A hollow cylinder of mass M and radius R is rotating about its axis of symmetry and a solid sphere of same mass and radius is rotating about an axis passing through its centre. It torques of equal magnitude are applied to them, then the ratio of angular accelerations produced is a) 2/5 b) 5/2 c) 5/4 d) 4/5

Options

A. 43587
B. 43501
C. 43560
D. 43589

Explanation

To solve the problem, we need to find the ratio of angular accelerations produced in a hollow cylinder and a solid sphere when equal torques are applied to both. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship between torque, moment of inertia, and angular acceleration.
The relationship is given by the equation:
\[ \tau = I \alpha \]
where \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration.

### Step 2: Write the equations for the hollow cylinder and the solid sphere.
For the hollow cylinder, the moment of inertia \( I_c \) is given by:
\[ I_c = \frac{1}{2} M R^2 \]
For the solid sphere, the moment of inertia \( I_s \) is given by:
\[ I_s = \frac{2}{5} M R^2 \]

### Step 3: Set up the equations for torque for both objects.
Since equal torques are applied, we can write:
\[ \tau = I_c \alpha_c \quad \text{(for the hollow cylinder)} \]
\[ \tau = I_s \alpha_s \quad \text{(for the solid sphere)} \]

### Step 4: Express angular accelerations in terms of torque.
From the equations above, we can express angular accelerations as:
\[ \alpha_c = \frac{\tau}{I_c} \quad \text{and} \quad \alpha_s = \frac{\tau}{I_s} \]

### Step 5: Find the ratio of angular accelerations.
Now, we can find the ratio of angular accelerations \( \frac{\alpha_c}{\alpha_s} \):
\[ \frac{\alpha_c}{\alpha_s} = \frac{\frac{\tau}{I_c}}{\frac{\tau}{I_s}} = \frac{I_s}{I_c} \]

### Step 6: Substitute the moments of inertia.
Substituting the values of \( I_c \) and \( I_s \):
\[ \frac{\alpha_c}{\alpha_s} = \frac{\frac{2}{5} M R^2}{\frac{1}{2} M R^2} \]

### Step 7: Simplify the expression.
The \( M R^2 \) terms cancel out:
\[ \frac{\alpha_c}{\alpha_s} = \frac{\frac{2}{5}}{\frac{1}{2}} = \frac{2}{5} \times \frac{2}{1} = \frac{4}{5} \]

### Step 8: Conclusion.
Thus, the ratio of angular accelerations produced is:
\[ \frac{\alpha_c}{\alpha_s} = \frac{4}{5} \]
So, the correct answer is option **d) 4/5**.

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