A hollow sphere of radius 2 cm is attached to an 18 cm long thread to make a pendulum. Find the time period of oscillation of this pendulum . How does it differ from the time period calculated using the formula for a simple pendulum?

Explanation

To find the time period of oscillation of a pendulum consisting of a hollow sphere attached to a thread, we will follow these steps:

### Step 1: Identify the given values
- Radius of the hollow sphere (r) = 2 cm = 0.02 m
- Length of the thread (L) = 18 cm = 0.18 m
- Total distance from the pivot point to the center of mass of the hollow sphere = L + r = 0.18 m + 0.02 m = 0.20 m

### Step 2: Calculate the moment of inertia (I) of the hollow sphere about the pivot point
Using the parallel axis theorem:
\[ I = I_{cm} + m d^2 \]
Where:
- \( I_{cm} \) is the moment of inertia about the center of mass, which for a hollow sphere is given by:
\[ I_{cm} = \frac{2}{3} m r^2 \]
- \( d \) is the distance from the center of mass to the pivot point, which we calculated as 0.20 m.

Substituting the values:
\[ I = \frac{2}{3} m (0.02)^2 + m (0.20)^2 \]
\[ I = \frac{2}{3} m (0.0004) + m (0.04) \]
\[ I = \frac{2}{3} m (0.0004) + \frac{120}{3} m (0.0004) \]
\[ I = \frac{2 + 120}{3} m (0.0004) \]
\[ I = \frac{122}{3} m (0.0004) \]
\[ I = 0.016267 m \]

### Step 3: Calculate the torque (τ)
The torque (τ) due to the weight of the hollow sphere is given by:
\[ τ = -mg \cdot x \]
Where \( x \) is the distance from the pivot to the center of mass in terms of angular displacement (θ):
\[ x = 0.20 \theta \]

Thus:
\[ τ = -mg (0.20 \theta) \]

### Step 4: Relate torque to angular displacement
For small oscillations, we can relate torque to angular displacement:
\[ τ = -k \theta \]
Where \( k = mg \cdot 0.20 \).

### Step 5: Find the time period (T)
The time period T for small oscillations is given by:
\[ T = 2\pi \sqrt{\frac{I}{k}} \]
Substituting for k:
\[ T = 2\pi \sqrt{\frac{I}{mg \cdot 0.20}} \]

### Step 6: Calculate the time period for a simple pendulum (T')
For a simple pendulum, the time period is given by:
\[ T' = 2\pi \sqrt{\frac{L}{g}} \]
Where \( L = 0.20 m \).

### Step 7: Compare the two time periods
Now we can compare the time periods calculated for the hollow sphere pendulum and the simple pendulum.

### Final Calculation
1. For the hollow sphere pendulum:
- Substitute \( I \) and \( k \) into the formula for T.
2. For the simple pendulum:
- Substitute \( L \) into the formula for T'.

### Conclusion
The time period of the hollow sphere pendulum will differ from that of a simple pendulum due to the additional moment of inertia associated with the hollow sphere.

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