A pendulum has a string of length 99.39 cm. how much length of the pendulum must be shortened to keep the current time of the pendulum if it loses 4 s a day?

Options

A. 0.0009 cm
B. 0.009 cm
C. 0.09 cm
D. 0.9 cm

Explanation

To solve the problem of how much length of the pendulum must be shortened to keep the current time if it loses 4 seconds a day, we can follow these steps:

### Step 1: Understand the time lost
The pendulum loses 4 seconds in one day. We need to find out how this loss relates to the length of the pendulum.

### Step 2: Calculate the total number of seconds in a day
There are 24 hours in a day, and each hour has 3600 seconds. Therefore, the total number of seconds in a day is:
\[ \text{Total seconds in a day} = 24 \times 3600 = 86400 \text{ seconds} \]

### Step 3: Set up the relationship between time and length
The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

### Step 4: Differentiate the period with respect to length
To find how a change in length \( dL \) affects the period \( dT \), we differentiate the period equation:
\[ dT = \frac{dT}{dL} dL = \frac{\pi}{\sqrt{gL}} dL \]
From the formula, we can express the relationship as:
\[ \frac{dT}{T} = \frac{1}{2} \frac{dL}{L} \]

### Step 5: Substitute known values
We know:
- \( dT = -4 \) seconds (time lost)
- \( T = 86400 \) seconds (total seconds in a day)
- \( L = 99.39 \) cm = 0.9939 m

Now we can substitute these values into the relationship:
\[ \frac{-4}{86400} = \frac{1}{2} \frac{dL}{0.9939} \]

### Step 6: Solve for \( dL \)
Rearranging the equation gives:
\[ dL = -2 \cdot \frac{-4}{86400} \cdot 0.9939 \]
Calculating this:
\[ dL = 2 \cdot \frac{4 \cdot 0.9939}{86400} \]
\[ dL = \frac{7.9512}{86400} \approx 0.000092 \text{ m} = 0.00920 \text{ cm} \]

### Final Answer
The length of the pendulum must be shortened by approximately **0.00920 cm** to keep the current time.

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