# How Do You Calculate the Phase Constant in Simple Harmonic Motion?

• ubbaken
In summary, to find the phase constant for simple harmonic motion using a cosine function, you can use the graph provided and find when the amplitude is at its maximum or minimum. By setting the argument of the cosine function to this value, you can solve for the phase constant. However, it is important to note that the phase constant is measured in radians and may require a negative value to accurately represent the phase shift.
ubbaken
What is the phase constant? Use a cosine function to describe the simple harmonic motion.

http://capa.physics.mcmaster.ca/figures/kn/Graph14/kn-pic1416_new.png

t1=40.0 s and A=20.0 cm

I'm really lost on how to get this done. Using x=Acos(ωt+φ); my approach has been to find when cos()=1 or 0 and then setting the phase equal to either 0 or pi/2 respectively.

So far, I have ω=pi/10 rad and I know pi<φ<3pi/2.

I'm stuck past this though (didn't get very far)...I tried it assuming the graph was accurate in scale and that it was 225deg (5pi/4 rad) but it was incorrect. I've also tried -10=20cos((pi/10)0+φ) --> φ=120deg; then adding a fraction of the period to get it in the proper phase, getting 240deg or 210deg (tried all, incorrect).

I am sure there is a more mathematical way to do it, that I am just missing or that I am starting, but not finishing correctly. Any help would be appreciated. Thanks!

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Okay, well I would suggest using the less mathematical way, and use the graph they have given you because it will help more than fiddeling around with numbers. At time=0 they show you that the amplitude is -1/2 of its max value. At time=0 a cosine function without a phase constant will be at the maximum amplitude. For a moment forget about the time and amplitude they give you. Phase constants, if we use the conventional definition of a positive phase constant, shift the function to the left.

You can formulate an equation to figure out the argument that belongs in the cosine function, and find the phase constant from there.

Hopefully that helps without giving you a direct formula.

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tried that, though I didn't word it that well there it seems.

with that logic, I work through to get: -1/2=cos(φ); which gives me 120deg - which isn't in the correct phase, so continuing along the curve until 240deg gives the right phase, but the answer is coming up as incorrect.

I am not really sure where we went wrong then. At t=0 x=-(1/2)A, so $$-\frac{1}{2} A = Acos(\omega*(0) + \phi)$$. It should be at 2(pi)/3. That doesn't work? Anyone else understand where we wrong?

Are you using an online thing that wants you to use radians?

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but it can't be 2pi/3 rads, cause that isn't the phase, it's the next one equal to -1/2. But I've tried them both. It accepts deg or rad answers.

Why can't it be 2pi/3? We want when the first -1/2 comes about, which is at the first phase shift. The other one, at 4pi/3, will go past the lower amplitude, and then go back up to -1/2.

Are you supposed to compose the whole waveform, solving for the angular frequency and all?

Heya, I have this exact same problem. I played around a bit. Recall "The quantity (wt + φ) is called the phase, and is measured in radians. The cosine function traces out one complete cycle when the phase changes by 2p radians. The phase is not a physical angle!" So, by that and the graph, all you do is make your answer for the question negative. I hope that helps. I had a different question and I kept getting 2.09 rad as the answer, I made it negative and it worked. On the 7th try. Darn CAPA.

Sorry for the formatting(Equation and grammar wise), I was googling around for an answer and I found this, then I solved the question, figured I'd post where I went wrong.

I had to solve the same question for physics class. Solution: 4 pi - 2 pi /3 = 10.5 rad (using the formula wt - initial phase).

## 1. What is the phase constant in physics?

The phase constant, also known as the phase angle, is a term used in physics to describe the starting point of a periodic function. It represents the initial position of the wave as it begins to oscillate or repeat itself.

## 2. How is the phase constant measured?

The phase constant is measured in radians or degrees and is typically denoted by the symbol φ. It can be calculated by finding the angle between the maximum value of the function and the horizontal axis, or by comparing the function to a reference point or function.

## 3. What is the difference between phase constant and phase shift?

The phase constant and phase shift are often used interchangeably, but they have slightly different meanings. The phase constant is the initial angle of a periodic function, while the phase shift is the amount by which the function is shifted in time or space compared to a reference point or function.

## 4. What is the importance of phase constant in practical applications?

The phase constant is important in many fields of science and engineering, including physics, electrical engineering, and signal processing. It is used to describe the behavior of waves, such as sound or electromagnetic waves, and to determine the relationship between different waves or signals.

## 5. Can the phase constant have a negative value?

Yes, the phase constant can have a negative value. This indicates that the function or wave is shifted in the opposite direction from the reference point or function. However, in some cases, the phase constant is defined to be between 0 and 2π, and a negative value may be represented as a positive value greater than 2π.

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