# Equations of the harmonic oscillator

1. Mar 2, 2014

### KodRoute

Hello, my book explains detailed the proofs of these three formulas:

y = Asin(ωt + φo)
v = ωAcos(ωt + φo)
a = -ω²Asin(ωt + φo)

Where a is acceleration, v is velocity, ω is angular velocity, A is amplitude.

The book uses the following figures:
Figure a) --> http://tinypic.com/view.php?pic=30m8f9j&s=8
Figure b) --> http://tinypic.com/view.php?pic=2hrejde&s=8
Figure c) --> http://tinypic.com/view.php?pic=a0f9te&s=8

In the first case a) what I'm seeing is that the sine in the triangle whose hypotenuse is R and opposite side vector y is (vector y)/(vector R), in this case R is the amplitude.

The rest b) and c) are just complicated, I don't understand them. Anyone out there to help me here?
Thank you!!

Edit: Sorry for posting this here, it's supposed to go to homework help, my fault.

2. Mar 2, 2014

### Staff: Mentor

Conceptual questions that arise from not understanding your textbook or lecturer's notes, are fine here. The homework-help forums are for getting help with solving specific exercises.

The simplest way to derive the formulas for v and a is by using calculus. v is the derivative of y with respect to t, and a is the derivative of v with respect to t. I suppose you haven't had calculus yet?

Last edited: Mar 2, 2014
3. Mar 2, 2014

### KodRoute

Hi! Thank you for replying, no I haven't done calculus yet. I thought these formulas came from looking at the geometry of these vectors.

4. Mar 2, 2014

### Staff: Mentor

For figure (b), focus on the small right-triangle that has $\vec v_t$ as its hypotenuse. Which trig function (sin or cos) is associated with the y-component of $\vec v_t$, and is that component + or -?

To get the amplitude (maximum value) of v, remember that the point moves around the circle at constant angular speed ω (radians/sec). What linear speed (m/sec) does that correspond to? (This is the magnitude of the vector $\vec v_t$.)

Last edited: Mar 2, 2014