# Simple harmonic motion>Origin?

1. Apr 12, 2015

### LogU16

Hello there!

I have a simple question; Why simple harmonic motion is called simple harmonic motion? What's the origin of this name? I could not find the answer despite browsing so many websites on the internet.

Could anybody tell me the answer?

2. Apr 12, 2015

### PWiz

Simple harmonic motion is a type of harmonic motion which can be approximated to a sinusoidal wave (where the acceleration is always proportional to the displacement and always acts towards the mean position). Other forms of harmonic motion may have more complicated functions which describe their motion.

3. Apr 12, 2015

### LogU16

Could you please explain it to me why has we given this name (simple harmonic motion) to this oscillatory motion? and how does acceleration act in an opposite direction to the displacement?
Thanks a lot!

4. Apr 12, 2015

### PWiz

Maybe because harmonic motion doesn't get simpler than this?
Think of a spring (held horizontally) with one end held by a clamp and a load attached to the other end. Now if you pull the load to one side and let go, you will observe oscillatory motion. What would happen if the acceleration acted in the direction of displacement? The mass would continue to accelerate indefinitely! Clearly, acceleration in the direction of displacement cannot result in regular motion (and it would defy CoE too). But if the acceleration always acted in the opposite direction to displacement, then as the mass would go further away from the mean position, it would slow down, stop and reverse direction, and then gain velocity as it would start moving to the average position. This gain would reduce as it would approach the mean position (where a=0) and then as the mass would overshoot to the other side, it would start slowing down (acceleration reverses direction as displacement is on the other side of the mean position now) until it stops again, then reverse direction... so on and so forth. This would also ensure that the total energy of the system stays constant.

5. Apr 12, 2015

### LogU16

Thanks. CoE??

6. Apr 12, 2015

### PWiz

Conservation of Energy.

7. Apr 12, 2015

### LogU16

Thanks. Could you please explain why does acceleration act always in the opposite direction to displacement in the case of horizontal mass-spring system? Why not they act in the same direction?
I'm unable to understand your previous reply (post#4), kindly make it easier for me to grasp it.

Thank you so much!!

8. Apr 12, 2015

### brainpushups

When you compress a spring, which way does it push/pull? What about when it is stretched?

9. Apr 12, 2015

### Staff: Mentor

In PWiz's example in #4, the acceleration is produced by the force of the spring acting on the oscillating mass. Which direction is that force acting when the spring is compressed? When it is stretched? That's the direction the acceleration will be acting.

10. Apr 12, 2015

### LogU16

Thanks, Nugatory.
When spring is stretched towards right, the force (restoring force) will be towards the left.
When spring is compressed towards left, the force (restoring force) will be towards the right.
What will be the direction of acceleration in both cases?

11. Apr 12, 2015

### PWiz

The acceleration must be along the direction of the force. (Newton's second law)

12. Apr 12, 2015

### Staff: Mentor

What does Newton's second law $\vec{F}=m\vec{a}$ say it will be?

13. Apr 12, 2015

### LogU16

When mass goes to mean position from extreme position, its velocity increases
When mass goes to extreme position, ts velocity decreases.

Could you please explain it to me why is it so?

14. Apr 12, 2015

### Staff: Mentor

You've already worked out which way the force is acting: when the mass is to the left of the mean position the force is acting towards the right and when the mass is to the left of the mean position the force is acting to the right. So what happens when the mass is to left of the mean position and moving left? The force tends to slow it down until it stops at the extreme left hand position, then as the the force continues to pull the mass to the right, the mass starts moving to the right and gaining speed until it reaches the mid-position. Once it passes the mid-position, the force starts pulling the mass to the left, slowing its rightwards motion.

15. Apr 13, 2015

### vanhees71

According to Fourier's theorem you can expand any (sufficiently good-mannered) function as a Fourier series,
$$f(t)=\sum_{n=-\infty}^{\infty} a_n \exp(-\mathrm{i} n \omega_0 t),$$
with the Fourier coefficients given by
$$a_n=\frac{\omega_0}{2 \pi} \int_{-\pi/\omega_0}^{\pi/\omega_0} \mathrm{d} t \exp(\mathrm{i} n \omega_0 t) f(t).$$
Harmonic motion is defined as being described by a single frequency, i.e., the Fourier series consists of only the two terms with one $n \in \mathbb{n}$ and the term with $-n$.

16. Apr 13, 2015

### Staff: Mentor

On the more linguistic side, from the Oxford English Dictionary:

17. Apr 13, 2015

### PWiz

Which is basically the same as saying that the motion is approximately sinusoidal, or the solution to the homogenous second order ordinary differential equation $\ddot x =-\omega^2 x$ with a characteristic equation which has complex roots [edited]

Last edited by a moderator: Apr 13, 2015
18. Apr 13, 2015

### Staff: Mentor

The name was given because it approximates the simplest vibration of sounding bodies, e.g., musical instruments, whence the name "harmonic."

19. Apr 13, 2015

### Staff: Mentor

With respect to guitars, plucking a string will produce a fundamental tone and higher level harmonics:

http://en.wikipedia.org/wiki/Guitar_harmonics

So simple harmonic would refer to the fundamental tone, the one the ear picks out most clearly.

20. Apr 13, 2015

### Staff: Mentor

I think this is a good point to end this thread. Thank you all for contributing.