# Simple Harmonic Motion- From Uniform Circular Motion

• chantalprince
In summary, the conversation discusses the equations for angular frequency and period in the context of circular motion. The equations are W = 2 pi/T = 2(pi)(f) and W = square root of (k/m) and T = (2 pi) x the square root of (m/k). The difference between measuring angular frequency in Hz and measuring frequency in radians per second is also mentioned. The conversation ends with a clarification on the correct equation for frequency.
chantalprince

## Homework Statement

I don't have a homework question exactly, but I need help with an equation please.

Angular frequency: W= 2 pi/T = 2(pi)(f) f= frequency

And- W = square root of (k/m) k = spring constant m= mass

So, wouldn't T = 2 pi / square root of (k/m) ??

My instructor has given us the following equations in class a few times. I cannot figure out what the heck is going on!

W = (2 pi) x the square root of(k/m)
T = (2 pi) x the square root of (m/k) -------> m/k this time

Any help is appreciated. I am so confused right now.

## The Attempt at a Solution

This is simply because of the fact that:

$$\frac{1}{\frac{a}{b}}=\frac{b}{a}$$

Ok- but in the book it gives: W = sq. root of (k/m)

Instructor gives: W = 2pi x sq. root (k/m)

Whats with the 2 pi??

Thanks-

What's the difference between measuring angular frequency in Hz and measuring frequency in radians per second? That might give you an idea where the conversion comes from.

Ok...I'll sit down with that thought. So, either one works right? They are the same thing??

Ah I see what you're getting at now. There's a mistake in the equation for frequency:

$$\omega_{n} = \sqrt{\frac {k}{m}}$$ (1)

where frequency is in radians per second.

But what if you want to express the frequency in Hertz? Well, we know that 1 Hz is equal to one cycle per second. In the case of circular motion, one cycle is equal to $$2\pi$$ radians.

So to convert from radians per second to Hertz, one must divide by $$2\pi$$. Hence:

$$\omega = \frac {1}{2\pi} \sqrt{\frac{k}{m}}$$ (2)

where frequency is now in Hertz.

Now let's express this in terms of the period of one cycle, T. Bear in mind that if you were simply to reciprocate the expression for frequency when expressed in radians per second (equation 1), you would be stating the length of time of rotation for one radian alone. Hence you have to multiply the expression by $$2\pi$$ now to obtain the period for a single cycle. This is now the same equation as you would obtain by reciprocating equation 2.

Hope this helps.

## 1. What is simple harmonic motion?

Simple harmonic motion refers to the back-and-forth oscillatory motion of an object around a fixed point, where the acceleration of the object is directly proportional to its displacement from the fixed point and is always directed towards the fixed point.

## 2. How is simple harmonic motion related to uniform circular motion?

Simple harmonic motion can be thought of as a projection of uniform circular motion along a single axis. This means that the circular motion can be broken down into a horizontal and vertical component, with the vertical component exhibiting simple harmonic motion.

## 3. What is the difference between simple harmonic motion and periodic motion?

Simple harmonic motion is a type of periodic motion, but not all periodic motion is simple harmonic. Periodic motion refers to any motion that repeats itself at regular intervals, while simple harmonic motion specifically follows a sinusoidal pattern.

## 4. What are some examples of simple harmonic motion in everyday life?

Some examples of simple harmonic motion include a pendulum, a mass on a spring, and a swinging door. These objects exhibit back-and-forth motion around a fixed point, with their acceleration always directed towards the fixed point.

## 5. How is simple harmonic motion used in technology and engineering?

Simple harmonic motion is used in a variety of technologies and engineering applications, such as in the design of shock absorbers, musical instruments, and clock pendulums. It is also used in the study of vibrations and waves, which play a crucial role in fields like acoustics and optics.

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