Trouble understanding aspect of SHM

[PsychonautQQ]

Homework Statement

When deriving the basic equations for SHM, you get the
$$a_c = /omega^2 A$$
and then continue on to derive
$$a_x = -a_c cos/theta$$
I was wondering where the negative sign came from in the equation above. I don't see the need for it, the x component of the centripetal acceleration seems to point in the same direction as the acceleration "projected onto a diameter" already.

ps why aren't my latex commands for omega and theta not working?

 

[hjelmgart]

I think you need to provide some of the steps, if I am to be of any use. However, I am guessing it's from differentiating cosine twice?

 

[PsychonautQQ]

That is correct. However I was wondering if anybody knew offhand why the sign on the projected acceleration is the opposite sign of the centripetal acceleration

 

[hjelmgart]

I think it is because, it is described as a standing wave. If you look at the formula and imagine a spring going up and down. It will start by falling (since else it can't bounce up again), so of course the acceleration must be negative.

Then look at your cosine function, which gives a positive result until it reaches Pi/2, at that point it reaches the bottom, and it will start to bounce upwards again, and thus the entire expression gives a positive result, since the cosine becomes negative. It could be helpful to look at the cosine function, while reading this somewhat "bad" explanation, but I hope you get it :-)

 

[Nickie]

I can understand your confusion about the negative sign in the equation for the x component of centripetal acceleration in SHM. The negative sign is actually necessary in this equation to account for the direction of the acceleration. In SHM, the object undergoes oscillatory motion, meaning it moves back and forth between two points. The negative sign indicates that the acceleration is directed towards the equilibrium position, where the object experiences the maximum restoring force. Without the negative sign, the equation would not accurately describe the motion of the object.

In terms of your Latex commands, it is likely that they did not work because they were not properly formatted. It is important to use the correct syntax and symbols when using Latex in order for it to display correctly. I suggest reviewing the Latex guide or seeking assistance from a tutor or classmate to ensure your commands are correct.