Energy and Oscillations

So far, we have quantitatively described the motion of a simple harmonic oscillator. We have also briefly talked about the spring force and its relationship to the oscillating motion. As a reminder, if a spring is stretched or compressed beyond its equilibrium position, it exerts a force that tries to pull or push the object back to equilibrium. The direction of that force Ñ always attempting to restore equilibrium Ñ is what leads to oscillatory motion. The spring force is

F = -ky

where F in the equation is the spring force measured in Newtons; y is the distance from the spring's equilibrium point - positive values correspond to the spring getting stretched and negative values correspond to the spring getting compressed; and k is the spring constant, measuring the stiffness of the spring. The spring constant k must have units of N/m.

What we now want to do is to discuss energy considerations of a mass and spring system. This discussion will apply more generally to any simple harmonic oscillator (a swinging pendulum or an acoustic oscillator) where the force varies proportionally to the distance from equilibrium. Let's start by sketching the force on M as it is oscillates. Remember, there is an amplitude to the oscillation, y_max causing the mass to move back and forth between two points: y = +y_max (the maximum positive displacement) and y = -y_max (the maximum negative displacement).

Let's think about the oscillation. Consider the quarter cycle of the motion where the mass starts at the equlibrium position and is moving towards in the positive direction, stretching the spring. We know that the spring force does negative work on the mass because it is slowing down as it approaches its maximum displacement from equilibrium; i.e. y is increasing towards +y_max. The work isn't so easy to calculate because the force changes with position, getting more negative (i.e., a stronger force) thereby pulling harder on the mass towards the equilibrium as it moves.

To find the work, we need to find the average force exerted on the mass. This is very much like how we found the average speed of an object that has a constant acceleration, where the velocity is changing constantly with time. Here, the force is changing constantly with position. At equilibrium, the spring force is zero, at +y_max, the spring force is F_max = -ky_max. Since the force is changing linearly with position, the average force over this quarter cycle is half of the maximum force. So, the work done by the spring on the mass over this quarter cycle is

Work over quarter cycle = (average force)(distance) or

Work over quarter cycle = (1/2 F_max)(y_max) or

Work over quarter cycle = -1/2 k (y_max)²

We now have to ask if the work is recoverable. The answer should be clear - once the mass reaches +y_max, it comes to rest. It does this for just an instant; it then speeds up, gaining kinetic energy (KE) in the next quarter cycle of its motion, heading back to the equilibrium position. This means that the negative work done by the spring has reduced the KE of the mass, converting the energy into spring potential energy (PE) which can then be reconverted into KE, and so on. The end result is that we attribute a stored energy (PE) whenever a spring is displaced from its equilibrium by an amount y

spring potential energy = +1/2 ky²

The positive sign for the spring PE means that it increases whenever the spring is either compressed or stretched. Now, we can give a physicists description of oscillatory motion; we'll restrict ourselves to simple harmonic oscillations where the PE increases as the square of the distance of the object from its equilibrium.

1st quarter cycle The object starts at the equilibrium position moving towards positive y. At the equilibrium position it has its maximum value of KE. As it moves toward the maximum positive displacement, it slows down, reducing its KE. The energy is converted into PE: the PE of the system increases
2nd quarter cycle The object starts at its maximum positive displacement, for a brief instant, it is at rest. At that point its KE is zero and its PE is as large as it will be. The mass moves back towards equilibrium. As it moves away from the maximum displacement, its PE is decreased. The energy is converted into KE: the mass speeds up heading towards equilibrium.
3rd quarter cycle The object starts at the equilibrium position moving towards negative y. At the equilibrium position it has its maximum value of KE. As it moves toward the maximum positive displacement, it slows down, reducing its KE. The energy is converted into PE: the PE of the system increases
4th quarter cycle The object starts at its maximum negative displacement, for a brief instant, it is at rest. At that point its KE is zero and its PE is as large as it will be. The mass moves back towards equilibrium. As it moves away from the maximum negative displacement, its PE is decreased. The energy is converted into KE: the mass speeds up heading towards equilibrium.

The energy stored in the system can either be expressed by its maximum KE (at equilibrium) or by its maximum PE (at the maximum positive or negative displacement)

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Last updated: 14 Sep 1999Comments: bland@indiana.edu