A clock with a metal pendulum beating seconds keeps correct time at `0^(@)C.` If it loses 12.5 s 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 will follow these steps:

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

### Step 2: Use the formula for time loss
The change in time (time lost) due to temperature change can be expressed as:
\[ \Delta t = \frac{1}{2} \alpha \Delta T \cdot T \]
where:
- \(\Delta t\) is the time lost,
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature,
- \(T\) is the time period of the pendulum.

### Step 3: Identify the variables
From the problem:
- \(\Delta t = 12.5\) seconds (time lost in a day),
- \(\Delta T = 25^\circ C - 0^\circ C = 25^\circ C\),
- \(T = 86400\) seconds (total seconds in a day).

### Step 4: Rearrange the formula to solve for \(\alpha\)
Rearranging the formula gives:
\[ \alpha = \frac{2 \Delta t}{\Delta T \cdot T} \]

### Step 5: Substitute the values
Substituting the known values into the equation:
\[ \alpha = \frac{2 \times 12.5}{25 \times 86400} \]

### Step 6: Calculate the value
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
Calculating the final value:
\[ \alpha = \frac{1}{86400} \text{ per } ^\circ C \]

### Final Answer
The coefficient of linear expansion of the metal pendulum is:
\[ \alpha = \frac{1}{86400} \text{ per } ^\circ C \]

---