Torques of equal magnitude are applied to a thin hollow cylinder and a solid sphere, both having the same mass and radius. Both of them are free to rotate about their axis of symmetry. If `alpha_(c) and alpha_(s)` are the angular accelerations of the cylinder and the sphere respectively, then the ratio `(alpha_(c))/(alpha_(s))` will be

Options

A. `5/2`
B. `5/4`
C. `4/5`
D. `2/5`

Answer

To find the ratio of the angular accelerations of a thin hollow cylinder and a solid sphere when equal torques are applied to both, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between torque, moment of inertia, and angular acceleration**:
The torque (τ) applied to an object is related to its moment of inertia (I) and angular acceleration (α) by the equation:
\[ \tau = I \cdot \alpha \]

2. **Identify the moment of inertia for both objects**:
- For a thin hollow cylinder rotating about its axis, the moment of inertia is given by:
\[ I_c = m r^2 \]
- For a solid sphere rotating about its axis, the moment of inertia is given by:
\[ I_s = \frac{2}{5} m r^2 \]
Here, \(m\) is the mass and \(r\) is the radius of both objects.

3. **Set up the equations for torque**:
Since equal torques are applied to both the cylinder and the sphere, we can write:
\[ \tau_c = I_c \cdot \alpha_c \quad \text{(for the cylinder)} \]
\[ \tau_s = I_s \cdot \alpha_s \quad \text{(for the sphere)} \]

4. **Equate the torques**:
Since \(\tau_c = \tau_s\), we have:
\[ I_c \cdot \alpha_c = I_s \cdot \alpha_s \]

5. **Substitute the moments of inertia**:
Substituting the expressions for \(I_c\) and \(I_s\):
\[ (m r^2) \cdot \alpha_c = \left(\frac{2}{5} m r^2\right) \cdot \alpha_s \]

6. **Cancel out common terms**:
We can cancel \(m\) and \(r^2\) from both sides (assuming they are not zero):
\[ \alpha_c = \frac{2}{5} \alpha_s \]

7. **Find the ratio of angular accelerations**:
Rearranging gives us:
\[ \frac{\alpha_c}{\alpha_s} = \frac{2}{5} \]

### Final Answer:
Thus, the ratio of the angular accelerations is:
\[ \frac{\alpha_c}{\alpha_s} = \frac{2}{5} \]