The pendulum of certain clock has time period 2.04 s. How fast or slow does the clock run during 24 hours?

Explanation

To solve the problem of how fast or slow the clock runs during 24 hours given that the time period of the pendulum is 2.04 seconds, we can follow these steps:

### Step 1: Calculate the total time in seconds for 24 hours.
We know that:
- 1 hour = 3600 seconds
- Therefore, 24 hours = 24 × 3600 seconds.

**Calculation:**
\[ 24 \text{ hours} = 24 \times 3600 = 86400 \text{ seconds} \]

### Step 2: Determine the number of oscillations the pendulum makes in 24 hours.
The number of oscillations can be calculated by dividing the total time in seconds by the time period of the pendulum.

**Given:**
- Time period \( T = 2.04 \text{ seconds} \)

**Calculation:**
\[ \text{Number of oscillations} = \frac{86400 \text{ seconds}}{2.04 \text{ seconds}} \approx 42352.94 \]
Since the number of oscillations must be a whole number, we round it down to 42352 oscillations.

### Step 3: Determine the expected time period for a "perfect" clock.
For a perfect clock, the time period of the pendulum is typically 2 seconds.

### Step 4: Calculate the difference in time period.
Now, we calculate the difference in time period between the actual pendulum and the ideal pendulum.

**Calculation:**
\[ \Delta T = 2.04 \text{ seconds} - 2 \text{ seconds} = 0.04 \text{ seconds} \]

### Step 5: Calculate the total time lost in 24 hours.
To find out how much time the clock runs slow in a day, we multiply the number of oscillations by the difference in time period.

**Calculation:**
\[ \text{Total time lost} = \text{Number of oscillations} \times \Delta T = 42352 \times 0.04 \text{ seconds} \approx 1694.08 \text{ seconds} \]

### Step 6: Convert the total time lost into minutes.
To convert seconds into minutes, we divide by 60.

**Calculation:**
\[ \text{Time lost in minutes} = \frac{1694.08 \text{ seconds}}{60} \approx 28.23 \text{ minutes} \]

### Conclusion:
The clock runs approximately 28.23 minutes slower in one day.

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