MHT CET · Physics · Work Power Energy

The bob of a pendulum of length ' \(l\) ' is pulled aside from its equilibrium position through an angle ' \(\theta\) ' and then released. The bob will then pass through its equilibrium position with speed ' \(v\) ', where ' \(v\) ' equal to ( \(g=\) acceleration due to gravity)

A \(\sqrt{2 g l(1-\cos \theta)}\)
B \(\sqrt{2 g l(1+\sin \theta)}\)
C \(\sqrt{2 g l(1-\sin \theta)}\)
D \(\sqrt{2 \mathrm{~g} l(1+\cos \theta)}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\sqrt{2 g l(1-\cos \theta)}\)

Step-by-step Solution

Detailed explanation

When bob of a pendulum rises up a height ' \(h\) ' potential energy at extreme position becomes kinetic energy of mean position.
\(\mathrm{mgh}=\frac{1}{2} \mathrm{mv}_{\max }^2 \)
\( \therefore \mathrm{v}_{\max }=\sqrt{2 \mathrm{gh}} \)
\(l=\mathrm{h}+l \cos \theta \)
\( \therefore \mathrm{~h}=l(1-\cos \theta) \)
\( \therefore \mathrm{v}_{\max }=\sqrt{2 \mathrm{gl}(1-\cos \theta)}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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