Simple Harmonic motion

Help would be appreshiated on the following question.

The diagram shows a trolley of mass m attached to a spring of force constant K. When the trolly is displaced to one side and then released, the trolly executes simple harmonic motion.

Show that the accerleration of the trolley is given by

a=-(K/M)x

where x is the displacement of the trolley from its equilibrium position and m is the trolleys mass.

I assume K/M is the same as the angular frequency but I'm not really sure what the question wants.

thanks for taking the time to read this.

Reply 1

Morbo

Use Hooke's law - the tension in the spring, F = -kx (where k is a positive constant)

Then use Newton's 2nd Law: F = ma

Combine the two, and...

ma = -kx
a = -(k/m)x

k/m is actually the angular frequency squared.

Reply 2

iknowmole

Thanks for the help. Its been a while since i did hook's law so the answer was a bit unobvious.

Reply 3

teachercol

Assuming the amplitude is not too big ...

Reply 4

Morbo

Using Hooke's law implies that you are staying within the elastic limit.

Reply 5

teachercol

I know - but there are often marks for putting this assumption in explicitly.

Reply 6

phys981

Also, isn't it also sometimes req'd as part of the definition of SHM?

Period independent of amplitude, for small amplitudes?

Reply 7

Morbo

rsk

Also, isn't it also sometimes req'd as part of the definition of SHM?

Period independent of amplitude, for small amplitudes?

This is necessary only in the case of something like a pendulum, for example, which can be approximated to SHM for small amplitudes. For the a spring obeying Hooke's law, you can have any amplitude you like.

All that is required to define SHM is that the acceleration is proportional to, and in the opposite direction to, the displacement.

Reply 8

phys981

Ah yes, thanks.

It's that the Edexcel book defines shm as

"shm: oscillatory motion where the period does not depend on the amplitude" (Things you need to know, p72 in the version I have).

However, just underneath that it gives this

"Definition of shm: motion where the acceleration or force) is directly proprtional ......" as you say.

I have a feeling I've seen an exam answer scheme require both of these for full marks but don't have the answer schemes at home to check tonight.

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