Modeling Harmonic Motion
Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.
Simple Harmonic Motion
A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0[/latex].
simple harmonic
We see that simple harmonic motion equations are given in terms of displacement:
[latex]d=a\cos \left(\omega t\right)\text{ or }d=a\sin \left(\omega t\right)[/latex]
where [latex]|a|[/latex] is the amplitude, [latex]\frac{2\pi }{\omega }[/latex] is the period, and [latex]\frac{\omega }{2\pi }[/latex] is the frequency, or the number of cycles per unit of time.
- Find the maximum displacement of an object.
- Find the period or the time required for one vibration.
- Find the frequency.
- Sketch the graph.
- [latex]y=5\sin \left(3t\right)[/latex]
- [latex]y=6\cos \left(\pi t\right)[/latex]
- [latex]y=5\cos \left(\frac{\pi }{2}t\right)[/latex]
A weight on a spring oscillates vertically with position given by [latex]y = 7\cos\left(\frac{\pi}{3}t\right)[/latex] cm, where [latex]t[/latex] is time in seconds. Find:
- The maximum displacement of the weight
- The period of oscillation
- The frequency
- At what time during the first period does the weight first reach [latex]y = 3.5[/latex] cm?