B Correct SHM Equation: Does € Matter?

AI Thread Summary
The discussion centers on the equations of motion for simple harmonic motion (SHM), specifically the forms X = Asin(wt + €) and V = Awcos(wt + €). It is clarified that both sine and cosine functions can represent SHM, depending on the phase constant €; thus, neither form is inherently correct or incorrect. The relationship between sine and cosine is established through a phase shift of π/2, allowing for conversion between the two based on initial conditions. The choice of sine or cosine often relates to the starting position of the oscillator, influencing the phase constant's value. Understanding these conventions is essential for accurately modeling SHM in various applications.
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A textbook I am using gives the basic eqn of motion of shm as follows :
X = Asin(wt + €)
V =Awcos(wt+€)
But other textbooks and online sources are interchanging sin and cos in above equations, so which is the correct one? Or does it depend on the phase constant €?
 
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Curiosity_0 said:
A textbook I am using gives the basic eqn of motion of shm as follows :
X = Asin(wt + €)
V =Awcos(wt+€)
But other textbooks and online sources are interchanging sin and cos in above equations, so which is the correct one? Or does it depend on the phase constant €?
To paraphrase Gertrude Stein: A sine function is a sine function is a sine function. The best way to write the SHO solution is to let ##X(t) = A ~sin( \omega t + \phi )## where the phase angle ## \phi ## is left to the boundary conditions. Yes, the phase angle makes the solution general to all SHO. Applications like a spring or pendulum tend to use cosine because we usually start the motion (t = 0) at an extreme extension and cos(0) = 1 is the maximum of the cosine function.

-Dan
 
Curiosity_0 said:
which is the correct one
There is no correct or incorrect one. It is simply a matter of convention.
 
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Sin and cos are related by a pi/2 phase. Both works
 
Curiosity_0 said:
A textbook I am using gives the basic eqn of motion of shm as follows :
X = Asin(wt + €)
V =Awcos(wt+€)
But other textbooks and online sources are interchanging sin and cos in above equations, so which is the correct one? Or does it depend on the phase constant €?
Note that $$\cos(wt + \phi) = \sin(wt + \phi + \frac \pi 2)$$In other words, every sine function is also a cosine function with a different phase and vice versa.
 
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Also note that, using a standard trig identity, $$\begin{align} & \cancel{A\cos(\omega t+\delta)=A\cos\delta \sin\omega t+A\sin\delta \cos\omega t} \nonumber \\ & A\sin(\omega t+\delta)=A\cos\delta \sin\omega t+A\sin\delta \cos\omega t \nonumber \end{align}$$With the definitions $$C_1\equiv A\cos\delta~;~~C_2\equiv A\sin\delta$$you have $$A\cos(\omega t+\delta)=C_1\sin\omega t + C_2\cos\omega t$$so the expressions are equivalent. Note that the amplitude and constant phase can be found from the definitions, $$A=\sqrt{C_1^2+C_2^2}~;~~\delta = \arctan\left(\frac{C_2}{C_1}\right)$$so you can go back and forth from one convention to the other.

On edit: Fixed wrong trig function on the LHS of the identity. Also, the phase has a sign ambiguity as noted in posts #7, #9 and #10.
 
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It's more safe to use
$$\delta=\arccos \left (\frac{C_1}{\sqrt{C_1^2+C_2^2}} \right ) \mathrm{sign} \, C_2.$$
 
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Curiosity_0 said:
Or does it depend on the phase constant €?
It does indeed. You can turn sine into cosine or vice-versa by changing the value of the phase constant.

Physically, this is equivalent to starting the oscillator in different positions and different velocities. In other words, when ##t=0## you can make ##x## have any value (in the closed interval ##[-A, A]##) by adjusting the value of the phase constant.
 
kuruman said:
$$A\cos(\omega t+\delta)=C_1\sin\omega t + C_2\cos\omega t$$
I use:
$$A\sin x + B\cos x = sgn(A)\sqrt{A^2+B^2} \sin(x + \delta) \ \ \ (\delta = \tan^{-1}\big (\frac B A \big))$$And
$$A\sin x + B\cos x = sgn(B)\sqrt{A^2+B^2} \cos(x - \delta) \ \ \ (\delta = \tan^{-1}\big (\frac A B\big))$$
 
  • #10
Also here, be careful with the use of the arctan-formula. You have to make sure to get the phase right. It's much easier to use the formula, adapted to your other conventions, given in #7. For this reason, when programming for that purpose you rather use the function atan2, rather than atan!
 
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  • #12
Sigh. I just want to say that you need to be careful with the arctan formula. The result is always a value in the interval ##(-\pi/2,\pi/2)##. What you obviously need to get a unique map between the Cartesian components of an ##\mathbb{R}^2## vector and its polar coordinates is a value within an interval of the length ##2 \pi##. Choosing the interval ##(-\pi,\pi]##, you get ##(x,y)=r(\cos \varphi,\sin \varphi)##, using
$$r=\sqrt{x^2+y^2}, \quad \varphi=\begin{cases} \arccos \left (\frac{x}{r} \right) \text{sign} \, y, &\text{for} \quad y \neq 0, \\ 0 & \text{for} \quad x>0, \quad y=0 \\ \pi & \text{for} \quad x<0, \quad y=0. \end{cases}$$
 
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