A uniform metal rod is used as a bar pendulum. If the room temperature rises by `10^(@)C`, and the coefficient of linear expansion of the metal of the rod is `2 xx 10^(-6) per^(@)C`, the period of the pendulum will have percentage increase of

Options

A. (a)`-2xx10^-3`
B. (b)`-1xx10^-3`
C. (c)`2xx10^-3`
D. (d)`1xx10^-3`

Explanation

To solve the problem of the percentage increase in the period of a bar pendulum due to a rise in temperature, we will follow these steps:

### Step 1: Understand the relationship between temperature change and period change
The period \( T \) of a pendulum is affected by the length of the pendulum. When the temperature increases, the length of the metal rod also increases due to thermal expansion. The fractional change in the period \( \frac{\Delta T}{T} \) can be related to the change in length due to temperature change.

### Step 2: Use the formula for fractional change in period
The fractional change in the period \( \frac{\Delta T}{T} \) due to a change in temperature \( \Delta t \) is given by the formula:
\[ \frac{\Delta T}{T} = \frac{1}{2} \alpha \Delta t \]
where:
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta t \) is the change in temperature.

### Step 3: Substitute the known values
From the problem, we have:
- \( \alpha = 2 \times 10^{-6} \, \text{per } ^\circ C \)
- \( \Delta t = 10 \, ^\circ C \)

Substituting these values into the formula:
\[ \frac{\Delta T}{T} = \frac{1}{2} \times (2 \times 10^{-6}) \times 10 \]

### Step 4: Calculate the fractional change
Calculating the right-hand side:
\[ \frac{\Delta T}{T} = \frac{1}{2} \times 2 \times 10^{-6} \times 10 = 10^{-5} \]

### Step 5: Convert the fractional change to percentage
To find the percentage increase in the period, we multiply the fractional change by 100:
\[ \text{Percentage change} = \frac{\Delta T}{T} \times 100 = 10^{-5} \times 100 = 10^{-3} \]

### Step 6: Express the result
The percentage increase in the period of the pendulum is:
\[ 1 \times 10^{-3} \, \text{or} \, 0.1\% \]

### Conclusion
Thus, the percentage increase in the period of the pendulum due to a rise in temperature of \( 10 \, ^\circ C \) is \( 0.1\% \).