Question

Two simple pendulums have (A) mass $M_1$, length $L_1$ and (B) mass $M_2$, length $L_2$. Given $M_1 = M_2$ and $L_1 = 2L_2$. If their total energies are same, then \dots

• A. amplitude of B is greater than amplitude of A.
• B. amplitude of B is smaller than amplitude of A.
• C. amplitude of both will be same.
• D. amplitude of B is twice that of A.

Hint:

Energy in a pendulum scales with $(A^2 / L)$. If you cut the string length in half ($L$ goes down), the pendulum gets "stiffer" and swings faster. To maintain the exact same total energy, its physical swing distance (amplitude) must shrink.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are comparing the oscillation amplitudes of two different simple pendulums that have identical masses and identical total mechanical energies, but different string lengths.

Step 2: Key Formula or Approach:
1. The total energy ($E$) of a simple harmonic oscillator is:
$$E = \frac{1}{2} M \omega^2 A^2$$
2. The angular frequency ($\omega$) of a simple pendulum is:
$$\omega = \sqrt{\frac{g}{L}} \implies \omega^2 = \frac{g}{L}$$
Substitute $\omega^2$ into the energy equation:
$$E = \frac{1}{2} M \left( \frac{g}{L} \right) A^2$$

Step 3: Detailed Explanation:
We are given that $E_A = E_B$ and $M_A = M_B = M$.
Set up the energy equality:
$$\frac{1}{2} M \left( \frac{g}{L_A} \right) A_A^2 = \frac{1}{2} M \left( \frac{g}{L_B} \right) A_B^2$$
Cancel the common constants ($\frac{1}{2}$, $M$, and $g$):
$$\frac{A_A^2}{L_A} = \frac{A_B^2}{L_B}$$
Rearrange to solve for the ratio of the amplitudes squared:
$$\frac{A_B^2}{A_A^2} = \frac{L_B}{L_A}$$
We are given $L_A = 2L_B$. Substitute this into the ratio:
$$\frac{A_B^2}{A_A^2} = \frac{L_B}{2L_B} = \frac{1}{2}$$
Taking the square root:
$$A_B = \frac{A_A}{\sqrt{2}}$$
Since $\sqrt{2} \approx 1.414 > 1$, we clearly see that $A_B < A_A$.
The amplitude of pendulum B is definitively smaller than the amplitude of pendulum A.

Step 4: Final Answer:
Amplitude of B is smaller than amplitude of A, matching option (b).