# How to prove that a motion is Simple Harmonic Motion (SHM)?

I would like to know how one could show and prove that a given motion is simple harmonic motion.

Once given an answer, I'll apply that technique to an example I am trying to figure out.

I believe a motion can be proved simple harmonic, if the relation between its is as such: $$a_x = - \omega^2\cdot x$$

And as such the period time is: $$T =\frac{2\pi}{\omega}$$

Question-so-far: How do you prove such for a given force $F = \frac{G\cdot m_e \cdot M}{R_E} \cdot r$ ? Or any force that has non-trivial constants?

• You should specify what information you are given. For example, the Brachistochrone motion is simple harmonic, but showing this is nontrivial, and requires solving the Brachistochrone. – Ron Maimon Jan 17 '12 at 5:51
• Related: physics.stackexchange.com/q/1018/2451 and links therein. – Qmechanic Jan 20 '13 at 17:35

If the total force $F$ on a mass $m$ follows Hooke's law,

$$F~=~-kx,$$

then one can use Newton's 2nd law

$$F=ma,$$

to infer that the motion is a simple harmonic motion

$$a =-\omega^2x, \qquad\qquad \frac{2\pi}{T}~=~\omega~=~ \sqrt{\frac{k}{m}}~,$$

cf. OP's correct belief. Now it only remains to solve the ODE

$$\frac{d^2x(t)}{dt^2}~=~-\omega^2x(t),$$

which is a pure math exercise.

• Suppose a mass attached to a spring intially compressed by some distance is released and then at the equilibrium point it suddenly reverses direction without loss of kinetic energy due to let's take elastic collision with another mass following the same path in opposite direction then can motion of both of them still be called simple harmonic motion – Hydrous Caperilla Jan 31 '18 at 7:08
• @Hydrous Caperilla: That seems to be a new question. – Qmechanic Jan 31 '18 at 14:08

In mechanics and physics, simple harmonic motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement .

So proving the restoring force F is directly proportional would be enough. For example, let's say there is mass m mounted between 2 spring with spring constant $k_1$, $k_2$ which are attached to wall and have its original length when system is at equilibrium. When the mass is disturbed slightly by displacement $x$, the restoring force will be $F = -(k_2 + k_1)x$ which proves the motion will be SHM.

$$F = ma$$ $$F = -mg \sin \theta$$ $$ma = -\frac{mgx}{L}$$ $$a = -\frac{gx}{L}$$ Since $-g/L$ is a constant, $a \propto -x$. Hence proved.

• Almost nothing this terse is ever helpful. We expect better quality than that. -1 – Brandon Enright Feb 24 '14 at 16:53