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Home >> AP Physics I >> Simple Harmonic Motion
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Simple harmonic motion (SHM) is the oscillation of springs
and pendulums. The motion follows a sinusoidal pattern. The term “simple” comes from the fact that it
is the most basic of all modes of oscillation, which can get very complicated
indeed! In this context, the term “harmonic”
means a regular oscillation. SHM is found in a surprisingly large number of situations. For the AP course we consider the oscillation of the pendulum and that of a spring. The regularity of these vibrations led to the development of the first truly accurate clocks - first using a pendulum and later a spring mechanism.
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The YouTube clip shows a mass-spring oscillator alongside an object moving in a circle at constant angular speed. Note that the vertical position of both the objects is the same as viewed from the side. At an advanced level we can use the similarities between SHM and circular motion to accurately model oscillations.
The objectives for the AP course are:
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Assignments 2018
A Mass on a Spring Oscillator
The energy that drives the oscillation of a spring comes from stretching the spring in the first place. We met this back in Unit 5: Work, Energy and Power as the elastic potential energy. The object at the end of the spring achieves its maximum speed when all of this potential energy has been transferred to kinetic energy. This also gives us a relationship between the stiffness of the spring (spring constant, k), the amplitude, x, and the maximum speed, v, of the mass.
From the principle of conservation of energy:
\[EPE_{lost} = KE_{gained}\]
\[\frac{1}{2}kx^{2}=\frac{1}{2}mv^{2}\]
\[v = \sqrt{\frac{k}{m}}x\]
Note: this equation is only valid for a horizontal oscillator. If the spring is mounted vertically, then gravitational potential energy becomes a factor as well!
\[GPE_{top}+EPE= GPE_{middle} + KE = GPE_{bottom}+EPE\]
The energy that drives the oscillation of a spring comes from stretching the spring in the first place. We met this back in Unit 5: Work, Energy and Power as the elastic potential energy. The object at the end of the spring achieves its maximum speed when all of this potential energy has been transferred to kinetic energy. This also gives us a relationship between the stiffness of the spring (spring constant, k), the amplitude, x, and the maximum speed, v, of the mass.
From the principle of conservation of energy:
\[EPE_{lost} = KE_{gained}\]
\[\frac{1}{2}kx^{2}=\frac{1}{2}mv^{2}\]
\[v = \sqrt{\frac{k}{m}}x\]
Note: this equation is only valid for a horizontal oscillator. If the spring is mounted vertically, then gravitational potential energy becomes a factor as well!
\[GPE_{top}+EPE= GPE_{middle} + KE = GPE_{bottom}+EPE\]
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The energy transfers for a spring oscillator. Image from hyperphysics. This page has a simulation that enables you to change parameters and see the effect on the oscillation.
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The period, T, of an oscillator is the time taken to complete one oscillation back and forth. We will be carrying out a detailed lab on this as well as deriving the relationship from first principles. Essentially the period (and hence the frequency) are governed by a) the mass being oscillated and b) the stiffness of the spring. The frequency is the number of oscillations per second and is simply the reciprocal of the period.
\[T = 2\pi \sqrt{\frac{m}{k}}\]
\[T = 2\pi \sqrt{\frac{m}{k}}\]
Pendulums
A simple pendulum is a mass hanging on the end of a string that is fixed at the top so that it can swing. The equations that model pendulums are only really valid for small angles of swing. As the swing angle increases above 10 degrees, the equations start to become inaccurate. The person that first noticed the constancy of the period of a pendulum was a very bored Galileo sitting in a church watching incense burners. These must have been on very long chains as his only method of timing the swings was his pulse! The energy changes that occur here have been met before: gravitational potential energy to kinetic energy. The height involved is the vertical height of the swing, not the arc length. Galileo realised that the amplitude of swing had no effect on the period - and hence discovered the principle that governed the first really accurate clocks.
The variables that govern the period and hence frequency of a pendulum are a) the length of the pendulum and b) the gravitational field strength (usually constant).
Another important discovery is that the conservation of angular momentum means that a pendulum will continue to swing in the same plane - even if the surroundings rotate about the pivot. This led to a scientist, called Foucault, in Paris using a huge pendulum to demonstrate the rotation of the Earth - an example of this can be seen in Boston Science museum as well as Foucault's original one in Paris.
\[T = 2\pi \sqrt{\frac{l}{g}}\]
The energy transfers for a simple pendulum. Image from google....
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Image from "How to build a Foucault Pendulum" - click to lead to the website.
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Other Resources
