Question

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $(E_{\text{sphere}} / E_{\text{cylinder}})$ Will be

• A. $ 2 : 3$
• B. $1 : 5$
• C. $1 : 4$
• D. $ 3 : 1$

Answer 1

The Correct Option is B

To find the ratio of the kinetic energies of rotation of the sphere and the cylinder, we need to calculate the kinetic energy for each of them and then compare their values.

Rotational Kinetic Energy of a Solid Sphere:

The moment of inertia $I_{\text{sphere}}$ of a solid sphere rotating about its diameter is given by:

$$I_{\text{sphere}} = \frac{2}{5} m R^2$$

where $m$ is the mass and $R$ is the radius of the sphere.

The rotational kinetic energy $E_{\text{sphere}}$ of the sphere is calculated using the formula:

$$E_{\text{sphere}} = \frac{1}{2} I_{\text{sphere}} \omega^2$$

Substituting $I_{\text{sphere}}$:

$$E_{\text{sphere}} = \frac{1}{2} \left(\frac{2}{5} m R^2\right) \omega^2 = \frac{1}{5} m R^2 \omega^2$$

Rotational Kinetic Energy of a Solid Cylinder:

The moment of inertia $I_{\text{cylinder}}$ of a solid cylinder rotating about its geometrical axis is given by:

$$I_{\text{cylinder}} = \frac{1}{2} m R^2$$

Given that the angular speed of the cylinder $2\omega$ is twice that of the sphere, the rotational kinetic energy $E_{\text{cylinder}}$ is calculated as:

$$E_{\text{cylinder}} = \frac{1}{2} I_{\text{cylinder}} (2\omega)^2$$

Substituting $I_{\text{cylinder}}$ and simplifying:

$$E_{\text{cylinder}} = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (2\omega)^2 = \frac{1}{2} \times \frac{1}{2} m R^2 \times 4\omega^2 = \frac{1}{2} m R^2 \omega^2$$

Ratio of Rotational Kinetic Energies:

Now, calculate the ratio $(E_{\text{sphere}} / E_{\text{cylinder}})$:

$$\text{Ratio} = \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{\frac{1}{5} m R^2 \omega^2}{\frac{1}{2} m R^2 \omega^2} = \frac{1}{5} \div \frac{1}{2} = \frac{2}{5}$$

Thus, the ratio is 1:5.

Therefore, the correct answer is 1 : 5.