A pendulum 1 m long makes 20 vibrations in 40s. Find the time taken to make 30 vibrations, if its length is increased to 4 m.

Answer

To solve the problem, we need to find the time taken for a pendulum of length 4 m to make 30 vibrations, given that a pendulum of length 1 m makes 20 vibrations in 40 seconds.

### Step 1: Calculate the time period of the original pendulum
The time taken for 20 vibrations is 40 seconds. Therefore, the time period (T) for one vibration can be calculated as:
\[ T = \frac{\text{Total time}}{\text{Number of vibrations}} = \frac{40 \text{ s}}{20} = 2 \text{ s} \]

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

For the original pendulum (1 m long):
\[ 2 = 2\pi \sqrt{\frac{1}{g}} \]

### Step 3: Solve for \(g\)
Dividing both sides by \(2\):
\[ 1 = \pi \sqrt{\frac{1}{g}} \]
Squaring both sides:
\[ 1 = \pi^2 \cdot \frac{1}{g} \]
Rearranging gives:
\[ g = \pi^2 \approx 9.87 \, \text{m/s}^2 \]

### Step 4: Calculate the time period for the new pendulum (4 m long)
Now, we need to find the time period for the new pendulum of length 4 m:
\[ T' = 2\pi \sqrt{\frac{4}{g}} = 2\pi \sqrt{\frac{4}{\pi^2}} = 2\pi \cdot \frac{2}{\pi} = 4 \text{ s} \]

### Step 5: Calculate the time taken for 30 vibrations
Now that we have the time period for the new pendulum, we can find the total time for 30 vibrations:
\[ \text{Total time for 30 vibrations} = 30 \times T' = 30 \times 4 = 120 \text{ s} \]

### Final Answer
The time taken to make 30 vibrations with the pendulum of length 4 m is **120 seconds**.

---