The length of a second pendulum is

Options

A. 99.8 cm
B. 99 cm
C. 100 cm
D. None of these

Explanation

To find the length of a second pendulum, we can follow these steps:

### Step 1: Understand the definition of a second pendulum
A second pendulum is defined as a pendulum that has a time period of 2 seconds. This means it takes 1 second to swing in one direction (tick) and 1 second to return (tock).

### Step 2: Write the formula for the time period of a simple pendulum
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

### Step 3: Set the time period for a second pendulum
Since we know that the time period \( T \) for a second pendulum is 2 seconds, we can substitute this value into the formula:
\[ 2 = 2\pi \sqrt{\frac{L}{g}} \]

### Step 4: Simplify the equation
We can simplify the equation by dividing both sides by \( 2 \):
\[ 1 = \pi \sqrt{\frac{L}{g}} \]

### Step 5: Isolate the square root term
Next, we isolate the square root term by dividing both sides by \( \pi \):
\[ \sqrt{\frac{L}{g}} = \frac{1}{\pi} \]

### Step 6: Square both sides to eliminate the square root
Now, we square both sides to eliminate the square root:
\[ \frac{L}{g} = \left(\frac{1}{\pi}\right)^2 \]

### Step 7: Solve for \( L \)
Now we can solve for \( L \):
\[ L = g \left(\frac{1}{\pi}\right)^2 \]

### Step 8: Substitute the value of \( g \)
Substituting \( g = 9.8 \, \text{m/s}^2 \):
\[ L = 9.8 \left(\frac{1}{\pi}\right)^2 \]

### Step 9: Calculate the length
Calculating the value:
\[ L = 9.8 \left(\frac{1}{3.14159}\right)^2 \approx 9.8 \times 0.10132 \approx 0.993 \, \text{m} \approx 99.4 \, \text{cm} \]

### Step 10: Conclusion
Thus, the length of a second pendulum is approximately \( 99.4 \, \text{cm} \).

### Final Answer
The length of a second pendulum is \( 99.4 \, \text{cm} \).

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