A solid cylinder and a solid sphere having same mass and same rad… - Vidyadip

MCQ

A solid cylinder and a solid sphere having same mass and same radius roll down on the same inclined plane. The ratio of the acceleration of the cylinder ' $a_c$ ' to that of sphere ' $a_s$ ' is

A
$\frac{11}{15}$
B
$\frac{13}{14}$
C
$\frac{15}{14}$
✓
$\frac{14}{15}$

✓

Answer

Correct option: D.

$\frac{14}{15}$

(d) : Acceleration of rolling body on inclined plane,
$
a=\frac{g \sin \theta}{1+\frac{K^2}{R^2}}
$
Here, $K$ is radius of gyration.
For solid cylinder, $I=\frac{1}{2} M R^2=M K_C^2 \Rightarrow K_C=\frac{R}{\sqrt{2}}$
For solid sphere, $I=\frac{2}{5} M R^2=M K^2{ }_S \Rightarrow K_{ S }=\sqrt{\frac{2}{5}} R$
$
\begin{aligned}
& a_c=\frac{g \sin \theta}{1+\frac{R^2}{2 R^2}}=\frac{2 g \sin \theta}{3} ...(i)\\
& a_s=\frac{g \sin \theta}{1+\frac{2 R^2}{5 R^2}}=\frac{5}{7} g \sin \theta ...(ii)\\
& \frac{a_c}{a_s}=\frac{2 \times 7}{3 \times 5}=\frac{14}{15}
\end{aligned}
$

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