Three bodies , a ring, a solid cylinder and a | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

According to conservation of mechanical

energy

$\mathrm{mgh}=\frac{1}{2} \mathrm{mv}^{2}\left(1+\frac{\mathrm{k}^{2}}{\mathrm{R}^{2}}\right)$

Vedclass · Accepted Answer

Solid sphere

Vedclass · Answer

According to conservation of mechanical

energy

$\mathrm{mgh}=\frac{1}{2} \mathrm{mv}^{2}\left(1+\frac{\mathrm{k}^{2}}{\mathrm{R}^{2}}ight)$

or $v^{2}=\left(\frac{2 g h}{1+\frac{k^{2}}{R^{2}}}ight)$

$Note $$:$ It is independent of the mass of the rolling body.

For a ring. $\mathrm{K}^{-2}=\mathrm{R}^{2}$

$v_{ring}=\sqrt{\frac{2 g h}{1+1}}=\sqrt{g h}$

For a solid cylinder, $\mathrm{k}^{-2}=\frac{\mathrm{R}^{2}}{2}$

$V_{	ext {cylinder }}=\sqrt{\frac{2 \mathrm{gh}}{1+\frac{1}{2}}}=\sqrt{\frac{4 \mathrm{gh}}{3}}$

For a solid sphere, $\mathrm{k}^{-2}=\frac{2}{5} \mathrm{R}^{2}$

$V_{	ext {sphere }}=\sqrt{\frac{2 g h}{1+\frac{2}{5}}}=\sqrt{\frac{10 g h}{7}}$

Among the given three bodies the solid sphere has the greatest and the ring has the least velocity at the bottom of the inclined plane.

Three bodies , a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are identical. Which of the bodies reaches the ground with maximum velocity?

Similar Questions

The centre of mass of a body

The moment of inertia of a thin circular lamina of mass $1\,kg$ and diameter $0.2\,metre$ rotating about one of its diameter is

Two particles having mass $'M'$ and $'m'$ are moving in circular paths having radii $R$ and $ r.$ If their time periods are same then the ratio of their angular velocities will

A solid cylinder rolls without slipping down an inclined plane of height $h$. The velocity of the cylinder when it reaches the bottom is

Ratio of masses and radii of two circular rings are $1 : 2$ and $2 : 1$ respectively then ratio of moment of inertia will be