The time period of a physical pendulum is given by `T=2pisqrt(I/(mgl))`. Where m= mass of the pendulum I= moment of inertia about the axis of suspension, l= distance of centre of mass of bob from the centre of suspension. Calculate the change in time period when temperature changes by `DeltaT`. The coefficient of linear expansion of the material of pendulum is `alpha`.

Answer

To calculate the change in the time period of a physical pendulum when the temperature changes by \(\Delta T\), we will follow these steps:

### Step 1: Understand the Formula for Time Period
The time period \(T\) of a physical pendulum is given by:
\[ T = 2\pi \sqrt{\frac{I}{mgl}} \]
where:
- \(I\) = moment of inertia about the axis of suspension
- \(m\) = mass of the pendulum
- \(g\) = acceleration due to gravity
- \(l\) = distance from the center of mass to the center of suspension

### Step 2: Determine the Effect of Temperature Change
When the temperature changes by \(\Delta T\), the dimensions of the pendulum will change due to thermal expansion. The linear expansion of a material is given by:
\[ L' = L(1 + \alpha \Delta T) \]
where:
- \(L\) = original length
- \(\alpha\) = coefficient of linear expansion
- \(L'\) = new length after expansion

### Step 3: Calculate the New Moment of Inertia
The moment of inertia \(I\) for a pendulum can be expressed as:
\[ I = mL^2 \]
After the temperature change, the new moment of inertia \(I'\) becomes:
\[ I' = m(L')^2 = m[L(1 + \alpha \Delta T)]^2 = mL^2(1 + \alpha \Delta T)^2 \]

### Step 4: Calculate the New Length from the Center of Suspension
The new length \(l'\) from the center of suspension is also affected by the temperature change:
\[ l' = l(1 + \alpha \Delta T) \]

### Step 5: Substitute New Values into the Time Period Formula
The new time period \(T'\) can be expressed as:
\[ T' = 2\pi \sqrt{\frac{I'}{mg l'}} = 2\pi \sqrt{\frac{mL^2(1 + \alpha \Delta T)^2}{mg l(1 + \alpha \Delta T)}} \]
Simplifying this gives:
\[ T' = 2\pi \sqrt{\frac{L^2(1 + \alpha \Delta T)^2}{gl(1 + \alpha \Delta T)}} \]
This can be further simplified to:
\[ T' = 2\pi \sqrt{\frac{L^2}{gl}}(1 + \alpha \Delta T)^{1/2} \]

### Step 6: Relate New Time Period to Original Time Period
We know that:
\[ T = 2\pi \sqrt{\frac{L^2}{gl}} \]
Thus, we can write:
\[ T' = T(1 + \alpha \Delta T)^{1/2} \]

### Step 7: Calculate the Change in Time Period
The change in time period \(\Delta T\) is given by:
\[ \Delta T = T' - T = T(1 + \alpha \Delta T)^{1/2} - T \]
Using the binomial expansion for small \(\alpha \Delta T\):
\[ (1 + x)^{1/2} \approx 1 + \frac{x}{2} \text{ for small } x \]
we can approximate:
\[ \Delta T \approx T\left(\frac{\alpha \Delta T}{2}\right) = \frac{T \alpha \Delta T}{2} \]

### Final Result
Thus, the change in time period when the temperature changes by \(\Delta T\) is:
\[ \Delta T \approx \frac{T \alpha \Delta T}{2} \]