Simple Harmonic Motion problem

In summary: Let's start with x(t) = Asin(wt + ø). If we take the derivative, we get v(t) = Awcos(wt + ø). Now if we use the trig identity sin^2(x) + cos^2(x) = 1, we can rearrange to get w^2 = k/m. This is equation (1) in the picture. So, to go from equation (1) to x(t) = Asin(wt + ø), we use the trig identity and take the derivative. Hope this helps!In summary, deriving x as a function of time for a simple harmonic oscillator involves using Newton's second law and Hooke's law to get a second-order differential equation. To
  • #1
Behroz
5
0
I'm supposed to derive x as a function of time for a simple
harmonic oscillator (ie, a spring). According to my textbook
this is done by using Newton's second law and hooke's law
as this: ma=-kx and one gets a differential equation in
the second order. I can follow the calculations until this
happens: (see attached picture)

(where omega is the frequency)

I do get the equation (1) when I solve the differential
equation myself but I don't understand how equation (1) translates
to (2)?
I assume this must be done by using some trigonometric law?
if so then which one and how??
Thanks
 

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  • #2
Spring mass systems often use omega to represent sqrt(k/m). It isn't a trigonometric law, though if your textbook eventually (I can't see the picture so I don't know) represents the motion as x(t) = Acos(wt + ø) then you will need to use trig.
 
  • #3
Mindscrape said:
Spring mass systems often use omega to represent sqrt(k/m). It isn't a trigonometric law, though if your textbook eventually (I can't see the picture so I don't know) represents the motion as x(t) = Acos(wt + ø) then you will need to use trig.

That's right.. but exactly which trig law do I use and how do I use it to go from equation (1) above in the attached picture to x(t) = Asin(wt + ø).

Or in other words HOW do I go FROM x(t)=x0cos(wt)+(v0/w)sin(wt) ---- (w being = sqrt(k/m) TO x(t) = Asin(wt + ø)
how? HOW? HOW?!??!? HOW?!?
 
  • #4
That is for you to find out. :p

Try working backwards, it might be a little easier.
 

What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is a type of oscillatory motion in which an object moves back and forth in a periodic manner, with a restoring force that is directly proportional to the displacement from its equilibrium position.

What are the characteristics of an object in Simple Harmonic Motion?

An object in SHM exhibits sinusoidal motion, meaning its displacement, velocity, and acceleration can be described by sine or cosine functions. It also has a constant period and amplitude, and its maximum displacement is equal to its amplitude.

What is the equation for Simple Harmonic Motion?

The equation for SHM is x = A*cos(ωt + φ), where x is the displacement from equilibrium, A is the amplitude, ω is the angular frequency, and φ is the phase constant.

How does the mass of an object affect its Simple Harmonic Motion?

For a given spring and amplitude, the period of SHM is independent of the mass of the object. However, a larger mass will result in a smaller angular frequency and a longer period. This means that a heavier object will oscillate slower than a lighter object.

What is the relationship between Simple Harmonic Motion and energy?

In SHM, the total mechanical energy (kinetic energy + potential energy) remains constant. As the object moves back and forth, it alternates between kinetic and potential energy, but the total energy remains the same. This is known as the principle of conservation of energy.

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