The speed of a homogeneous solid sphere after | vedclass

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

Question

(A) For a solid sphere,the moment of inertia about its center is $I = \frac{2}{5} m R^2$.According to the work-energy theorem,the potential energy los

Vedclass · Accepted Answer

$\sqrt{\frac{10}{7}gh}$

Vedclass · Answer

(A) For a solid sphere,the moment of inertia about its center is $I = \frac{2}{5} m R^2$.According to the work-energy theorem,the potential energy lost equals the sum of translational and rotational kinetic energy: $mgh = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$.Since the sphere rolls without sliding,the condition for pure rolling is $v = R\omega$,which implies $\omega = \frac{v}{R}$.Substituting the values into the energy equation: $mgh = \frac{1}{2} m v^2 + \frac{1}{2} (\frac{2}{5} m R^2) (\frac{v}{R})^2$.Simplifying the expression: $mgh = \frac{1}{2} m v^2 + \frac{1}{5} m v^2$.$mgh = (\frac{1}{2} + \frac{1}{5}) m v^2 = \frac{7}{10} m v^2$.Solving for $v$: $v^2 = \frac{10}{7} gh$,therefore $v = \sqrt{\frac{10}{7} gh}$.

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

Similar Questions

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?

$A$ body of mass $m$ slides down an incline and reaches the bottom with a velocity $V$. If the same mass were in the form of a disc which rolls down this same incline,the velocity of the disc at the bottom would be

$A$ cylinder is pure rolling up an incline plane. It stops momentarily and then rolls back. The force of friction

$A$ 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?

$A$ 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?

Vedclass Test Series

Exam Paper Generator

Online Exam Module