MHT CET · Physics · Laws of Motion

The bob of a simple pendulum of length ' \(L\) ' has a mass ' \(m\) ' and charge ' \(q\) '. The pendulum is suspended between the plates of a charged parallel plate capacitor. The direction of electric field is shown in figure. The period of oscillations of the simple pendulum is (acceleration due to gravity \(g>q E / m\) )

A \(2 \pi \sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\)
B \(2 \pi\left[\frac{\mathrm{L}}{\frac{\mathrm{qE}}{\mathrm{m}}-\mathrm{g}}\right]^{\frac{1}{2}}\)
C \(2 \pi\left[\frac{\mathrm{L}}{\mathrm{g}-\frac{\mathrm{qE}}{\mathrm{m}}}\right]^{\frac{1}{2}}\)
D \(2 \pi\left[\frac{\mathrm{L}}{\mathrm{g}+\frac{\mathrm{qE}}{\mathrm{m}}}\right]^{\frac{1}{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(2 \pi\left[\frac{\mathrm{L}}{\mathrm{g}-\frac{\mathrm{qE}}{\mathrm{m}}}\right]^{\frac{1}{2}}\)

Step-by-step Solution

Detailed explanation

Electric force, \(\mathrm{F}_{\text {electric }}=\mathrm{qE}\)
The effective force,
\(\begin{aligned}& \mathrm{mg}_{\text {eff }}=\mathrm{mg}-\mathrm{F}_{\text {electric }} \\& \mathrm{g}_{\text {eff }}=\mathrm{g}-\frac{\mathrm{qE}}{\mathrm{m}}\end{aligned}\)
The period of oscillation for a simple pendulum,
\(\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\)
Time period when pendulum is suspended between the plates,
\(\begin{aligned}& \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{L}}{\mathrm{g}_{\text {eff }}}} \\& \mathrm{T}=2 \pi\left[\frac{\mathrm{L}}{\mathrm{g}-\frac{\mathrm{qE}}{\mathrm{m}}}\right]^{\frac{1}{2}}\end{aligned}\)

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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