Figure (8-E15) shows a smooth track, a part of which is a...

Figure (8-E15) shows a smooth track, a part of which is a circle of radius $R .$ A block of mass $m$ is pushed against a spring of spring constant $k$ fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force $m g$ when it reaches the point $P$, where the radius of the track is horizontal.

Key Concepts

Conservation of Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. In problems like this, the energy stored in a compressed spring is converted into kinetic energy and, if applicable, gravitational potential energy as the object moves, allowing one to relate the initial spring compression to the speed of the object at a different point along its path.

Centripetal Force

Centripetal force is the net force required to keep an object moving along a circular path, directed toward the center of the circle. In this context, it is crucial for determining the conditions that allow the block to follow the curved track; specifically, the condition that the force with which the block presses against the track provides the necessary centripetal acceleration.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is either compressed or stretched from its equilibrium position, given by the formula 1/2 k x², where k is the spring constant and x is the displacement. This energy is the initial source that propels the block, and its precise value is important when equating it to the kinetic energy of the block later in its motion.

Normal Force in Circular Motion

Normal force is the perpendicular force exerted by a surface on an object in contact with it. In circular motion problems, it often contributes to the net force required for centripetal acceleration. Here, specifying that the block presses against the track with a force equal to its weight (mg) sets a particular condition that helps link the dynamics of the motion with the energy transformations occurring from the release of the spring.

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