Understanding Simple Harmonic Motion: Homework Equations and Graph Analysis

[Calpalned]

Homework Statement

My textbook states that for simple harmonic motion, the sinusoidal graph of the x (displacement) as a function of time can be created using the "Equation of motion".

Homework Equations

The equation of motion $A \cos (\omega t + \phi)$

The Attempt at a Solution

I know that $A$ stretches the cosine graph vertically, but that the frequency is unaffected. How do $\omega$ and $\phi$ affect the graph?

 

[SammyS]

Homework Statement

My textbook states that for simple harmonic motion, the sinusoidal graph of the x (displacement) as a function of time can be created using the "Equation of motion".

Homework Equations

The equation of motion $A \cos (\omega t + \phi)$

The Attempt at a Solution

I know that $A$ stretches the cosine graph vertically, but that the frequency is unaffected. How do $\omega$ and $\phi$ affect the graph?

Try some values and see.

or ...

What do you know about shifting and stretching/shrinking of graphs in general?

What is the period of y = cos(x) ?

How does the graph of y = f(x + k) compare with the graph of y = f(x) ?

etc.

 

[Calpalned]

Try some values and see.

or ...

What do you know about shifting and stretching/shrinking of graphs in general?

What is the period of y = cos(x) ?

How does the graph of y = f(x + k) compare with the graph of y = f(x) ?

etc.

Thank you so much! Is my understanding below valid?

$A$ stretches/compresses the graph vertically
$\omega$ affects the period. The larger $\omega$ is, the shorter the period.
$\phi$ is the horizontal shift and it is negative. That is, a positive value of $\phi$ will shift the graph to the left.

 

[SammyS]

Thank you so much! Is my understanding below valid?

$A$ stretches/compresses the graph vertically
$\omega$ affects the period. The larger $\omega$ is, the shorter the period.
$\phi$ is the horizontal shift and it is negative. That is, a positive value of $\phi$ will shift the graph to the left.

Correct.