Deriving equaions for harmonic oscillations

In summary, a particle of mass 0.240 kg undergoes simple harmonic motion with an amplitude of 0.200 m and a period of 1.20 s. The equations for position, velocity, and acceleration are x(t) = A sin(ωt + ϕ), v(t) = ωA cos(ωt + ϕ), and a(t) = -ω^2A sin(ωt + ϕ), respectively. The angular frequency is ω = 2π/1.20 s ≈ 5.24 rad/s and the spring constant can be calculated using the formula k = mω^2 ≈ 0.325 N/m. At t = 0, the particle
  • #1
frozenguy
192
0

Homework Statement


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?


Homework 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?


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?
 
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  • #2
yes solve

[tex]\frac{d^2x}{dt^2}+\frac{k}{m}y=0[/tex]

and find x(t)
 
  • #3
rock.freak667 said:
yes solve

[tex]\frac{d^2x}{dt^2}+\frac{k}{m}y=0[/tex]

and find x(t)

So I use the equation [tex]\frac{d^2x}{dt^2}+\frac{k}{m}y=0[/tex] ? 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 [tex]\frac{dx}{dt}+\frac{kxt}{m}[/tex] if they are the same variable ya? Because I don't know what y is supposed to be in that other one.

And then to get position vector, I integrate [tex]\frac{dx}{dt}+\frac{kxt}{m}[/tex]

And get [tex]x+\frac{kxt^2}{2m}[/tex]


Or V(t)=[tex]-\frac{kxt}{m}[/tex]


and X(t)=[tex]-\frac{kxt^2}{2m}[/tex] ?

But I can only use omega and t.

So instead of [tex]\frac{k}{m}[/tex] should i just use [tex]\omega^2[/tex] which would be a constant?
 
Last edited:
  • #4
frozenguy said:
So I use the equation [tex]\frac{d^2x}{dt^2}+\frac{k}{m}y=0[/tex] ? 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 [tex]\frac{dx}{dt}+\frac{kxt}{m}[/tex] if they are the same variable ya? Because I don't know what y is supposed to be in that other one.

And then to get position vector, I integrate [tex]\frac{dx}{dt}+\frac{kxt}{m}[/tex]

And get [tex]x+\frac{kxt^2}{2m}[/tex]


Or V(t)=[tex]-\frac{kxt}{m}[/tex]


and X(t)=[tex]-\frac{kxt^2}{2m}[/tex] ?

But I can only use omega and t.

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

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

Do you know how to solve a 2nd order ODE with constant coefficients?
 
  • #5
rock.freak667 said:
Sorry, it should be d2x/dt2+(k/m)x=0

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

No I don't. We have only begun 1st order linear in Calc 2. Are my integrals wrong?
 
  • #6
You do not need to solve any differential equations. It is given that the particle undergoes simple harmonic motion. You certainly have learned 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

[tex] x(t) = A sin (\omega t + \phi) [/tex].

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

ehild
 
  • #7
ehild said:
You do not need to solve any differential equations. It is given that the particle undergoes simple harmonic motion. You certainly have learned 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

[tex] x(t) = A sin (\omega t + \phi) [/tex].

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

ehild

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
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
 

Related to Deriving equaions for harmonic oscillations

1. What is a harmonic oscillator?

A harmonic oscillator is a system that exhibits a repetitive motion or oscillation in which the restoring force is directly proportional to the displacement from the equilibrium position. Examples of harmonic oscillators include a mass attached to a spring and a pendulum.

2. How do you derive equations for harmonic oscillations?

To derive equations for harmonic oscillations, we use the principles of Newton's laws of motion and Hooke's law. We start by setting up a free-body diagram and applying Newton's second law to find the net force acting on the object. Then, we use Hooke's law to relate the force to the displacement and solve for the equation of motion.

3. What is the equation of motion for a harmonic oscillator?

The equation of motion for a harmonic oscillator is F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position. This equation is also known as the simple harmonic motion equation.

4. How do we determine the period and frequency of a harmonic oscillator?

The period of a harmonic oscillator is the time it takes for one complete oscillation, while the frequency is the number of oscillations per unit time. To determine the period and frequency, we use the equation T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant.

5. Can we apply the equation of motion for a harmonic oscillator to real-world situations?

Yes, the equation of motion for a harmonic oscillator can be applied to real-world situations such as in the study of vibrations and waves. It is also used in engineering and design, for example in designing suspension systems for vehicles or in the construction of buildings to reduce the effects of earthquake vibrations.

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