A clock with a metallic pendulum gains 5 s each day at a temperature of `15^@C` and loses 10 s each day at a temperature of `30^@C`. Find the coefficient of thermal expansion of the pendulum metal.

Options

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

Explanation

To solve the problem, we need to determine the coefficient of thermal expansion (α) of the metallic pendulum based on the time gained and lost at different temperatures. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We know that:
- At 15°C, the clock gains 5 seconds per day.
- At 30°C, the clock loses 10 seconds per day.

### Step 2: Convert Time Gains and Losses to a Common Unit
Since the time is given in seconds per day, we need to express the total seconds in a day:
- Total seconds in a day = 24 hours × 60 minutes × 60 seconds = 86400 seconds.

### Step 3: Set Up the Equations
Using the formula for time change due to thermal expansion:
\[ \frac{\Delta T}{T} = \frac{1}{2} \alpha \Delta t \]
Where:
- \(\Delta T\) is the change in time (gain or loss).
- \(T\) is the total time in seconds (86400 seconds).
- \(\alpha\) is the coefficient of thermal expansion.
- \(\Delta t\) is the change in temperature.

#### For the first condition (15°C):
- Gain of 5 seconds:
\[ \frac{5}{86400} = \frac{1}{2} \alpha (T - 15) \]

#### For the second condition (30°C):
- Loss of 10 seconds:
\[ \frac{-10}{86400} = \frac{1}{2} \alpha (T - 30) \]

### Step 4: Rearranging the Equations
From the first equation:
\[ \alpha = \frac{10}{86400 (T - 15)} \]

From the second equation:
\[ \alpha = \frac{-20}{86400 (T - 30)} \]

### Step 5: Set the Two Expressions for α Equal
Setting the two expressions for α equal to each other:
\[ \frac{10}{86400 (T - 15)} = \frac{-20}{86400 (T - 30)} \]

### Step 6: Cross Multiply and Simplify
Cross-multiplying gives:
\[ 10 (T - 30) = -20 (T - 15) \]
Expanding both sides:
\[ 10T - 300 = -20T + 300 \]
Combining like terms:
\[ 30T = 600 \]
Thus:
\[ T = 20°C \]

### Step 7: Substitute T Back to Find α
Now substitute \(T = 20°C\) back into one of the equations for α:
Using the first equation:
\[ \alpha = \frac{10}{86400 (20 - 15)} = \frac{10}{86400 \times 5} = \frac{10}{432000} = \frac{1}{43200} \text{ per degree Celsius} \]

### Final Answer
The coefficient of thermal expansion of the pendulum metal is:
\[ \alpha = \frac{1}{43200} \text{ per degree Celsius} \]