Simple Harmonic Motion




Simple Harmonic Motion describes when a particle or object is in a constant motion that is repetitive in its path. It will oscillate back and forth on a particular path constantly, and theoretically forever. One property of the oscillation is its frequency, which is how many oscillations that are completed per second. There is also the period, T, which is time for one complete oscillation.


Equation One

x(t) = Acos(wt)
A = amplitude
w = angular frequency

Equation Two

v(t) = Awsin(wt)

++Equation Three
a(t) = -Aw^2cos(wt)

Equation Four

T = 1/f

Equation Five

w = 2(pi)/T = 2(pi)f

A 20 g particle moves in simple harmonic motion with a frequency of 3 oscillations per second and an amplitude of 5cm.
(a) Through what total distance does the particle move during one cycle of its motion?

A 20 g particle moves in simple harmonic motion with a frequency of 3 oscillations per second and an amplitude of 5cm.
(b) What is its maximum speed? Where does that occur?

Special Cases

A special case of simple harmonic motion is in the case of pendulums. Pendulums operate in the same manner, but include gravity in their calculations. Also, it is important to note that while theoretically, in a frictionless system something may oscillate forever, most systems do eventually come to rest due to friction.



The most obvious application is when looking a systems involving springs that have been given energy and released. These equations help you to calculate the positions, velocity, and acceleration of objects attached to the spring.

External Links


A mobster decides he wants to win a horse race, and drags three random guys out of a university to make it happen for him. Turns out they're a biologist, engineer, and a mathematician. Each is given an hour to come up with a plan, and the best one will be allowed to live to carry the plan through.

Biologist: I can concoct a mix of steroids, stimulants, and nutrients that will give the horse a 20% metabolic boost for 15 minutes, allowing it to win the race.

Engineer: We replace the horse's muscles with plastic contractile polymer and mount some dilithium crystals in its stomach, allowing it to run at 150km/h until it falls apart, which won't happen until after the race.

Mathematician: Well, now, if we consider a spherical horse in simple harmonic motion…

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