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

Question

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.

Thank you in advance!

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?

Answer 1

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.

Answer 2

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.

Answer 3

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