A pendulum clock keeps correct time at `0^(@)C`. Its mean coefficient of linear expansions is `alpha//.^(@)C`, then the loss in seconds per day by the clock if the temperature rises by `t^(@)C` is

Options

A. `1/2 alpha t xx 86400`
B. `alpha t xx86400`
C. `(1/2 alpha t)/(1+alpha t//2)xx86400`
D. `(alphat)/(1+alpha t//2)`

Answer

To solve the problem of how much time a pendulum clock loses per day when the temperature rises by \( t \) degrees Celsius, we can follow these steps:

### Step 1: Understand the relationship between temperature change and timekeeping
A pendulum clock keeps accurate time at \( 0^\circ C \). The timekeeping of the clock is affected by the temperature due to the linear expansion of the pendulum. The mean coefficient of linear expansion is denoted as \( \alpha \).

### Step 2: Determine the formula for time loss
The time loss or gain (\( \Delta t \)) due to a temperature change can be expressed as:
\[ \Delta t = \frac{1}{2} \alpha t \]
where \( \alpha \) is the coefficient of linear expansion and \( t \) is the temperature rise in degrees Celsius.

### Step 3: Calculate the total time loss per day
Since there are \( 86400 \) seconds in a day (24 hours × 3600 seconds/hour), we need to multiply the time loss per second by the total number of seconds in a day:
\[ \Delta T_{\text{per day}} = \Delta t \times 86400 \]
Substituting the expression for \( \Delta t \):
\[ \Delta T_{\text{per day}} = \left(\frac{1}{2} \alpha t\right) \times 86400 \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \Delta T_{\text{per day}} = \frac{1}{2} \alpha t \times 86400 \]

### Final Answer
Thus, the loss in seconds per day by the clock if the temperature rises by \( t \) degrees Celsius is:
\[ \Delta T_{\text{per day}} = \frac{1}{2} \alpha t \times 86400 \]