Take the values of (prop) from table `20.2`.
A pendulum clock of time period `2 s` gives the correct time at `30^@ C`. The pendulum is made of iron. How many seconds will it lose or gain per day when the temperature falls tp `0^@ C`? `alpha _(Fe) = 1.2 xx 10^-5 (.^@ C)^-1`.

Explanation

To solve the problem of how many seconds a pendulum clock will lose or gain per day when the temperature falls to 0°C, we will follow these steps:

### Step 1: Understand the relationship between temperature change and time period
The time period \( T \) of a pendulum is affected by the length of the pendulum, which changes with temperature due to thermal expansion. The change in the time period \( \Delta T \) can be expressed as:
\[ \Delta T = \frac{1}{2} T \alpha \Delta \theta \]
where:
- \( T \) is the original time period,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta \theta \) is the change in temperature.

### Step 2: Identify the given values
From the problem, we have:
- Original time period \( T = 2 \, \text{s} \)
- Coefficient of linear expansion for iron \( \alpha = 1.2 \times 10^{-5} \, (°C)^{-1} \)
- Initial temperature \( \theta_1 = 30°C \)
- Final temperature \( \theta_2 = 0°C \)

### Step 3: Calculate the change in temperature
The change in temperature \( \Delta \theta \) is given by:
\[ \Delta \theta = \theta_1 - \theta_2 = 30°C - 0°C = 30°C \]

### Step 4: Calculate the change in time period
Now, substituting the values into the formula for \( \Delta T \):
\[ \Delta T = \frac{1}{2} \times 2 \, \text{s} \times 1.2 \times 10^{-5} \, (°C)^{-1} \times 30°C \]
\[ \Delta T = 1 \times 1.2 \times 10^{-5} \times 30 \]
\[ \Delta T = 3.6 \times 10^{-4} \, \text{s} \]

### Step 5: Calculate the total change in time over one day
To find out how many seconds the clock will lose or gain in one day, we need to convert the change in time period per second into a daily total:
\[ \text{Total change in time per day} = \Delta T \times \text{Number of seconds in a day} \]
There are \( 24 \times 60 \times 60 = 86400 \) seconds in a day. Thus:
\[ \text{Total change in time per day} = 3.6 \times 10^{-4} \, \text{s} \times 86400 \, \text{s} \]
\[ \text{Total change in time per day} = 31.104 \, \text{s} \]

### Conclusion
The pendulum clock will gain approximately **31.104 seconds** per day when the temperature falls to 0°C.

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