Question

Two (A and B) pendulums begin to swing simultaneously. The first pendulum (A) makes five oscillations when the other (B) makes three oscillations. The ratio of the lengths of pendulum \(A\) to that of \(B\) is

• A. \(3:5\)
• B. \(9:25\)
• C. \(5:3\)
• D. \(2:3\)

Hint:

Find the time period ratio from the number of oscillations in the same time, then use the fact that length is proportional to the square of the time period.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Both pendulums swing for the same total time. The one that makes more oscillations has the smaller time period.
For a simple pendulum \(T=2\pi\sqrt{\dfrac{l}{g}}\), so \(l\propto T^{2}\).

Step 2: Time Period Ratio
In the same time \(t\): \(T_A=\dfrac{t}{5}\) and \(T_B=\dfrac{t}{3}\).
So \(\dfrac{T_A}{T_B}=\dfrac{3}{5}\).

Step 3: Calculation
\(\dfrac{l_A}{l_B}=\left(\dfrac{T_A}{T_B}\right)^{2}=\left(\dfrac{3}{5}\right)^{2}=\dfrac{9}{25}\).

Step 4: Compare the options
The ratio \(9:25\) matches option (B).

Final Answer:
The ratio of the lengths of pendulum A to pendulum B is \(9:25\). \[ \boxed{\text{(B) }9:25} \]