The small cylinder of mass m and radius r is confined to rol

The small cylinder of mass m and radius r is confined to roll on the circular surface of radius R. By the methods of potential energy, prove that the cylinder is unstable in case (a) and stable in case (b).

Solution

Gravitational potential energy is energy stored in body due to its vertical position or height , it is calculated by multiplying force of gravity by height attained by object (h)

force of gravity = mass of object * acceleration due to gravity (g)= mg

potential energy (PE) = m*g*h

total energy of the system is the sum of kinectic energy and potential energy , The tendency of a system to reach equilibrium(stable) at a local minimum of the potential energy is termed as the principle of least potential energy

at condition (a) the total energy is the due to kinectic energy(KE) and potential energy(PE) due to height R (radius of curved surface

PE = m(mass of mass of small cylinder ) * g* R

total energy = KE+mgR

at condition (b) the cylinder in on the bottom of curved surface so the PE due to height is zero hence only KE is acting on the ball

total energy = KE

the small cylinder is stable when total energy and gravitational potential energy acting on it is low

at (a) total enery and potential energy is high so it is unstable

at (b) total energy and potential energy is low so it is stable