As shown in the figure, a smooth track includes a circular section of radius R . A block of mass m is pushed against a spring having a spring constant k , which is fixed at its left end, and is subsequently released. Determine the initial compression of the spring required for the block to press against the track with a force mg upon reaching point P , where the track's radius is horizontal.

Step-by-step solution

Let the initial compression of the spring be x . According to the law of conservation of mechanical energy, the initial elastic potential energy of the spring is converted into the gravitational potential energy and kinetic energy of the block at point P . 1 2 kx^2 = mgR + 1 2 mv^2 At point P , the radius is horizontal, so the normal force N provided by the track acts horizontally towards the center of the circular path. This normal force provides the necessary centripetal force. N = mv^2 R Given that the block pre