MHT CET · Physics · Oscillations

A sphere of radius ' \(r\) ' is Kept on a concave minor of radius of curvature ' \(R\) ' The arrangement is kept on a horizontal table If the sphere is displaced from its equilibrium position and left then it executes S.H.M. The period of oscillation will be ( \(\mathrm{g}=\) acceleration due to gravity.)

A \(2 \pi[(\mathrm{R} / \mathrm{gr})]^{\frac{1}{2}}\)
B \(2 \pi[(\mathrm{R}-\mathrm{r}) / \mathrm{g}]^{\frac{1}{2}}\)
C \(2 \pi[(\mathrm{R}-\mathrm{r}) 1.4 / \mathrm{g}]^{\frac{1}{2}}\)
D \(2 \pi[(\mathrm{Rr}) / \mathrm{g}]^{\frac{1}{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(2 \pi[(\mathrm{R}-\mathrm{r}) / \mathrm{g}]^{\frac{1}{2}}\)

Step-by-step Solution

Detailed explanation

Consider the figure below:

Restoring torque about axis of rotation \(\mathrm{O}\) is given by,
\(\begin{aligned} & \tau=-\mathrm{mgr}_{\perp}=-\mathrm{mg}(\mathrm{R}-\mathrm{r}) \sin \theta \\ & =\mathrm{m}(\mathrm{R}-\mathrm{r})^2 \alpha \\ & \Rightarrow \alpha=-\left(\frac{\mathrm{g}}{(\mathrm{R}-\mathrm{r})}\right) \theta=-\omega^2 \theta\end{aligned}\)
The angular frequency can be written as: \(\omega^2=\left(\frac{\mathrm{g}}{\mathrm{R}-\mathrm{r}}\right)\)
\(\therefore \mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{(\mathrm{R}-\mathrm{r})}{\mathrm{g}}}\)

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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