Understanding Simple Harmonic Motion: Explaining x=Acos(wt+phi)

[Jozefina Gramatikova]

Homework Statement

x=Acos(wt+phi)

Homework Equations

can somebody explain to me please when phi=0. I saw many different questions with many solutions and I can't understand when we have just x=Acos(wt) and when x=Acos(wt+phi)

The Attempt at a Solution

 

[PeroK]

Homework Statement

x=Acos(wt+phi)

Homework Equations

can somebody explain to me please when phi=0. I saw many different questions with many solutions and I can't understand when we have just x=Acos(wt) and when x=Acos(wt+phi)

The Attempt at a Solution

It depends where the system starts at ##t=0##.

 

[Jozefina Gramatikova]

So, if it starts from x=0 at t=0 => phi=0? and at t=0 x=1 => phi would be different from zero?

 

[PeroK]

So, if it starts from x=0 at t=0 phi=0? and at t=0 x=1 phi would be different from zero?

Yes. ##-A \le x(0) \le A##, where ##x(0)## is the displacement of the system at ##t=0##, and determines ##\phi##.

 

[vela]

So, if it starts from x=0 at t=0 => phi=0? and at t=0 x=1 => phi would be different from zero?

Plug t=0 and ##\phi=0## into ##x(t) = A\cos(\omega t + \phi)##. Is ##x(0)## equal to 0?

 

[Jozefina Gramatikova]

Plug t=0 and ##\phi=0## into ##x(t) = A\cos(\omega t + \phi)##. Is ##x(0)## equal to 0?

x(t)=Acos(0)
x(t)=A

 

[vela]

x(t)=Acos(0)
x(t)=A

Presumably ##A \ne 0## otherwise there'd be no oscillating going on, so clearly ##x(0)=0## and ##\phi=0## contradict each other.

 

[haruspex]

So, if it starts from x=0 at t=0 => phi=0?

No.
If it is at x₀ at t=0 and at x=Acos(ωt+φ) at time t, what equation can you write for x₀?

 

[Jozefina Gramatikova]

I don't know I am really confused right now.

 

[PeroK]

I don't know I am really confused right now.

##\cos(0) = 1##

 

[Jozefina Gramatikova]

##\cos(0) = 1##

I know that :D Thats why I said:
"x(t)=Acos(0)
x(t)=A"

 

[haruspex]

I don't know I am really confused right now.

Just substitute those t and x values in the general equation in post #8.

 

[haruspex]

x(t)=Acos(0)

When plugging in a value for t you should replace all occurrences of t.

 

[haruspex]

I know that :D Thats why I said:
"x(t)=Acos(0)
x(t)=A"

You did not seem to know it in post #3.

 

[Jozefina Gramatikova]

Just substitute those t and x values in the general equation in post #8.

x=Acos(ωt+φ)
x0=Acos(ω x 0+φ)
x0=Acos(ω x 0+φ)
x0=Acos(φ)?

 

[Jozefina Gramatikova]

Could you give me the right solution, please? I get more and more confused with every single post.

 

[Ray Vickson]

Could you give me the right solution, please? I get more and more confused with every single post.

Let's assume that ##A>0##, so when ##x(t) = A \cos( \omega t + \phi)##, the value of ##x(t)## oscillates back and forth between ##-A## and ##+A##. That is, ##-A \leq x(t) \leq A## for all ##t##. Furthermore, we have ##x = -A## for some ##t## and ##x = +A## for some other ##t##.

If ##\phi = 0## we have ##x(0) = A##, so that means that the motion starts at time zero from the upper end of the allowed interval. Then for a while ##x(t)## will decrease until it hits the lower end ##-A## of the allowed interval, at time ##\pi/\omega##. Then it will start to increase until it hits ##x=A## at time ##2 \pi/\omega,## etc., etc.

If you want ##x(0)## to be a point other than ##A## you need to have ##\phi \neq 0.## To help you to enhance your understanding, try the following question: "what would be the value of ##\phi## if you require ##x(0) = 0?## (Of course, because the ##\cos## function is periodic there will be many values of ##\phi## that will work, but among these there is one that is "simplest", and that is the one you should attempt to find.)

 

[haruspex]

x0=Acos(φ)?

Yes.