TS EAMCET · Physics · Rotational Motion

A solid sphere is rolling without slipping on a semi-circular track of radius \(10 \mathrm{~m}\) as shown in the figure. The radius of solid sphere is much smaller than the radius of semi-circular track. At the lowest point, it has a velocity \(10 \mathrm{~m} / \mathrm{s}\). To what maximum angle \(\theta\) from the vertical will the sphere travel before it comes back down? Neglect the rolling friction between the sphere and the track. (Take, \(g=10 \mathrm{~m} / \mathrm{s}^2\) )

A \(\sin ^{-1}\left(\frac{3}{5}\right)\)
B \(\sin ^{-1}\left(\frac{3}{7}\right)\)
C \(\cos ^{-1}\left(\frac{3}{10}\right)\)
D \(\cos ^{-1}\left(\frac{1}{3}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\cos ^{-1}\left(\frac{3}{10}\right)\)

Step-by-step Solution

Detailed explanation

As there is no slip, energy is conserved. So, mechanical energy of ball at bottom \(=\) mechanical energy of ball at top \(\Rightarrow \quad m g r+\frac{1}{2} m v_{\mathrm{COM}}^2+\frac{1}{2} I_{\mathrm{COM}} \omega^2=m g h\)…

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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Column I		Column II
A)	For ring	I)	$\frac{M g \sin θ}{2.5}$
B)	For solid sphere	II)	$\frac{M g \sin θ}{3}$
C)	For solid cylinder	III)	$\frac{M g \sin θ}{3.5}$
D)	For hallow cylinder	IV)	$\frac{M g \sin θ}{2}$