Question

please explain the answer

10) A small mass m moves along a hoop of radius R without friction. Attached to the mass is a spring with spring constant k. The other end of the spring is attached to the bottom of the hoop. The equilibrium length of the spring is 0 and the force from the spring on the bead is kr. The mass is placed at the bottom of the hoop and given an initial speed to the right of to. What is the minimum value of vo so that the mass can make full revolution of the hoop?

Answer 1

2R m k N

Work done by the gravitational force, F, (= -mg), as the mass moves from the bottom (z = 0) to top of the hoop (z = 2R) can be calculated as 2R 2R U= S F, dz = | (-mg) dz = -mg[2]* = mg/2R] =-2mgr 0 0 (z denotes the vertical displacement of the mass) Similarly, the work done by the spring force, F = -kr) on the mass as the mass moves from the bottom (r = 0) to top of the hoop (r = 2R) can be calculated as 2R «. -14-16- (-kr) dz= -k [4R21 = -k 2 -2kR2 2 0 Initial kinetic energy of the mass is given by 1 T zmv> Since, it is asked to find the minimum initial velocity for the mass to reach the top of the hoop, we can set the kinetic energy (velocity) of the mass at the top of the hoop to zero T2 = 0 Applying the work-kinetic energy theorem T. + u=T, Or T+ Up + U = T2 Substituting the known expressions z muž – 2mgR – 2kR² = 0

Or 1 ż mvě = 2mgR + 2k R2 Or vo 4mgR + 4kR2 m Or 4(mgR+ kR2) = 2 gR + KR2 Vo = m m