Question

Consider a simple pendulum undergoing simple harmonic motion with a time period \(T\), and a fixed amplitude \(\theta_0\) of angular oscillation. Its angular momentum about the point of suspension exhibits an oscillatory behavior with an amplitude \(A\). Which of the following relations between \(A\) and \(T\) is correct?

• A. \(A \propto T^3\)
• B. \(A \propto T^2\)
• C. \(A \propto T\)
• D. \(A \propto T^4\)

Hint:

Express all variables in terms of the primary variable \(T\).
Length \(L \propto T^2\), which means moment of inertia \(I \propto L^2 \propto T^4\).
Since angular momentum amplitude is \(I \omega\) and \(\omega \propto T^{-1}\), we directly get \(T^4 \cdot T^{-1} = T^3\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the relationship between the amplitude of the oscillatory angular momentum of a simple pendulum and its time period.
We are given that the angular amplitude of the oscillation is fixed.

Step 2: Key Formula or Approach:
• The time period of a simple pendulum of length \(L\) is:
\[ T = 2\pi \sqrt{\frac{L}{g}} \implies L \propto T^2 \]

• The moment of inertia of the pendulum bob of mass \(m\) about the suspension point is:
\[ I = m L^2 \]

• The angular displacement is given by \(\theta(t) = \theta_0 \sin(\omega t)\), where the angular frequency is:
\[ \omega = \frac{2\pi}{T} \]

• The angular velocity is \(\dot{\theta}(t) = \theta_0 \omega \cos(\omega t)\).

• The angular momentum is:
\[ L(t) = I \dot{\theta} = I \theta_0 \omega \cos(\omega t) \]

Step 3: Detailed Explanation:

• The amplitude of the angular momentum oscillation is the maximum value of \(L(t)\), which we denote as \(A\).:
\[ A = I \theta_0 \omega \]

• Substitute the expression for the moment of inertia \(I = m L^2\).:
\[ A = (m L^2) \theta_0 \omega \]

• Since the mass \(m\) and the angular amplitude \(\theta_0\) are constant, we have:
\[ A \propto L^2 \omega \]

• From the relation for the time period, we know that \(L \propto T^2\).

• Also, the angular frequency is related to the time period as \(\omega \propto T^{-1}\).

• Substituting these proportionality relations into the expression for \(A\).:
\[ A \propto (T^2)^2 \cdot T^{-1} \]
\[ A \propto T^4 \cdot T^{-1} = T^3 \]

Step 4: Final Answer:
Therefore, the relation between \(A\) and \(T\) is \(A \propto T^3\).