(A) The time taken by an object to roll down an inclined plane is given by $t = \sqrt{\frac{2h}{g \sin^2 \theta} (1 + \frac{K^2}{R^2})}$,where $K$ is

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The solid sphere will reach the bottom first.

(A) The time taken by an object to roll down an inclined plane is given by $t = \sqrt{\frac{2h}{g \sin^2 \theta} (1 + \frac{K^2}{R^2})}$,where $K$ is the radius of gyration and $R$ is the radius of the object.For a solid sphere,$\frac{K^2}{R^2} = \frac{2}{5} = 0.4$.For a disc,$\frac{K^2}{R^2} = \frac{1}{2} = 0.5$.For a solid cylinder,$\frac{K^2}{R^2} = \frac{1}{2} = 0.5$.Since the solid sphere has the minimum value of $\frac{K^2}{R^2}$,it will have the minimum time of descent.Therefore,the solid sphere reaches the bottom first.

Class 11 Physics System of Particles and Rotational Motion Rolling On Inclined Plane

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$A$ solid sphere,a disc,and a solid cylinder start rolling down an inclined plane from rest. All three objects are made of the same material and have the same mass. Then,

A
The solid sphere will reach the bottom first.
B
The solid sphere will reach the bottom last.
C
The disc will reach the bottom first.
D
All objects will reach the bottom at the same time.

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System of Particles and Rotational Motion

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Rolling On Inclined Plane

$A$ solid cylinder of mass $M$ and radius $R$ rolls down a smooth inclined plane about its own axis and reaches the bottom with velocity $v$. What is the height of the inclined plane? (Assume $g$ is the acceleration due to gravity.)

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Two identical solid cylinders run a race starting from rest at the top of an inclined plane. If one cylinder slides and the other rolls,which of the following statements is true?

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$A$ solid sphere is rolling on a frictionless surface,as shown in the figure,with a translational velocity $v \, m/s$. If it is to climb the inclined surface to a height $h$,then $v$ should be:

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$A$ disc of mass $M$ and radius $R$ is rolling on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v$,what is the maximum height $h$ reached by the disc?

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$A$ solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length $60\,cm$. If the cylinder rolls without slipping,its speed upon reaching the bottom of the inclined plane is $...........\,ms^{-1}$. (Given $g = 10\,ms^{-2}$)

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$A$ solid sphere,a disc,and a solid cylinder start rolling down an inclined plane from rest. All three objects are made of the same material and have the same mass. Then,

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$A$ solid cylinder of mass $M$ and radius $R$ rolls down a smooth inclined plane about its own axis and reaches the bottom with velocity $v$. What is the height of the inclined plane? (Assume $g$ is the acceleration due to gravity.)

Two identical solid cylinders run a race starting from rest at the top of an inclined plane. If one cylinder slides and the other rolls,which of the following statements is true?

$A$ solid sphere is rolling on a frictionless surface,as shown in the figure,with a translational velocity $v \, m/s$. If it is to climb the inclined surface to a height $h$,then $v$ should be:

$A$ disc of mass $M$ and radius $R$ is rolling on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v$,what is the maximum height $h$ reached by the disc?

$A$ solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length $60\,cm$. If the cylinder rolls without slipping,its speed upon reaching the bottom of the inclined plane is $...........\,ms^{-1}$. (Given $g = 10\,ms^{-2}$)

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