Question

A stationary marble with mass M rolls down a hill with height H. Determine the velocity of the marble at the bottom of the ramp.

Answer 1

Answer:$v=\sqrt{\frac{10}{7}gH}$

Explanation:

Given

Marble rolls down the hill

Considering it to be a solid sphere of radius R

Conserving the total Energy as it is constant

$E_0=E_f\quad [\text{as initial energy=Final energy}]$

$MgH=\frac{1}{2}Mv^2+\frac{1}{2}I\omega ^2$

where I=moment of inertia of marble

$(\frac{2}{5}MR^2)$

Considering pure rolling  

so $v=\omega \times R$

$MgH=\frac{1}{2}Mv^2+\frac{1}{2}\times \frac{2}{5}MR^2\times (\frac{v}{R})^2$

$gH=\frac{7}{10}v^2$

$v=\sqrt{\frac{10}{7}gH}$