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Simple Harmonic Motion

"The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction"

-Sidney Coleman

Main Concept

Hooke's Law states that the force F exerted by a stretched spring is proportional to its displacement, x, from the equilibrium position. The constant of proportionality, k, is known as the spring constant:


F=k x.


If an object of mass m is attached to the end of the spring, extended and then released (possibly with a nonzero initial velocity), it will oscillate periodically according to the formula:


xt = A sinkmt+ϕ,


where the amplitude A and phase ϕ depends on the initial velocity and position of the mass at the time of release. This system is called the simple harmonic oscillator, and the associated motion is called simple harmonic motion.



By combining Newton's second law F=m a  with Hooke's law and noting that the acceleration a is just ⅆ2xⅆt2, we obtain


mⅆ2xⅆt2 =k x.

The solution of this differential equation for xt can be expressed as

xt = Asinkmt+ϕ,


where A is is the amplitude of the oscillation, i.e. its maximum displacement, and ϕ is the initial phase. In the animations below, we have set  A=1 and ϕ=0, thus


xt = sinkmt.


Try adjusting the spring constant and the mass. What happens to the motion of the block? Does increasing k speed up the motion? Does increasing m also speed up the motion?

Spring Constant, k 

 Mass of particle, m 




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