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. 20.52 s
B. 15.00 s
C. 52.10 s
D. 63.10 s

Answer

To solve the problem, we need to determine how many seconds a day the clock will gain when the ambient temperature falls from 25°C to 0°C. The pendulum of the clock is made of brass, which has a coefficient of linear expansion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Coefficient of linear expansion for brass, \( \alpha = 0.000019 \, ^\circ C^{-1} \)
- Initial temperature, \( T_1 = 25 \, ^\circ C \)
- Final temperature, \( T_2 = 0 \, ^\circ C \)
- Change in temperature, \( \Delta \theta = T_2 - T_1 = 0 - 25 = -25 \, ^\circ C \)

2. **Determine the Change in Length of the Pendulum:**
The change in length \( \Delta L \) due to temperature change can be expressed as:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta \theta \]
where \( L_0 \) is the original length of the pendulum. However, we will not need the actual length since it will cancel out later.

3. **Relate Change in Length to Change in Time Period:**
The time period \( T \) of a pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( g \) is the acceleration due to gravity. The change in time period \( \Delta T \) can be approximated for small changes in length as:
\[ \Delta T \approx \frac{1}{2} \cdot \alpha \cdot \Delta \theta \cdot T \]

4. **Calculate the Time Period at 25°C:**
The time period \( T \) for a day (24 hours) in seconds is:
\[ T = 24 \times 60 \times 60 = 86400 \, \text{seconds} \]

5. **Substitute Values into the Change in Time Period Equation:**
Now, substituting the values into the equation for \( \Delta T \):
\[ \Delta T = \frac{1}{2} \cdot 0.000019 \cdot (-25) \cdot 86400 \]

6. **Calculate \( \Delta T \):**
\[ \Delta T = \frac{1}{2} \cdot 0.000019 \cdot (-25) \cdot 86400 \]
\[ = \frac{1}{2} \cdot 0.000019 \cdot (-2160000) \]
\[ = -20.52 \, \text{seconds} \]

7. **Interpret the Result:**
The negative sign indicates that the clock will gain time. Therefore, the clock will gain approximately **20.52 seconds** in a day when the temperature drops from 25°C to 0°C.

### Final Answer:
The clock will gain approximately **20.52 seconds per day**.