# Trouble understanding aspect of SHM

• PsychonautQQ
In summary, the conversation discusses the derivation of equations for Simple Harmonic Motion (SHM) and the question of where the negative sign in the equation for the x component of centripetal acceleration comes from. The experts suggest that it may be due to differentiating cosine twice and the nature of SHM as a standing wave. They also suggest looking at the cosine function for better understanding.

## 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?

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?

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

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 :-)

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.

## What is simple harmonic motion (SHM)?

Simple harmonic motion (SHM) is a type of periodic motion in which a system oscillates back and forth around an equilibrium position with a constant amplitude and a constant period. It is often described as a "back-and-forth" motion.

## What is the difference between amplitude and period in SHM?

Amplitude refers to the maximum displacement of a system from its equilibrium position during SHM. On the other hand, period refers to the time it takes for a system to complete one full cycle of oscillation.

## What factors affect the amplitude and period of SHM?

The amplitude and period of SHM are affected by the mass, stiffness, and initial displacement of the system. A higher mass or stiffness will result in a larger amplitude and longer period, while a larger initial displacement will result in a larger amplitude and shorter period.

## How is SHM related to circular motion?

SHM and circular motion are related through the concept of uniform circular motion, in which an object moves along a circular path with a constant speed. If we consider a point on this circular path, its projection onto a diameter of the circle will exhibit SHM.

## What are some real-life examples of SHM?

Some examples of SHM in real life include the motion of a pendulum, the vibrations of a guitar string, the motion of a mass-spring system, and the swinging of a playground swing.