Question

Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $\theta_{1}$ and $\theta_{2}$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

• A. $\theta_{1} l_{2}^{2}=\theta_{2} l_{1}^{2}$
• B. $\theta_{1} l_{1}=\theta_{2} l_{2}$
• C. $\theta_{1} l_{1}^{2}=\theta_{2} l_{2}^{2}$
• D. $\theta_{1} l_{2}=\theta_{2} l_{1}$

Hint:

The angular acceleration of a pendulum depends on its length and angular displacement.

Answer 1

The Correct Option is D

1. Angular acceleration: \[ \alpha = -\omega^2 \theta \] \[ \omega = \sqrt{\frac{g}{l}} \]
2. Equating angular accelerations: \[ \frac{g}{l_1} \theta_1 = \frac{g}{l_2} \theta_2 \] \[ \theta_1 l_2 = \theta_2 l_1 \] Therefore, the correct answer is (4) $\theta_{1} l_{2}=\theta_{2} l_{1}$.

Answer 2

We are given two simple pendulums having lengths \( l_1 \) and \( l_2 \), with angular displacements \( \theta_1 \) and \( \theta_2 \), respectively. It is given that both pendulums have the same angular acceleration. We need to find the correct relationship between their displacements and lengths.

Concept Used:

For a simple pendulum performing small oscillations, the restoring torque per unit moment of inertia gives the angular acceleration:

\[ \alpha = -\frac{g}{l} \sin \theta \]

For small angles (in radians), \( \sin \theta \approx \theta \). Thus, the angular acceleration can be approximated as:

\[ \alpha = -\frac{g}{l} \theta \]

Step-by-Step Solution:

Step 1: Write angular acceleration for the two pendulums.

\[ \alpha_1 = -\frac{g}{l_1} \theta_1, \quad \alpha_2 = -\frac{g}{l_2} \theta_2 \]

Step 2: Given that the angular accelerations are equal in magnitude.

\[ \alpha_1 = \alpha_2 \]

Step 3: Substitute the expressions for \( \alpha_1 \) and \( \alpha_2 \).

\[ -\frac{g}{l_1} \theta_1 = -\frac{g}{l_2} \theta_2 \]

Step 4: Simplify the equation by canceling \( g \) and the negative signs.

\[ \frac{\theta_1}{l_1} = \frac{\theta_2}{l_2} \]

Final Computation & Result:

The correct relation between angular displacements and lengths is:

\[ \boxed{\frac{\theta_1}{l_1} = \frac{\theta_2}{l_2}} \]

Final Answer: \( \dfrac{\theta_1}{l_1} = \dfrac{\theta_2}{l_2} \)