A solid sphere of mass m and radius r is released from rest from the given position. If it rolls without sliding on the circular track of radius R, its speed when it reaches its lowest position will be

Question

A solid sphere of mass m and radius r is released from rest from the given position. If it rolls without sliding on the circular track of radius R, its speed when it reaches its lowest position will be (A) √(8/7) g(R-r) (B) √(10/7) g(R-r) (C) √(10/9) g(R-r) (D) √(5/7) g(R-r)

Tardigrade Admin · Accepted Answer

When the sphere reaches its lowest position it loses gravitational potential energy by

Δ U

and gains a kinetic energy by

Δ K

, where

Δ U = - m g (R - r)

and

Δ K = \frac{1}{2} m v_{c}^{2} (1 + \frac{K ^{2}}{R ^{2}}) = \frac{1}{2} m v^{2} (1 + \frac{2}{5}) = \frac{7}{10} m v^{2}

According to conservation of energy,

Δ U + Δ K = 0

or

- m g (R - r) + \frac{7}{10} m v^{2} = 0

or

v = \frac{10}{7} g (R - r)

Q. A solid sphere of mass $m$ and radius $r$ is released from rest from the given position. If it rolls without sliding on the circular track of radius $R$ , its speed when it reaches its lowest position will be

$\frac{8}{7} g (R - r)$

$\frac{10}{7} g (R - r)$

$\frac{10}{9} g (R - r)$

$\frac{5}{7} g (R - r)$

Q. A solid sphere of mass m and radius r is released from rest from the given position. If it rolls without sliding on the circular track of radius R, its speed when it reaches its lowest position will be

78​g(R−r)​

710​g(R−r)​

910​g(R−r)​

75​g(R−r)​

Q. A solid sphere of mass $m$ and radius $r$ is released from rest from the given position. If it rolls without sliding on the circular track of radius $R$ , its speed when it reaches its lowest position will be

$\frac{8}{7} g (R - r)$

$\frac{10}{7} g (R - r)$

$\frac{10}{9} g (R - r)$

$\frac{5}{7} g (R - r)$