Finding amplitude from simple harmonic equation function

[vetgirl1990]

Homework Statement

The periodic motion is given in the form: f(t) = Acos(wt+φ)
What is the amplitude and phase constant for the harmonic oscillator when:

(a) f(t) represents position function x(t)
(b) f(t) represents velocity function v(t)
(c) f(t) represents acceleration function a(t)

Homework Equations

x(t) = Acos(wt+φ)
v(t) = -wAsin(wt+φ)
a(t) = -w²Acos(wt+φ)

The Attempt at a Solution

(a) To find amplitude from a position equation, I know that amplitude is the maximum displacement of the particle in harmonic oscillation, so A=x(t)
To get A=x(t), I would need my phase of motion to be zero, so that cos(wt+φ)=1. This would occur when φ=0 and t=0.
Therefore A=x and φ=0

However, I'm not really sure why it's relevant to ask the amplitude and phase constant for the velocity and acceleration functions. Both amplitude and phase constant (φ) are determined from initial conditions, so wouldn't the amplitude and phase constant be the same for x(t), v(t) and a(t), given that it's based off the same function?

 

[BvU]

What about that factor $\omega$ or $\omega^2$ ?

Advice: replace the A in your relevant equations by some other letter. It interferes with the A in the problem statement !

Actually: same for the $\phi$. The $\phi$ in the problem statement is to be treated as a given. You can't require it to be zero afterwards...

 

[vetgirl1990]

What about that factor $\omega$ or $\omega^2$ ?

Advice: replace the A in your relevant equations by some other letter. It interferes with the A in the problem statement !

Actually: same for the $\phi$. The $\phi$ in the problem statement is to be treated as a given. You can't require it to be zero afterwards...

Sorry, I don't quite understand your reply. I just know that Amplitude and Phase constant need to be determined from initial conditions.

 

[haruspex]

I fear you have not understood what you are asked to do.
For a), you are to take the position as specified by x(t)=A cos(ωt+φ). In terms of the symbols in that equation, what is the amplitude, and what is the phase? Yes, it's an extremely simple question, don't try to make it complicated.

b) and c) are where the interest lies. In b), the motion is now defined by v(t)=A cos(ωt+φ). This is still SHM, but clearly the constants in it no longer have their usual meanings. 'Amplitude' still refers to the variation in x(t), so in terms of the symbols in the v(t) equation given, what is the amplitude now?

 

[vetgirl1990]

I fear you have not understood what you are asked to do.
For a), you are to take the position as specified by x(t)=A cos(ωt+φ). In terms of the symbols in that equation, what is the amplitude, and what is the phase? Yes, it's an extremely simple question, don't try to make it complicated.

b) and c) are where the interest lies. In b), the motion is now defined by v(t)=A cos(ωt+φ). This is still SHM, but clearly the constants in it no longer have their usual meanings. 'Amplitude' still refers to the variation in x(t), so in terms of the symbols in the v(t) equation given, what is the amplitude now?

Ah I see what you mean... amplitude would still be "A". As in the same amplitude that was specified in the position equation.

 

[BvU]

I can't follow. If v(t)=A cos(ωt+φ), then surely x(t) is not A cos(ωt+φ), so the amplitude is not equal to A.

 

[vetgirl1990]

I can't follow. If v(t)=A cos(ωt+φ), then surely x(t) is not A cos(ωt+φ), so the amplitude is not equal to A.

Then I'm afraid I still don't understand.

 

[cnh1995]

If I understand the question correctly, you are supposed to obtain position function from each given function and then find the amplitude and phase constant.

 

[haruspex]

Then I'm afraid I still don't understand.

In b), you are given v(t)=A cos(ωt+φ). This defines the motion (up to a point) but do not assume that A stands for amplitude, etc.
Suppose x(t) is still SHM. Pick some new symbols to represent its amplitude, frequency and phase, then write out the equation for x(t) in terms of those. From that, obtain an equation for v(t), and compare it with the given equation.