15.7: Problem Solving: Energy in Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion in time and position, in which an object oscillates back and forth around an equilibrium position with a constant amplitude and frequency. In SHM, there is a continuous exchange between the potential and kinetic energy, which results in the oscillation of the object.

Consider the spring in a shock absorber of a car. The spring attached to the wheel executes simple harmonic motion while the car is moving on a bumpy road. The force on the spring is conservative, and the potential energy is stored when the spring is extended or compressed. In this case, the wheel attached to the spring oscillates in one dimension, with the force of the spring acting parallel to the motion. At the equilibrium position, the potential energy stored in the spring is zero. If there are no dissipative forces, the total energy is the sum of the potential energy and the kinetic energy and is expressed as follows:

The total energy in simple harmonic motion remains conserved for the system at every point during the motion and is proportional to the square of the amplitude.

The total energy equation in simple harmonic motion presents a useful relationship between velocity, position, and total mechanical energy. This equation can be used if the problem requires a relation between position, velocity, and acceleration without reference to time. Since the energy conservation equation involves displacement and velocity, one must infer the signs of the displacement and velocity from the situation. For instance, if the body moves from the equilibrium position toward the point of the greatest positive displacement, the displacement and velocity are positive.

Studying the energy in simple harmonic motion is vital for understanding the behavior of oscillating systems in physics and engineering.

Consider a car moving over a bumpy road. Here, the suspension of the car acts like a spring connecting the mass of the car to the wheels.

During its motion, the spring compresses and expands, executing simple harmonic motion.

Suppose the car bounces with a certain amplitude; what is its maximum velocity, and how does its total energy change if it is halfway between its initial and equilibrium positions?

The total energy in simple harmonic motion remains conserved. Rearrangement of terms gives the velocity at any position during the motion.

The maximum velocity occurs when the car's spring passes through the equilibrium position. On substituting the known quantities, the maximum velocity is obtained. The total energy at the equilibrium position is then obtained by substituting the known and calculated quantities.

When the car's spring moves from maximum to half amplitude, the velocity is considered negative. The total energy is calculated by substituting the velocity with other known quantities.

In both cases, the total energy of simple harmonic motion is the same.