Question

A simple pendulum oscillates with an angular amplitude \(θ\). If the maximum tension in the string is twice the minimum tension then \(θ\) is

• A. \(cos^{-1}(0.75)\)
• B. \(cos^{-1}(0.5)\)
• C. \(sin^{-1}(0.5)\)
• D. \(sin^{-1}(0.75)\)

Hint:

Find tension at the lowest point using energy conservation and centripetal force, and at the extreme point where speed is zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The tension is smallest at the extreme position (speed zero) and largest at the mean position (speed maximum).

Step 2: Minimum tension.
At the extreme position, the speed is zero, so there is no centripetal force needed along the string. The tension balances the component of weight along the string:
\[ T_{min} = mg\cos\theta \]

Step 3: Maximum tension.
Energy conservation from the extreme to the bottom: \(\dfrac{1}{2}mv^2 = mgL(1 - \cos\theta)\), so \(v^2 = 2gL(1 - \cos\theta)\). At the bottom, \(T - mg = \dfrac{mv^2}{L}\):
\[ T_{max} = mg + 2mg(1 - \cos\theta) = mg(3 - 2\cos\theta) \]

Step 4: Use the condition.
\[ T_{max} = 2T_{min} \Rightarrow 3 - 2\cos\theta = 2\cos\theta \Rightarrow \cos\theta = \frac{3}{4} \]

Step 5: Check the options.
So \(\theta = \cos^{-1}(0.75)\), option (A). The options with \(\cos^{-1}(0.5)\), \(\sin^{-1}(0.5)\) and \(\sin^{-1}(0.75)\) do not satisfy the equation.

Final Answer:
The angular amplitude is \(\cos^{-1}(0.75)\). \[ \boxed{\theta = \cos^{-1}(0.75)} \]