Simple Harmonic Motion – SHM for a mass/spring oscillator

Study this applet of a mass/spring oscillator.

Plotting the displacement, velocity and acceleration on the same axes looks like this.

Dynamics of SH oscillator
Dynamics of SH oscillator

Print out a copy of the graph and measure the slopes of the x/t and v/t graphs to find instantaneous velocities and accelerations. Check your answers by calculation (the best way to do it – exams sometimes penalize if you try to measure slopes.)

The graph shows that for a body moving with SHM, displacement is proportional to acceleration, but opposite in sign. In an exam, it’s better to say that a is proportional to x and always directed to the zero of displacement.

Screen Shot 1.png

Since max v = rω, velocity varies with time as:

Screen Shot 2.png

The slope of the v/t graph gives the acceleration, so:

Screen Shot 3.png

but x = rsinωt, so

Screen Shot 4.png

The constant of proportionality is ω

 

Screen Shot 1.png

f being the frequency of oscillation (Hz) = 1/time period, T

This equationScreen Shot 4.png defines motion which is simple harmonic.

Use the graph  to find the time period of the oscillations.

Answer: (careful – vertical and horizontal axes must match – they do here…)

Screen Shot 3.png

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s