# Simple Harmonic Motion: Kinetic Energy & Potential Energy

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• 0:05 What is Simple…
• 0:42 Energy in Simple…
• 1:48 Equations
• 2:53 Example Problem
• 4:08 Lesson Summary

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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video, you will be able to explain what simple harmonic motion is, describe the energy interplay involved in various situations, and use energy equations to solve simple harmonic motion problems. A short quiz will follow.

## What Is Simple Harmonic Motion?

Simple harmonic motion is any periodic, repetitive motion where a restoring force is applied that is proportional to the displacement, in the opposite direction of that displacement. For example, when you pull a spring, the force trying to yank the spring back to its original position increases and increases. If you pull that spring out and let go, it will bounce back and forth around a middle, equilibrium position. This motion back and forth is simple harmonic motion.

During simple harmonic motion, the velocity is greatest at the equilibrium position, and the acceleration (and net force) is greatest at the outer edges - the maximum displacement, or amplitude.

## Energy in Simple Harmonic Motion

Simple harmonic motion also involves an interplay between different types of energy: potential energy and kinetic energy. Potential energy is stored energy, whether stored in gravitational fields or stretched elastic materials. And kinetic energy is the energy any moving object has, which is proportional to the square of the velocity of that object.

When a horizontal spring oscillates back and forth, it's an interplay between elastic potential energy and kinetic energy. When a pendulum swings, it's an interplay between gravitational potential energy and kinetic energy. And when a vertical spring is oscillating, both gravitational and elastic potential energy are involved, with kinetic energy in the middle.

But whatever the specific types of potential and kinetic energy, the variation is similar mathematically. It can be described using this graph:

The total energy remains constant, but one type of energy goes up, while the other goes down. This constant total energy shows how energy in a closed system is never created or destroyed; it only moves from one place to another. This is called the law of conservation of energy.

## Equations

The energy equation for simple harmonic motion varies, depending on the exact circumstances. Kinetic energy is 1/2mv^2, where m is the mass of the object, and v is the velocity of the object. Gravitational potential energy is mgh, where g is the acceleration due to gravity, and h is the height of the object above the ground. And elastic potential energy is 1\2kx^2, where k is the spring constant (or stiffness) of the spring, and x is the displacement of the spring from its equilibrium position. The way you combine these equations together is based on the situation.

To figure this out, you need to look at the situation before and after. For example, with a horizontal spring, you can say that the kinetic energy in the middle of the motion is equal to the elastic potential energy at full stretch. So 1/2mv^2 in the middle equals 1\2kx^2 at the edges. Or for a pendulum, mgh at the top equals 1/2mv^2 in the middle. So whatever kind of energy is involved, you write out an equation, plug numbers in, and solve.

## Example Problem

Maybe this would be easier if we went through an example. Let's say a horizontal mass on a spring is oscillating such that it has a velocity of 3 meters per second when it passes through the equilibrium position. If the spring constant of the spring is 0.6, and the amplitude of the oscillation is 2 meters, what mass is attached to the spring?

First of all, we should write out what we know. The velocity v is 3, the spring constant, k is 0.6, and the amplitude A is 2. And we're trying to find m, so m equals question mark.

Now we need to set up our equation. The spring constant is horizontal, so gravitational potential energy isn't involved here. This situation is just an interplay between elastic potential energy and kinetic energy. So 1/2mv^2 equals 1\2kx^2. There isn't an amplitude in the equation, but when the spring is stretched to its maximum, the extension of the spring, x, will be equal to the amplitude of the oscillation. So A is really x in this equation. That means we have all we need to solve for m.

Do some algebraic rearrangement, plug numbers in, and solve for m, and we get 0.267 kilograms. And that's it; we're done.

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