A metal sphere of radius r and specific heat s is rotated about an axis passing through its centre at a speed of n rotation/s. It is suddenly stopped and `50%` of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is

Options

A. `(2pi^(2)n^(2)r^(2))/(5S)`
B. `(pi^(2)n^(2))/(10r^(2)S)`
C. `(7)/(8)pir^(2)n^(2)S`
D. `(5(pirn)^(2))/(14S)`

Answer

To solve the problem step-by-step, we can follow these calculations:

### Step 1: Calculate the Moment of Inertia (I) of the Sphere
The moment of inertia \( I \) of a solid sphere about an axis through its center is given by:
\[ I = \frac{2}{5} m r^2 \]
where \( m \) is the mass of the sphere and \( r \) is its radius.

### Step 2: Calculate the Angular Velocity (ω)
The angular velocity \( \omega \) in radians per second can be calculated from the number of rotations per second \( n \):
\[ \omega = 2 \pi n \]

### Step 3: Calculate the Kinetic Energy (KE)
The kinetic energy \( KE \) of the rotating sphere can be calculated using the formula:
\[ KE = \frac{1}{2} I \omega^2 \]
Substituting the values of \( I \) and \( \omega \):
\[ KE = \frac{1}{2} \left(\frac{2}{5} m r^2\right) (2 \pi n)^2 \]
\[ KE = \frac{1}{2} \cdot \frac{2}{5} m r^2 \cdot 4 \pi^2 n^2 \]
\[ KE = \frac{4}{5} m r^2 \pi^2 n^2 \]

### Step 4: Calculate the Energy Used to Increase Temperature
According to the problem, 50% of the kinetic energy is used to increase the temperature:
\[ \text{Energy used} = \frac{1}{2} KE = \frac{1}{2} \cdot \frac{4}{5} m r^2 \pi^2 n^2 = \frac{2}{5} m r^2 \pi^2 n^2 \]

### Step 5: Relate the Energy to Temperature Change
The energy used to increase the temperature is also given by the formula:
\[ Q = m s \Delta T \]
where \( Q \) is the heat energy, \( s \) is the specific heat, and \( \Delta T \) is the rise in temperature. Setting the two expressions for energy equal gives:
\[ \frac{2}{5} m r^2 \pi^2 n^2 = m s \Delta T \]

### Step 6: Solve for the Rise in Temperature (ΔT)
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{2}{5} r^2 \pi^2 n^2 = s \Delta T \]
Now, solving for \( \Delta T \):
\[ \Delta T = \frac{\frac{2}{5} r^2 \pi^2 n^2}{s} = \frac{2}{5} \frac{r^2 \pi^2 n^2}{s} \]

### Final Answer
The rise in temperature of the sphere is:
\[ \Delta T = \frac{2}{5} \frac{r^2 \pi^2 n^2}{s} \]

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