A simple pendulum, has a period of 5.77 seconds. When the pendulum is shortened by 3 m, the period is 4.60 seconds. Calculate the new length of the pendulum

1. Analyze the Question

We are given two different periods for a simple pendulum at two different lengths. We need to find the new length ($L_2$) after the original length ($L_1$) was shortened by 3 meters.

2. Explain the Concept

The period ($T$) of a simple pendulum is given by the formula:

Where:

• $T$ = Period in seconds
• $L$ = Length of the pendulum in meters
• $g$ = Acceleration due to gravity ($\approx 9.81 \, \text{m/s}^2$)

Since $2\pi$ and $\sqrt{g}$ are constants, we can establish a ratio:

Squaring both sides gives:

3. Show the Solution

Step 1: Set up the equations. Let $L_1$ be the original length. The new length $L_2 = L_1 - 3$. Given: $T_1 = 5.77 \, \text{s}$ and $T_2 = 4.60 \, \text{s}$.

Step 2: Substitute the values into the ratio formula.

Step 3: Solve for $L_1$.

Step 4: Calculate the new length ($L_2$).

4. Identify the Answer

Comparing our result to the given options, the calculated new length is 5.23 m. This matches option A.

5. Exam Tip

When dealing with pendulum period problems, always remember that the period is proportional to the square root of the length ($T \propto \sqrt{L}$). If you ever forget the full formula, just remember that if you increase the length, the period increases, and if you decrease the length, the period decreases. This helps you check if your final answer makes logical sense!