(A) The velocity $v$ of a body rolling down an inclined plane of height $h$ is given by the formula:$v = \sqrt{\frac{2gh}{1 + \frac{K^2}{R^2}}}$Here,$

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$\sqrt{\frac{4gL\sin\theta}{3}}$

(A) The velocity $v$ of a body rolling down an inclined plane of height $h$ is given by the formula:$v = \sqrt{\frac{2gh}{1 + \frac{K^2}{R^2}}}$Here,$h = L \sin\theta$ and for a disc,the radius of gyration $K$ is given by $K^2 = \frac{R^2}{2}$,so $\frac{K^2}{R^2} = \frac{1}{2}$.Substituting these values into the formula:$v = \sqrt{\frac{2g(L \sin\theta)}{1 + \frac{1}{2}}}$$v = \sqrt{\frac{2gL \sin\theta}{\frac{3}{2}}}$$v = \sqrt{\frac{4gL \sin\theta}{3}}$

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

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$A$ disc rolls without slipping down an inclined plane of length $L$ and inclination $\theta$. What will be its velocity at the bottom?

A
$\sqrt{\frac{4gL\sin\theta}{3}}$
B
$\sqrt{\frac{2gL\sin\theta}{3}}$
C
$\sqrt{\frac{10gL\sin\theta}{7}}$
D
$\sqrt{4gL\sin\theta}$

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System of Particles and Rotational Motion

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Rolling On Inclined Plane

$A$ solid sphere is rolling down an inclined plane without slipping. The ratio of its rotational kinetic energy to its total kinetic energy is:

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$A$ ball is rolling without slipping. The radius of gyration of the ball about an axis passing through its center of mass is $K$. If the radius of the ball is $R$,what fraction of the total energy is rotational kinetic energy?

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$A$ thin hollow cylinder is open at both ends.
$(i)$ It slides without rotation.
(ii) It rolls without slipping.
If the speed is the same in both cases,the ratio of the kinetic energies is:

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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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If a solid sphere is rolling,the ratio of its rotational kinetic energy to the total kinetic energy is given by

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$A$ disc rolls without slipping down an inclined plane of length $L$ and inclination $\theta$. What will be its velocity at the bottom?

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$A$ solid sphere is rolling down an inclined plane without slipping. The ratio of its rotational kinetic energy to its total kinetic energy is:

$A$ ball is rolling without slipping. The radius of gyration of the ball about an axis passing through its center of mass is $K$. If the radius of the ball is $R$,what fraction of the total energy is rotational kinetic energy?

$A$ thin hollow cylinder is open at both ends.$(i)$ It slides without rotation.(ii) It rolls without slipping.If the speed is the same in both cases,the ratio of the kinetic energies is:

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?

If a solid sphere is rolling,the ratio of its rotational kinetic energy to the total kinetic energy is given by

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$A$ thin hollow cylinder is open at both ends.
$(i)$ It slides without rotation.
(ii) It rolls without slipping.
If the speed is the same in both cases,the ratio of the kinetic energies is: