A solid sphere rolls without slipping along the track shown in the figure. The sphere starts from rest from a height h above the bottom of the loop of radius R which is much larger than the radius of the sphere r . The minimum value of h for the sphere to complete the loop is

Question

A solid sphere rolls without slipping along the track shown in the figure. The sphere starts from rest from a height h above the bottom of the loop of radius R which is much larger than the radius of the sphere r . The minimum value of h for the sphere to complete the loop is <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-flhntgaezhyephyx.jpg" /> (A) 2.1 R (B) 2.3 R (C) 2.7 R (D) 2.5 R

Tardigrade Admin · Accepted Answer

At the topmost point of the loop minimum value of the linear speed of the centre of the sphere should be :

v = g R

or translational kinetic energy

K_{T} = \frac{1}{2} m v^{2} = \frac{1}{2} m g R

.
In case of pure rolling of a solid sphere the ratio of rotational to translational kinetic energy is :

\frac{K _{R}}{K _{T}} = \frac{2}{5}

∴

Total kinetic energy at topmost point should be :

K = \frac{5 + 2}{5} \cdot (K)_{T} = \frac{7}{5} (\frac{1}{2} m g R) = \frac{7}{10} m g R

Now from conservation of mechanical energy:

\frac{7}{10} m g R = m g (h - 2 R)

∴ h = 2.7 R

Q. A solid sphere rolls without slipping along the track shown in the figure. The sphere starts from rest from a height h above the bottom of the loop of radius R which is much larger than the radius of the sphere r . The minimum value of h for the sphere to complete the loop is

2.1 R

2.3 R

2.7 R

2.5 R

Q. A solid sphere rolls without slipping along the track shown in the figure. The sphere starts from rest from a height $h$ above the bottom of the loop of radius $R$ which is much larger than the radius of the sphere $r$ . The minimum value of $h$ for the sphere to complete the loop is