answer is:
Option B: $$1: 5$$

Answer

2 : 3

Answer

$$1: 5$$

Answer

$$1: 4$$

Answer

$$3: 1$$

Centre of Mass and Rotational Motion

A solid sphere of mass '$M' and radius 'R$' is rotating about its diameter. A solid cylinder of same mass and same radius is also rotating about its geometrical axis with an angular speet twice that of the sphere. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder is

Choose the correct answer: