A solid sphere (initially at rest) of mass 7 kg and radius 40 cm which is free to rotate about its axis is given an angular impulse of `5kgm^(2)s^(-1)` followed by the similar impulse after every 5 second. Calculate the angular speed of sphere 50 s after the initial impulse.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given data
- Mass of the sphere (m) = 7 kg
- Radius of the sphere (r) = 40 cm = 0.4 m
- Angular impulse given (J) = 5 kg·m²/s
- Time interval for each impulse = 5 seconds
- Total time after which we need to find the angular speed = 50 seconds

### Step 2: Calculate the moment of inertia (I) of the solid sphere
The moment of inertia (I) of a solid sphere about its axis is given by the formula:
\[ I = \frac{2}{5} m r^2 \]
Substituting the values:
\[ I = \frac{2}{5} \times 7 \, \text{kg} \times (0.4 \, \text{m})^2 \]
\[ I = \frac{2}{5} \times 7 \times 0.16 = \frac{2 \times 7 \times 0.16}{5} = \frac{2.24}{5} = 0.448 \, \text{kg·m}^2 \]

### Step 3: Calculate the angular speed after the first impulse
The angular impulse is related to the change in angular momentum:
\[ J = I \Delta \omega \]
Since the sphere is initially at rest, we have:
\[ \Delta \omega = \frac{J}{I} \]
Substituting the values:
\[ \Delta \omega = \frac{5 \, \text{kg·m}^2/\text{s}}{0.448 \, \text{kg·m}^2} \approx 11.16 \, \text{rad/s} \]

### Step 4: Determine the number of impulses in 50 seconds
Since an impulse is applied every 5 seconds, the number of impulses in 50 seconds is:
\[ \text{Number of impulses} = \frac{50 \, \text{s}}{5 \, \text{s}} = 10 \]

### Step 5: Calculate the total angular speed after 50 seconds
The total change in angular speed after 10 impulses is:
\[ \text{Total change in angular speed} = 10 \times \Delta \omega = 10 \times 11.16 \, \text{rad/s} = 111.6 \, \text{rad/s} \]

### Step 6: Final angular speed
Since the sphere started from rest, the final angular speed (ω) after 50 seconds is:
\[ \omega = 0 + 111.6 \, \text{rad/s} = 111.6 \, \text{rad/s} \]

### Conclusion
The angular speed of the sphere 50 seconds after the initial impulse is approximately **111.6 rad/s**.

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