Question

A small block of mass m slides along the frictionless loop the loop track shown below. If it starts from rest at point A, what is the speed of the block at point B? (v = squareroot (10 g R)) What is the net force acting on the block at point C? (Don't forget the gravitational force. (F = -mg (8i + j) At what height above the bottom should the block be released so that the normal force exerted on it by the track at the top of the loop is equal to its weight? (h = 3R) A small mass, m starts from rest and slides along the surface of a frictionless, solid hemisphere of radius r as shown. Show that the angle at which the mass loses contact with the hemisphere is given by Theta = cos^-1 2/3 (Note that this is equivalent to showing that the height at which the mass loses contact with the hemisphere is h = (2/3)r)

Answer 1

a) Conserving energy 5 mgR = 1/2 mr^2 v = squareroot 10 gR b) conserving energy 5 mgR = 1/2 mv^2_1 + mgR v_1 = squareroot 8 gR F_net = N (- i) + mg (- j) N = mv^2/R F_net = - mg (8i + j) c) N + mg = mv^2/R N = my v = squareroot 2gR conserving energy mgh = 2mg R = 1/2 m v^2 h = 3 R N = 0 mv^2/R = mg cos theta rightarrow (1) mg R (1 - cos theta) = mv^2/2 rightarrow (2) theta = cos^- 1 2/3