a. When the displacement of a mass on a spring is (1)/(2) A, what fraction of the mechanical ene

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A. When the displacement of a mass on a spring is (1)/(2) A,...

a. When the displacement of a mass on a spring is $\frac{1}{2} A,$ what fraction of the mechanical energy is kinetic energy and what fraction is potential energy?
b. At what displacement, as a fraction of $A$, is the energy half kinetic and half potential?

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of systems like a mass attached to a spring, where the restoring force is directly proportional to the displacement and acts in the opposite direction. This leads to sinusoidal motion where the mass continuously exchanges energy between kinetic and potential forms as it moves through its equilibrium position and toward its turning points.

Conservation of Mechanical Energy

In an ideal oscillatory system with no external damping, the total mechanical energy remains constant throughout the motion. This energy is partitioned between kinetic energy, which is highest at the equilibrium position, and potential energy, which is highest at the extreme points of motion. Understanding this conservation principle is crucial when analyzing energy fractions at various displacements.

Potential and Kinetic Energy Expressions in a Mass-Spring System

In a mass-spring system, the potential energy stored in the spring is given by (1/2) k x², where x is the displacement from equilibrium, while the kinetic energy is given by (1/2) m v². At maximum displacement, the kinetic energy is zero and all the energy is stored as potential energy; conversely, at the equilibrium position, the potential energy is zero and the system’s energy is entirely kinetic. These relationships allow determination of energy fractions at any point during the motion.

Energy Partitioning at Specific Displacements

The analysis of how energy is divided in a mass-spring system at a specific displacement involves substituting the displacement into the potential energy expression and comparing it to the total mechanical energy. For example, if the displacement is a known fraction of the maximum amplitude, one can determine the corresponding fraction of potential energy (and hence kinetic energy) using the formula for potential energy. Setting the potential and kinetic energies equal allows the determination of the displacement at which the energies are equally divided.

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