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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
License: CC BY-NC 3.0

During one of the Skylab missions, astronaut Alan Bean measures his mass by sitting on a seat that oscillates back and forth. Through the use of a spring, the total mass related to the frequency of vibration by:

\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
License: CC BY-NC 3.0

The Pendulum below the clock is an example of oscillation [Figure2]

  • Oscillations are seen when a system is moved or disturbed from a stable equilibrium. Common examples of oscillations is the pendulum of a clock, the vibration of guitar strings while being played and even a boat moving up and down on a body of water. 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 relationship between the force felt by a mass that is attached to a spring and displaced through a distance \begin{align*}x\end{align*} is:

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

  • Therefore, the greater an object is displaced from its stable equilibrium position, the greater the restoring force will be. By using Newton's second law and the relationship between acceleration and displacement, the acceleration is given as

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

  • (For an object to be considered a simple harmonic oscillator, its acceleration is 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 osculations when Alan Bean sits on the chair in the above image?

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Image Attributions

  1. [1]^ Credit: NASA; Source: http://history.nasa.gov/EP-177/ch5-2.html; License: CC BY-NC 3.0
  2. [2]^ Credit: Craig Pennington; Source: http://www.flickr.com/photos/40528133@N06/5312595806; License: CC BY-NC 3.0

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