# Simple harmonic motion momentum leads restoring force by 90 degrees [closed]

What is meant by "in a simple harmonic motion, momentum leads the restoring force of elasticity by 90 degrees" ??

## closed as unclear what you're asking by Rory Alsop, JMac, Yashas, GiorgioP, Jon CusterApr 25 at 21:54

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• It means exactly what it says. Tell us which particular word(s) in the sentence you don't understand, and somebody will explain them. A full explanation of every word would be a very long answer.. – alephzero Apr 24 at 9:27

Momentum is mass times velocity $$m\,v\propto v$$ and force of elasticity is a spring force; spring constant times spring deflection $$c\,x\propto x$$

for harmonic motion :

$$x=\sin(\omega\,t)\quad$$

and

$$v=\frac{dx}{dt}=\omega\,\cos(\omega\,t)=\omega\,\sin(\omega\,t+\pi/2)$$

thus:

"Simple harmonic motion momentum leads restoring force by 90 degrees"

• For simple harmonic motion the direction of the restoring force is in the opposite direction to the displacement and for a spring $F=-kx$. – Farcher Apr 24 at 21:34

Momentum of a mass undergoing shm is mass times velocity.
Force on the mass due to elasticity (eg spring?) is mass times acceleration.

So you are asking about the phase difference between the velocity (momentum) and the acceleration (force) when a body undergoes simple harmonic motion.

You will note that the acceleration (force) leads the velocity (momentum) by a quarter of a period which is contrary to the statement that you made in your question.

PS After a cursory view of the graphs you may be under the impression that becuse the velocity graph is to the right of the acceleration graph the velocity is leading the acceleration but the horizontal axis is time not position.
This means that everything that the acceleration graph does (eg reach a peak) the velocity graph does a little later in time ie the velocity lags the acceleration.

• Great distinction about "leading" and the left-right positioning of the curves. – Bill N Apr 24 at 17:04