A solid cylinder of radius R and mass M rolls | vedclass

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

Question

(D) For a body rolling down an inclined plane of height $h$ without slipping,the conservation of energy gives: $Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I\o

Vedclass · Accepted Answer

$A$ hollow cylinder of same mass and same radius

Vedclass · Answer

(D) For a body rolling down an inclined plane of height $h$ without slipping,the conservation of energy gives: $Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$. Since $v = R\omega$,we have $\omega = v/R$. Substituting this,$Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I(v/R)^2 = \frac{1}{2}Mv^2(1 + \frac{I}{MR^2})$. For a solid cylinder,the moment of inertia $I = \frac{1}{2}MR^2$,so $Mgh = \frac{1}{2}Mv^2(1 + 1/2) = \frac{3}{4}Mv^2$,which gives $v = \sqrt{4gh/3}$. The speed $v$ depends on the factor $k^2/R^2$ where $I = Mk^2$. Specifically,$v = \sqrt{\frac{2gh}{1 + k^2/R^2}}$. For a hollow cylinder,$I = MR^2$,so $k^2/R^2 = 1$. Thus,$v_{hollow} = \sqrt{\frac{2gh}{1+1}} = \sqrt{gh}$. Since $\sqrt{gh} < \sqrt{4gh/3}$,the speed is less for a hollow cylinder. The speed is independent of mass $M$ and radius $R$ for the same geometry.

$A$ solid cylinder of radius $R$ and mass $M$ rolls down an inclined plane without slipping and reaches the bottom with a speed $v$. The speed would be less than $v$ if we use:

Similar Questions

Solid cylinders of radii $r_1, r_2$ and $r_3$ roll down an inclined plane from the same place simultaneously. If $r_1 > r_2 > r_3$,which one would reach the bottom first?

When a uniform solid sphere and a disc of the same mass and of the same radius roll down a rough inclined plane from rest to the same distance,then the ratio of the time taken by them is

$A$ solid sphere of mass $M$ and radius $R$ is rolling down an inclined plane of length $L$ and height $h$ without slipping. What will be the velocity of its center of mass upon reaching the bottom?

$A$ solid cylinder and a hollow cylinder,both of the same mass and same external diameter,are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?

Vedclass Test Series

Exam Paper Generator

Online Exam Module