Which equation to use in a Simple Harmonic Motion

[j00kz]

Im kind of confused on which acceleration equation to use.

A = -(kx)/m

or

A = -(w^2)Acos[(angular freq)(time) + phase constant]

as both of these contribute to SHM.

Im guessing I can use the first acceleration equation when i know how far the object stretched and if i don't i can use the second equation if i have k, m


Thanks!

 

[HallsofIvy]

The first is the "linear acceleration". The second is the "angular acceleration". That is suppose "x" is the x coordinate of an object on the edge of a wheel. As the wheel turns, the point accelerates upward (assuming it starts at the right and the wheel is rotating counterclockwise), moving only slowly to the left, accelerating in "leftward" motion as it goes up, then its motion to the left slowing to 0 as it goes back over to left side. That is, the x- speed, left being negative, goes from 0 to a minimum at the top, slowing to 0 at the right, then becomes positive, so that the object is moving back to the right. That's what the "cosine" does.

If instead we measure the angle the line from the center of rotation makes with the x axis, it is steadily increasing. That is the "kx/m" angle argument of the cosine.

 

[technician]

Essentially both equations are the same! Do you realize the meaning of the terms in the equations?
x = displacement
k = stiffness (N/m) of the system
m = mass of the oscillating object
ω is related to the time period of the oscillations. it is called 'angular frequency. ω = 2π/T where T is the time period
A = amplitude of the oscillation
t = time
the displacement,x, is given by ACos(ωt) [or ASin(ωt)...depending on which textbook you use]
The phase constant relates to when the timing is started

with a full analysis it is possible to show that ω^2 = k/m. If you use this information and compare the 2 equations can you see their equivalence?

Which one to use?...depends on how the information is given to you in a question.

 

[sophiecentaur]

The equation involving the trig function could be said to contain more information about the situation than the other one because it includes the 'constants of integration'. The shorter equation applies to an infinity of possible situations.

 

[technician]

The equations are identical/interchangeable.
I would like to see an example that can be solved using only one of these equations,!

 

[sophiecentaur]

A lot of the responses here are confused. Angular acceleration is not involved here; the only 'angle' involved is the so-called angular frequency (ω), which is constant, throughout.
The "A" on the RHS of the lower equation should have a suffix 'max' or somesuch as it is a constant which comes from the initial conditions, whereas the A on the left is time dependent.
@technician
The A in both cases stands for Acceleration. The version you are referring to would describe the displacement and it would not have the ω² in front (the result of double differentiation- to obtain the acceleration).

In the standard derivation of SHM, the first equation is used to show the acceleration is towards the zero position is proportional to the displacement.(We would normally start with the basic force / displacement equation). When this diff equation is solved, you then get the displacement with time:
A_time = A_maxcos[(angular freq)(time) + phase constant]

The second equation in the OP is just the result of double differentiation wrt time of the solution for displacement with time. It's more of an alternative description of the motion than a way of solving what happens in a given physical situation. It is of little use unless the 'spring constant' k and mass m are known. k and m are the important parameters in determining ω.

 

[technician]

I think I can see the cause of confusion. I think the original equations have a misprint.
The second equation should be (I would say)
a = -ω^2.ACos(ωt + ∅)
where 'a' = acceleration and 'A' = amplitude (max value of 'x')

my summary would be for SHM

x = ACos(ωt) [or x = ASin(ωt)]

v = dx/dt = -ωASin(ωt) [or v = ωACos(ωt)]

a = dv/dt = -ω^2.ACos(ωt) [or a = dv/dt = -ω^2.ASin(ωt)]

In older textbooks 'x' was used for displacement and 'r' for amplitude. I prefer this because 'r' conveys the idea of the radius of the circular motion that can be linked to SHM. but now we use 'A' !

For ease of typing I have not included the phase constant but can't guarantee that there are notyping errors