Simple Harmonic Motion

The following notation is used in SHM.

x - Displacement
a - Acceleration
T - Period
w - Angular Frequency
f - Frequency

My question is what does X0 stand for !

And another question for example I am given these equations
x= X0 sin (wt)
x= X0 cos (wt)

How do I know when to use cos or sin? Do I use cos when displacement starts at is max and sin when it starts at zero?

Reply 1

Morbo

davidsmith

You know cos and sin functions can go between +/- 1, right? So this means that x can go between +/- x₀. So x₀ is simply the maximum displacement of the oscillation, AKA the amplitude.

How do I know when to use cos or sin? Do I use cos when displacement starts at is max and sin when it starts at zero?

Simply put, yes. It depends on what the displacement is when t=0.

This is an example of a choice of phase for your oscillation. You might know that

\cos (\omega t) = \sin (\omega t + \pi/2)

, but even more generally you can write any oscillation as

\sin(\omega t + \phi)

, where

\phi

controls where the oscillation is in its cycle at

t=0

AKA the phase. You can write this explicitly as

\sin(\omega [t + \Delta t])

, where

\Delta t = \phi/\omega

Reply 2

davidsmith

Thanks I get it now! :smile:

Reply 3

Brooksy1800

My question is what is this formula finding? is the x for total displacement?

(edited 2 years ago)

Reply 4

Stonebridge

Original post by Brooksy1800

My question is what is this formula finding? is the x for total displacement?

In the formula
x= X₀ sin (wt)

x is the displacement of the particle from its equilibrium position at any time t (it's oscillating about that position)
X₀ is the maximum value it can be for the reason the other poster has given. This is because the maximum value of sin(wt) is 1
When t=0 the sin is zero and the displacement is zero (The particle is at its equilibrium position)
when t is such that the angle wt is 90 degrees, the sin is 1 and x = X₀which is the max displacement, called the amplitude.
At 180 degs x is zero again, and at 270 degs it's back to maximum.

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