Hello this is a question from A2 level mechanics, am struggling to get the book answer. Here goes:
A bead, of mass m is threaded onto a smooth circular ring, of radius r, which is fixed in a vertical plane. The bead is moving on the wire so that its speed at the lowest point of its path is four times its speed, v, at the highest point.
Find v in terms of r and g.
The usual technique is to work in terms of energy thus,
Loss in kinetic energy = gain in potential
1/2mv^2 - 1/2m(v/4)^2 = mgr
1/2m(v^2 - (v^2)/16) = mgr
(15/16)v^2 = 2gr
v^2 = 32gr/15
v = root(32gr/15)
But the book answer is v = root(4gr/15)
Any help offered is greatly appreciated.