The period of the pendulum is directly proportional to the square root of the length of the string. The period of such a pendulum with string of length 16 cm is 52 seconds. Find the length of the string if the period is 65 seconds

Options

A. 4.5 cm
B. 5 cm
C. 6 cm
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship
The period \( t \) of the pendulum is directly proportional to the square root of the length \( L \) of the string. This can be expressed mathematically as:
\[ t \propto \sqrt{L} \]
This means that:
\[ t = k \sqrt{L} \]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant \( k \)
We are given that when the length \( L = 16 \) cm, the period \( t = 52 \) seconds. We can substitute these values into the equation to find \( k \):
\[ 52 = k \sqrt{16} \]
Since \( \sqrt{16} = 4 \), we can simplify this to:
\[ 52 = 4k \]
Now, solving for \( k \):
\[ k = \frac{52}{4} = 13 \]

### Step 3: Use the constant to find the new length
Now we need to find the length \( L \) when the period \( t = 65 \) seconds. We use the same equation:
\[ 65 = 13 \sqrt{L} \]
To isolate \( \sqrt{L} \), we divide both sides by 13:
\[ \sqrt{L} = \frac{65}{13} \]
Calculating the right side gives:
\[ \sqrt{L} = 5 \]

### Step 4: Solve for \( L \)
Now we square both sides to find \( L \):
\[ L = 5^2 = 25 \]

### Conclusion
The length of the string when the period is 65 seconds is \( L = 25 \) cm.

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