Question

An inclined plane makes an angle $30^\circ$ with the horizontal. A solid sphere rolling down an inclined plane from rest without slipping has linear acceleration ( $\text{g} =$ acceleration due gravity) ( $\sin 30^\circ = 0.5$ )

• A. $\frac{5\text{ g}}{7}$
• B. $\frac{5\text{ g}}{14}$
• C. $\frac{2\text{ g}}{3}$
• D. $\frac{\text{g}}{3}$

Hint:

Solid sphere is "heavier" in the middle, so it rolls faster than a ring but slower than a block sliding without friction.

Answer 1

The Correct Option is B

Step 1: Concept
The acceleration of a body rolling down an incline is $a = \frac{g \sin \theta}{1 + K^2/R^2}$.

Step 2: Meaning
For a solid sphere, the moment of inertia is $I = \frac{2}{5}MR^2$, so $K^2/R^2 = 2/5$.

Step 3: Analysis
$a = \frac{g \sin 30^\circ}{1 + 2/5}$
$a = \frac{g(0.5)}{7/5} = \frac{5g}{14}$.

Step 4: Conclusion
The linear acceleration is $\frac{5g}{14}$. Final Answer: (B)