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 speed twice that of the sphere. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder is

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

Hint:

To solve this quickly, break the ratio into its separate components: $\frac{K_s}{K_c} = \left(\frac{I_s}{I_c}\right) \times \left(\frac{\omega_s}{\omega_c}\right)^2$.
The moment of inertia ratio is $\frac{2/5}{1/2} = \frac{4}{5}$. The angular velocity squared ratio is $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$. Multiply them together: $\frac{4}{5} \times \frac{1}{4} = \frac{1}{5}$ instantly!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the ratio of the rotational kinetic energy of a solid sphere to that of a solid cylinder. Both objects share identical masses and radii, but the cylinder rotates with an angular velocity that is twice that of the sphere.

Step 2: Key Formula or Approach:
1. The rotational kinetic energy ($K.E._{\text{rot}}$) of a rigid body is given by: $$K.E._{\text{rot}} = \frac{1}{2}I\omega^2$$ 2. Recall the standard moment of inertia formulas: For a solid sphere about its diameter: $I_{\text{sphere}} = \frac{2}{5}MR^2$ For a solid cylinder about its central geometric axis: $I_{\text{cylinder}} = \frac{1}{2}MR^2$

Step 3: Detailed Explanation:
Let's assign values based on the prompt instructions: Sphere: Moment of inertia $I_s = \frac{2}{5}MR^2$, Angular speed $\omega_s = \omega$ Cylinder: Moment of inertia $I_c = \frac{1}{2}MR^2$, Angular speed $\omega_c = 2\omega$ Write the rotational kinetic energy expression for the sphere: $$K_s = \frac{1}{2} I_s \omega_s^2 = \frac{1}{2} \left(\frac{2}{5}MR^2\right) \omega^2 = \frac{1}{5}MR^2\omega^2$$ Write the rotational kinetic energy expression for the cylinder: $$K_c = \frac{1}{2} I_c \omega_c^2 = \frac{1}{2} \left(\frac{1}{2}MR^2\right) (2\omega)^2 = \frac{1}{4}MR^2 \cdot 4\omega^2 = MR^2\omega^2$$ Now, calculate the ratio of the kinetic energy of the sphere to the cylinder ($\frac{K_s}{K_c}$): $$\text{Ratio} = \frac{\frac{1}{5}MR^2\omega^2}{1MR^2\omega^2} = \frac{1}{5} = 1 : 5$$

Step 4: Final Answer:
The ratio of their rotational kinetic energies is $1 : 5$, which corresponds to option (B).