# Deriving equaions for harmonic oscillations

1. Nov 25, 2009

### frozenguy

1. The problem statement, all variables and given/known data
Suppose a particle of mass 0.240 kg acted on by a spring undergoes simple harmonic
motion. We observe that the particle oscillates between x = 0.200 m and x = - 0.200 m
and that the period of the oscillation is 1.20 s. At time t = 0, the particle is at x = 0 and
has positive vx.
(a) Beginning with whatever general harmonic solution you prefer to use, derive
equations for x(t), vx(t), and ax(t). The only symbols your results may contain are ω and t.
Be sure to use SI units and include units in your results.
(b) Calculate the angular frequency of the oscillation and the spring constant of the
spring.
(c) At what time will the mass reach x = -0.100 m? Calculate vx at that time.
(d) Derive equations for K(t) and U(t). The only symbols your results may contain are ω
and t. Be sure to use SI units and include units in your results.
(e) What is the total energy of the motion?

2. Relevant equations
a.) I was thinking I could use the (d^2y)/(dt) + (k/m)y=0
Thats the harmonic oscillator differential equation.

d.) The total energy is governed by how much the spring is initially compressed or stretched right? So Ui + Ki = Uf + Kf?

3. The attempt at a solution
Can I derive the three equations by integrating the harmonic oscillator diff eq?
I think I'm sure I can get b. and c. once I can get these derived. When it asks to use any of the general solutions, is that referring to the equations using cosine and sine functions and phi?

2. Nov 25, 2009

### rock.freak667

yes solve

$$\frac{d^2x}{dt^2}+\frac{k}{m}y=0$$

and find x(t)

3. Nov 25, 2009

### frozenguy

So I use the equation $$\frac{d^2x}{dt^2}+\frac{k}{m}y=0$$ ? I thought the y should be an x or the x should be a y, as in the same.

well isn't that the acceleration equation?

So I integrate to get V(t). which would be $$\frac{dx}{dt}+\frac{kxt}{m}$$ if they are the same variable ya? Because I dont know what y is supposed to be in that other one.

And then to get position vector, I integrate $$\frac{dx}{dt}+\frac{kxt}{m}$$

And get $$x+\frac{kxt^2}{2m}$$

Or V(t)=$$-\frac{kxt}{m}$$

and X(t)=$$-\frac{kxt^2}{2m}$$ ?

But I can only use omega and t.

So instead of $$\frac{k}{m}$$ should i just use $$\omega^2$$ which would be a constant?

Last edited: Nov 25, 2009
4. Nov 25, 2009

### rock.freak667

Sorry, it should be d2x/dt2+(k/m)x=0

Do you know how to solve a 2nd order ODE with constant coefficients?

5. Nov 25, 2009

### frozenguy

No I don't. We have only begun 1st order linear in Calc 2. Are my integrals wrong?

6. Nov 25, 2009

### ehild

You do not need to solve any differential equations. It is given that the particle undergoes simple harmonic motion. You certainly have learnt that simple harmonic motion means that the displacement of the particle along a line is a sinusoidal function of time. The particle moves along the x axis now, so this function is of the form

$$x(t) = A sin (\omega t + \phi)$$.

You have to find the amplitude, A, the angular frequency, omega, and the phase constant, phi from the given data.

ehild

7. Nov 25, 2009

### frozenguy

So I can start with that? I just get confused because it says to derive x(t), v(t), a(t) starting from any general harmonic solution. Is what you posted a general harmonic solution for the position as a function of time? So I only have to derive two equations, v(t) and a(t) which would be the first and second derivatives, respectively, of the posted equation?

8. Nov 25, 2009

### ehild

The form of solution is given. "Beginning with whatever general harmonic solution you prefer to use" means that you can start either with sine or with a cosine function or with their combination.
The first task is to find the amplitude, angular frequency and phase constant.

At t=0, the particle is at x=0 and the derivative of x(t) ( the velocity) is positive. So what is the phase constant?
The period is given. Get the angular frequency.
The maximum displacement from equilibrium is 0.2 m, What is the amplitude of the oscillation?
And yes, the velocity is the first derivative of the x(t) function and the acceleration is the second derivative with respect to time. Give all three functions, with the appropriate numerical data.

ehild