(B) For a body rolling without slipping,the total kinetic energy $E_{total}$ is the sum of translational kinetic energy $E_t$ and rotational kinetic e

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(B) For a body rolling without slipping,the total kinetic energy $E_{total}$ is the sum of translational kinetic energy $E_t$ and rotational kinetic energy $E_r$.$E_t = \frac{1}{2}mv^2$$E_r = \frac{1}{2}I\omega^2 = \frac{1}{2}(mK^2)(\frac{v}{R})^2 = \frac{1}{2}mv^2(\frac{K^2}{R^2})$$E_{total} = E_t + E_r = \frac{1}{2}mv^2(1 + \frac{K^2}{R^2})$The fraction of total energy that is rotational is given by:$\frac{E_r}{E_{total}} = \frac{\frac{1}{2}mv^2(\frac{K^2}{R^2})}{\frac{1}{2}mv^2(1 + \frac{K^2}{R^2})}$$= \frac{K^2/R^2}{(R^2 + K^2)/R^2} = \frac{K^2}{K^2 + R^2}$

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

Medium

$A$ 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
$K^2 / R^2$
B
$K^2 / (K^2 + R^2)$
C
$R^2 / (K^2 + R^2)$
D
$(K^2 + R^2) / R^2$

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System of Particles and Rotational Motion

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Rolling On Inclined Plane

$A$ 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, then the speed when it reaches the bottom is (in $\,m/s$)

DifficultTS EAMCET 2020

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$A$ 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?

MediumAIPMT 2010

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$A$ uniform sphere of radius $R$ and mass $m$ is placed on an inclined plane which makes an angle $45^{\circ}$ to the horizontal. For which of the following values of the coefficient of friction does the sphere roll without slipping? Select the incorrect option.

MediumTS EAMCET 2019

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$A$ solid sphere of mass $4 \ kg$ and radius $28 \ cm$ is on an inclined plane. If the acceleration of the sphere when it rolls down without sliding is $3.5 \ m \ s^{-2}$,then the acceleration of the sphere when it slides down without rolling is (in $m \ s^{-2}$)

MediumAP EAMCET 2025

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The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$,from rest without sliding,is

DifficultAIPMT 1992

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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?

Similar Questions

$A$ 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, then the speed when it reaches the bottom is (in $\,m/s$)

$A$ 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?

$A$ uniform sphere of radius $R$ and mass $m$ is placed on an inclined plane which makes an angle $45^{\circ}$ to the horizontal. For which of the following values of the coefficient of friction does the sphere roll without slipping? Select the incorrect option.

$A$ solid sphere of mass $4 \ kg$ and radius $28 \ cm$ is on an inclined plane. If the acceleration of the sphere when it rolls down without sliding is $3.5 \ m \ s^{-2}$,then the acceleration of the sphere when it slides down without rolling is (in $m \ s^{-2}$)

The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$,from rest without sliding,is

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