Simple harmonic motion displacement equation confusion

In summary, the conversation discusses the use of sine and cosine equations in simple harmonic motion (SHM) problems. It is mentioned that any solution expressed as a cosine can also be expressed as a sine and vice versa. The conversation also highlights that the choice of which equation to use depends on the given values and the initial displacement at time 0, which sets the phase. Ultimately, both equations will yield the same physical answer.
  • #1
randomgamernerd
139
4
Member warned to use the homework template for posts in the homework sections of PF.
Okay, so I have just started with simple harmonic motion(SHM).
So the equation of displacement in my textbook is given as:
X= ACos(wt +x)
where A is the amplitude
X is displacememt from mean position at time t
w is angular frequency
x is phase constant.
Everything going good till now, then comes an illustration where suddenly the equation becomes:
X= ASin(wt +x)
where A is the amplitude
X is displacememt from mean position at time t
w is angular frequency
x is phase constant.
and then onwards in most of the sums they solved with the sin equation and not cos.
I don't get which equation to use when, how and why?
I know every Sin function can be expressed in terms of cos and vice versa, also every SHM is either in the form of Sin or Cos.
But I can't figure out which one to use and why and when.
One observation i made- usually in case of linear SHM about X axis, cos equation is used and in case of linear SHM about y axis, sin equation is used. I used the word usually beacuse that was not the case everytime. Also i don't think this can be a logic as its upto me which one will i take as X axis and which one y if its just a linear SHM.
Someone please help!
 
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  • #2
randomgamernerd said:
Someone please help!

If you look at the graphs of ##\sin## and ##\cos## you'll see they are the same function displaced by ##\pi/2##. In other words:

##\cos(x) = \sin(x + \pi/2)##

Hence:

##A\cos(wt + \phi) = A\sin(wt + \phi + \pi/2)##

Therefore, any solution to any equation that can be expressed as a cosine can also be expressed as a sine and vice versa.
 
  • #3
yeah, i know this fact as I have already mentioned above, but what I mean is which formula to use when we just need to plug in the given values into the formula.
I mean let's suppose we want to find the displacement of a particle undergoing SHM after 2s and its given amplitude is 2m, w= 5rad/s and x= 3 rad.
Then my answer will vary as if I use the sin formula my answer is going to be 2*sin13 which is equal to 0.84m
but if i use the cos equation my answer will be 2*cos13 which is 1.8m
so which one is the correct answer then?and why??
 
  • #4
randomgamernerd said:
yeah, i know this fact as I have already mentioned above, but what I mean is which formula to use when we just need to plug in the given values into the formula.
I mean let's suppose we want to find the displacement of a particle undergoing SHM after 2s and its given amplitude is 2m, w= 5rad/s and x= 3 rad.
Then my answer will vary as if I use the sin formula my answer is going to be 2*sin13 which is equal to 0.84m
but if i use the cos equation my answer will be 2*cos13 which is 1.8m
so which one is the correct answer then?and why??

If you have a solution of ##2 \cos(13)## then if you look for a sine solution you would get ##2 \sin(13 + 90)##.

You will get the same physical answer whether you look for a sine solution or a cosine solution.
 
  • #5
hows that possible, in one case it is having a displacement of 1.8m and in other case it is 0.84m?
 
  • #6
randomgamernerd said:
lets suppose we want to find the displacement of a particle undergoing SHM after 2s and its given amplitude is 2m, w= 5rad/s and x= 3 rad.
That is not enough information. You need to know the displacement at time 0. That sets the phase. If you use sin you will calculate one phase, if cos you will calculate a different phase. The resulting two equations will be completely equivalent.
 
  • #7
randomgamernerd said:
hows that possible, in one case it is having a displacement of 1.8m and in other case it is 0.84m?

##\cos(13) = \sin(13 + 90) = 0.97##

You don't seem to understand that:

##A\cos(wt + \phi) = A\sin(wt + \phi + \pi/2)##

These are precisely the same function; they have precisely the same graph; and, they are precisely the same solution to a SHM problem.
 
  • #8
haruspex said:
That is not enough information. You need to know the displacement at time 0. That sets the phase. If you use sin you will calculate one phase, if cos you will calculate a different phase. The resulting two equations will be completely equivalent.

ok...now i got it
 
  • #9
thnx everyone
 

Related to Simple harmonic motion displacement equation confusion

1. What is the equation for simple harmonic motion displacement?

The equation for simple harmonic motion displacement is x = A cos(ωt + φ), where x is the displacement, A is the amplitude, ω is the angular frequency, and φ is the phase constant.

2. How is simple harmonic motion displacement different from regular displacement?

Simple harmonic motion displacement is a specific type of displacement where the object moves back and forth in a periodic motion, while regular displacement refers to the overall change in position of an object over time.

3. Can you explain the significance of the cosine function in the displacement equation?

The cosine function in the displacement equation represents the oscillatory nature of simple harmonic motion, where the object moves back and forth between two extremes.

4. How do I determine the period of a simple harmonic motion using the displacement equation?

The period of a simple harmonic motion can be determined by calculating the time it takes for the object to complete one full cycle, which can be found by solving for t in the displacement equation x = A cos(ωt + φ).

5. What factors affect the amplitude and phase constant in the displacement equation for simple harmonic motion?

The amplitude and phase constant in the displacement equation can be affected by factors such as the initial conditions of the object, any external forces acting on the object, and the properties of the system itself, such as mass and stiffness.

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