Practice Applying Simple Harmonic Motion Formulas

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  • 0:04 Harmonic Motion: Overview
  • 0:36 Potential Energy
  • 2:48 Kinetic Energy
  • 5:43 Calculating the Period
  • 7:59 Calculating Maximum Velocity
  • 8:55 Lesson Summary
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Lesson Transcript
Instructor: Laura Foist

Laura has a Masters of Science in Food Science and Human Nutrition and has taught college Science.

The simple harmonic motion formulas can be used to determine the kinetic and potential energies, period, and maximum velocity of a spring in motion. In this lesson, we'll look at some examples of these formulas at work.

Harmonic Motion: Overview

When a spring is compressed and bouncing up and down, the height of the spring and the time passed can be graphed with the motion repeating sinusoidally. In other words, the graph follows the same pattern as a sine graph:

The sine graph completes one cycle at 2 pi.
sine graph

On a graph following the movement of a spring, the y-axis is the position of the spring and the x-axis is the time that has passed. Using this graph, we can calculate the potential energy, kinetic energy, period (pr, time to complete one oscillation), and maximum velocity.

Potential Energy

The equation to calculate potential energy for a spring, or elastic potential energy, is:

Potential energy formula

PE = potential energy

k = Hooke's spring constant (specific for each spring)

A = amplitude (position of the spring and how far it has been stretched)

Hooke's constant changes, depending on the material, shape, and size of the spring, and is equal to the force divided by the distance a spring is stretched. The dimensions are N/m (Newtons per meter).

Let's say that you're asked to calculate the potential energy of a spring with an amplitude of 3 meters. This spring has a Hooke's constant of 10 N/m:

Potential energy example

So the potential energy is 45 Joules (J).

Sometimes, you may need to calculate Hooke's constant. For example, let's say you use a force of 5 Newtons to stretch a spring out to its full length of 20 cm. What is the potential energy of this spring?

PE = ?

A = 0.2 m

k = F / A = 5 N / 0.2 m = 25 N/m

Notice that we calculated Hooke's constant by taking the force used divided by the distance stretched.

Now we can calculate the potential energy:

PE example equation

This spring has a potential energy of 0.5 J.

Note that when we're referring to the ''potential energy of a spring,'' we're referring to its maximum potential energy. However, the potential energy can be calculated at any point during an oscillation.

Kinetic Energy

The equation to calculate kinetic energy is:

Kinetic energy formula

This formula probably looks really familiar, because we use it to calculate the kinetic energy of other objects, such as cars, in motion. As long as we know the mass of a spring and how fast it is going, we can calculate the kinetic energy of the spring in that instant.

For example, what is the kinetic energy of a spring with a mass of 536 grams traveling at 5 m/s?

KE = ?

m = 0.536 kg

v = 5 m/s

KE example equation

Here, the kinetic energy of the spring is 6.7 J.

Typically, we don't know the mass of the spring. So we may need to use the potential energy to calculate the kinetic energy:

Energy final and initial

If we know that a spring with a Hooke's constant of 85 N/m can stretch to a total of 12 cm, what is the kinetic energy when it is at the halfway point?

KE = ?

k = 85 N/m

A1 = 0.12 m

A2 = 0.06 m

When a spring is stretched as far as possible there is a moment when it has no velocity, which means that the initial kinetic energy is 0 J. If we use the initial and final kinetic/potential energies formula, we can calculate the potential energies to calculate the final kinetic energy:

Potential energy initial

The initial potential energy is 0.612 J.

Final potential formula

The final potential energy is 0.153J.

Final kinetic energy

The final kinetic energy is 0.459 J.

Calculating the Period

The formula for calculating the period, or time to complete one oscillation, is:

Period formula

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