A clock with a metal pendulum beating seconds keeps correct time at `0^(@)C.` If it loses 12.5s a day at `25^(@)C` the coefficient of linear expansion of metal pendulum is

Options

A. `(1)/(86400)/^(@)C`
B. `(1)/(43200)/^(@)C`
C. `(1)/(14400)/^(@)C`
D. `(1)/(28800)/^(@)C`

Explanation

To find the coefficient of linear expansion of the metal pendulum, we can follow these steps:

### Step 1: Understand the problem
The clock keeps correct time at 0°C but loses 12.5 seconds a day at 25°C. We need to find the coefficient of linear expansion (α) of the metal pendulum.

### Step 2: Identify the formula
The time lost (ΔT) due to a change in temperature (Δθ) can be expressed as:
\[ \Delta T = \frac{1}{2} \alpha \Delta \theta T \]
where:
- ΔT = time lost (12.5 seconds)
- α = coefficient of linear expansion
- Δθ = change in temperature (25°C - 0°C = 25°C)
- T = total time in seconds for one day (24 hours)

### Step 3: Calculate the total time in seconds for one day
Convert 24 hours into seconds:
\[ T = 24 \times 3600 = 86400 \text{ seconds} \]

### Step 4: Rearrange the formula to solve for α
From the formula, we can rearrange it to find α:
\[ \alpha = \frac{2 \Delta T}{\Delta \theta T} \]

### Step 5: Substitute the known values into the formula
Substituting the values we have:
- ΔT = 12.5 seconds
- Δθ = 25°C
- T = 86400 seconds

Now substituting these values into the equation:
\[ \alpha = \frac{2 \times 12.5}{25 \times 86400} \]

### Step 6: Perform the calculations
Calculating the numerator:
\[ 2 \times 12.5 = 25 \]
Calculating the denominator:
\[ 25 \times 86400 = 2160000 \]
Now substituting back:
\[ \alpha = \frac{25}{2160000} \]

### Step 7: Simplify the fraction
\[ \alpha = \frac{1}{86400} \text{ °C}^{-1} \]

### Conclusion
The coefficient of linear expansion of the metal pendulum is:
\[ \alpha = \frac{1}{86400} \text{ °C}^{-1} \]

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