TS EAMCET · Physics · Rotational Motion

A solid sphere of radius $R$ makes a perfect rolling down on a plane which is inclined to the horizontal axis at an angle $θ$ . If the radius of gyration is $k$ , then its acceleration is

A $\frac{g \sin θ}{(1 + \frac{k^{2}}{R^{2}})}$
B $\frac{g \sin θ}{(R^{2} + k^{2})}$
C $\frac{g \sin θ}{2 (R^{2} + k^{2})}$
D $\frac{g \sin θ}{2 (1 + \frac{k^{2}}{R^{2}})}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{g \sin θ}{(1 + \frac{k^{2}}{R^{2}})}$

Step-by-step Solution

Detailed explanation

Referring to above diagram, balancing forces along the inclined plane, ma=mgsinθ-f ...1 torque is given by, τ=Iα=fR but α=aR & l=mk2 ∴ mk2×aR=fR⇒f=amk2R2 Putting in equation 1,…

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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A solid sphere of radius R makes a perfect rolling down on a plane which is inclined to the horizontal axis at an angle θ. If the radius of gyration is k, then its acceleration is

Step-by-step Solution

A solid sphere of radius $R$ makes a perfect rolling down on a plane which is inclined to the horizontal axis at an angle $θ$ . If the radius of gyration is $k$ , then its acceleration is