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# Simple Harmonic Motion

## Relate velocity, acceleration, and displacement in simple harmonic motion

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The Vibrating Chair

### The Vibrating Chair

Credit: NASA
Source: http://history.nasa.gov/EP-177/ch5-2.html

During one of the Skylab missions, astronaut Alan Bean measured his mass by sitting on a seat that oscillates back and forth. Using a spring, mass can be related to the frequency of vibration:

1f=(12π)km\begin{align*}\frac{1}{f} = \left(\frac{1}{2\pi}\right)\sqrt{\frac{k}{m}}\end{align*}

#### News You Can Use

Credit: Craig Pennington
Source: http://www.flickr.com/photos/40528133@N06/5312595806

The pendulum below the clock oscillates and swings back and forth [Figure2]

• Oscillations are seen when a system is moved or disturbed from a stable equilibrium. The vibration of guitar strings being played or a boat bobbing up and down in water are common examples of oscillations. Even new buildings that are exposed to high velocity winds are being built with massive springs at their base to account for the natural sway of the building.
• The following equation represents the relationship between the force felt by a mass that is attached to a spring and displaced through a distance x\begin{align*}x\end{align*}:

F=kx\begin{align*}F = -kx\end{align*}

• Therefore, the more an object is displaced from its stable equilibrium position, the greater the restoring force will be. The following equation defines the relationship between acceleration and displacement:

a=kmx\begin{align*}a = -\frac{k}{m}x\end{align*}

• For an object to be considered a simple harmonic oscillator, its acceleration must be proportional to its displacement but oppositely directed.

#### Show What You Know?

Using the information provided above, answer the following questions.

1. Is the spring constant proportional or inversely proportional to the restoring force of a simple harmonic oscillator?
2. Looking at the image of Alan Bean, would you expect the rate of oscillations to increase or decrease if his mass were doubled?
3. What parameter could be changed to speed up the rate of oscillations when Alan Bean sits on the chair in the above image?

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