A clock which keeps correct time at `25^@C` , has a pendulum made of brass. The coefficient of linear expansion for brass is `0.000019.^@C^(-1)` . How many seconds a day will it gain if the ambient temperature falls to `0.^@C` ?

Options

A. (a)20.52 s
B. (b)15.00 s
C. (c)52.10 s
D. (d)63.10 s

Answer

To solve the problem, we need to determine how much time a clock will gain in seconds per day when the temperature drops from 25°C to 0°C due to the change in the length of the pendulum made of brass. The steps are as follows:

### Step 1: Understand the relationship between the pendulum length and temperature
The time period \( T \) of a pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Determine the change in length due to temperature change
The change in length \( \Delta L \) of the pendulum due to temperature change can be calculated using the formula:
\[ \Delta L = L_0 \alpha \Delta \theta \]
where:
- \( L_0 \) is the original length of the pendulum,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta \theta \) is the change in temperature.

### Step 3: Calculate the change in temperature
The change in temperature \( \Delta \theta \) is:
\[ \Delta \theta = 0°C - 25°C = -25°C \]

### Step 4: Calculate the change in time period
The change in time period \( \Delta T \) due to the change in length can be approximated as:
\[ \Delta T = \frac{1}{2} \alpha \Delta \theta T \]
where \( T \) is the original time period at 25°C.

### Step 5: Substitute the values
Given:
- \( \alpha = 0.000019 \, °C^{-1} \)
- \( \Delta \theta = -25°C \)
- \( T \) for a pendulum of length \( L_0 \) at 25°C is approximately 1 day (86400 seconds).

Now substituting the values:
\[ \Delta T = \frac{1}{2} \times 0.000019 \times (-25) \times 86400 \]

### Step 6: Calculate \( \Delta T \)
Calculating the above expression:
\[ \Delta T = \frac{1}{2} \times 0.000019 \times (-25) \times 86400 \]
\[ = \frac{1}{2} \times 0.000019 \times (-2160000) \]
\[ = -20.52 \, \text{seconds} \]

### Step 7: Interpret the result
The negative sign indicates that the clock will gain time, meaning it will run faster by approximately 20.52 seconds per day.

### Final Answer
The clock will gain approximately **20.52 seconds per day** when the temperature falls to 0°C.

---