Periodic Motion and Simple Harmonic Motion Terminology Question

[Yosty22]

In lecture, we are beginning to learn about waves and periodic motion under simple harmonic motion. We were given the equations:
x=Acosθ and θ=ωt+\phi -- Substituting, we get x=Acos(ωt+\phi).

This is simple enough; however what is Phi? All I was told is that "phi is a constant that allows us to start anywhere on the sine or cosine graph," meaning that the phi term allows the equation to work with both sine and cosine functions.

This being said, what exactly is phi? In using this equation to solve problems, how do I know what to set for phi if it seems to be just a phase shift to compensate for the sine and cosine functions? Is it something that I can arbitrarily choose to be 0 if I am not given a specific value to use in the question?

Thank you in advance for your responses.

 

[jtbell]

$\omega t + \phi$ is the phase of the oscillation ("where it is in its cycle") at time t. $\phi$ is the initial phase of the oscillation ("where it is in its cycle") at time 0.

"Where it is in its cycle" is described by an angle between 0 and $2\pi$ radians.

If you're using a cosine to represent the oscillation, then $\phi = 0$ gives you an oscillation that starts off at t = 0 with maximum displacement in the + direction.

$\phi = \pi$ gives you an oscillation that starts off at t = 0 with maximum displacement in the - direction.

$\phi = \pi/2$ gives you an oscillation that starts off at t = 0 passing through the equliibrium (central) point while moving in the - direction.

Etc.

If you're using a sine to represent the oscillation, then the "meaning" of the different phase angles is different: $\phi = 0$ means passing through the equilibrium point while moving in the + direction, $\phi = \pi/2$ means maximum displacement in the + direction, etc.

 

[cepheid]

In lecture, we are beginning to learn about waves and periodic motion under simple harmonic motion. We were given the equations:
x=Acosθ and θ=ωt+\phi -- Substituting, we get x=Acos(ωt+\phi).

This is simple enough; however what is Phi? All I was told is that "phi is a constant that allows us to start anywhere on the sine or cosine graph," meaning that the phi term allows the equation to work with both sine and cosine functions.

This being said, what exactly is phi? In using this equation to solve problems, how do I know what to set for phi if it seems to be just a phase shift to compensate for the sine and cosine functions? Is it something that I can arbitrarily choose to be 0 if I am not given a specific value to use in the question?

Thank you in advance for your responses.

Phi is the initial phase of the oscillation. Phase just determines where in the cycle you are. You don't get to set it: it is determined by the initial conditions of the system (i.e. what's going on at at t = 0).

Consider a mass on a spring. Suppose at t = 0, the mass is at maximum positive extent (x = +A) and the speed is 0. This corresponds to the mass having reached one extreme end of the oscillation, come to rest, and the restoring force being a maximum (meaning that the mass is about to start moving back towards the equilibrium position at x = 0). So if you think about it, this oscillation corresponds perfectly to a pure cosine wave. A cosine function starts out at the maximum value, with slope (derivative) zero, and then declines towards 0 (and beyond). So, these initial conditions correspond to an initial phase of $\phi = 0$, making x(t) a pure cosine wave.

Suppose at t = 0, the mass is at the equilibrium position (x = 0) and the speed is a maximum in the positive direction (the mass is traveling at max speed through x = 0 towards the direction of the positive limit x = +A). If you think about it, this motion corresponds to a pure sine wave. A sine wave starts out at zero and is increasing at the maximum rate (slope) towards the maximum positive value. So, these initial conditions correspond to an initial phase of $\phi = -\pi/2$, in order to make x(t) a pure sine wave. A sine function is just a cosine function phase shifted (delayed) by pi/2.

Suppose at t = 0, the mass is at *some arbitary point* in the oscillation. This would just correspond to the oscillation being phase shifted (from the pure cosine case) by *some arbitrary* amount, which you could do by setting the value of phi arbitrarily.

 

[Yosty22]

Phi is the initial phase of the oscillation. Phase just determines where in the cycle you are. You don't get to set it: it is determined by the initial conditions of the system (i.e. what's going on at at t = 0).


Suppose at t = 0, the mass is at the equilibrium position (x = 0) and the speed is a maximum in the positive direction (the mass is traveling at max speed through x = 0 towards the direction of the positive limit x = +A). If you think about it, this motion corresponds to a pure sine wave. A sine wave starts out at zero and is increasing at the maximum rate (slope) towards the maximum positive value. So, these initial conditions correspond to an initial phase of $\phi = -\pi/2$, in order to make x(t) a pure sine wave. A sine function is just a cosine function phase shifted (delayed) by pi/2.

In this case, since the wave function could be modeled by a pure sine function, but the equation uses a cosine function, you add the -pi/2 so that it matches the sine function. That is, if it the scenario matches a pure sine curve, the equation used would be:
x=Acos(ωt+(-pi/2)), correct?

 

[mikeph]

A*cos(wt - pi/2) = A*sin(wt), yes.

 

[cepheid]

In this case, since the wave function could be modeled by a pure sine function, but the equation uses a cosine function, you add the -pi/2 so that it matches the sine function. That is, if it the scenario matches a pure sine curve, the equation used would be:
x=Acos(ωt+(-pi/2)), correct?

Yeah, the key point is that ANY sinusoidal oscillation of any starting phase can be represented as a sine wave (or a cosine wave) PHASE SHIFTED by the right amount, and that's what phi does.

 

[jtbell]

And you have to use a different amount of shift depending on whether you're using a sine or cosine. The difference between the amount for a sine and the amount for a cosine is $\pi/2$.

 

[sophiecentaur]

The absolute phase of an oscillation is often not particularly relevant. It becomes very important when two or more waves add together or when you are interested in the PD between two parts of a circuit with AC. On those occasions, the 'phi' will tell you how near 'in phase or out of phase' the two oscillations are.
The significance of this may not be apparent until you actually have to solve a problem which involves it and then it becomes more obvious.