# Phase space diagram for a spring in simple harmonic motion

• Yosty22
In summary, the problem involves a mass connected to a spring with given values for mass and spring constant. The phase space diagram for the oscillator can be accurately sketched using the equations x(t) = 0.4cos(wt) and x'(t) = 0.4wsin(wt). The total energy can be expressed as a function of x and x', and by plugging in values and simplifying, the shape of the trajectory can be identified.
Yosty22

## Homework Statement

A mass m = 750 g is connected to a spring with spring constant k = 1.5 N/m. At t = 0 the mass is set into simple harmonic motion (no damping) with the initial conditions represented by the point P in the phase space diagram at the right. **(This phase space diagram has nothing on it, that is my job to fill in. The point p is located at (x, x dot) = (0.4,0).

a. Using the given information, sketch an accurate phase space plot for the oscillator. Explain your reasoning and show all work.

## The Attempt at a Solution

I know from studying simple harmonic motion in the past that the solution comes of the form x'' = -w2x, giving the solution x(t) = Acos(wt) + Bsin(wt)

Since I know that at t = 0, it is released at x = 0.4, so x(0) = 0.4. This gives A = 0.4. Next, I know that at t = 0, x dot (dx/dt) = 0. This gives B = 0.

Therefore, our equations become: x(t) = 0.4cos(wt) and x'(t) = 0.4wsin(wt).

Next, I know that the total energy (call it E) is equal to .5mv2 + .5kx2. Since V = x'(t), I can square my x'(t) equation to get the kinetic energy and square my x(t) equation to get the potential energy term. This gives me:

E = (.5m)(.16cos2(wt)) + (.5k)(.16w2sin2(wt))

However, I am unsure as of how to make this into a phase space diagram. Any help would be appreciated.[/sup]

Yosty22 said:
Therefore, our equations become: x(t) = 0.4cos(wt) and x'(t) = 0.4wsin(wt).

These equations give enough information to plot points on the trajectory in phase space.

If you want to identify the shape of the trajectory, then write the energy equation in terms of the symbols x and x'. (x' is the same as "x dot".) Plug in values for all parameters and simplify to get a relation between x and x'. Try to identify the shape of the trajectory from this relation.

## 1. What is a phase space diagram?

A phase space diagram is a graphical representation of the motion of a system in a multi-dimensional space, where each axis represents a different variable or parameter of the system. It is often used to visualize the behavior of a system over time.

## 2. How is a phase space diagram useful for understanding simple harmonic motion?

In a phase space diagram for a spring in simple harmonic motion, the x-axis represents the displacement of the spring from its equilibrium position, and the y-axis represents the velocity of the spring. This allows us to see the relationship between these two variables and how they change over time, providing a deeper understanding of the dynamics of the system.

## 3. What does the shape of a phase space diagram for a spring in simple harmonic motion tell us?

The shape of a phase space diagram for a spring in simple harmonic motion is an ellipse. This tells us that the motion of the spring is periodic and follows a sinusoidal pattern, with the amplitude and frequency determined by the initial conditions of the system.

## 4. Can a phase space diagram be used to predict the future behavior of a spring in simple harmonic motion?

Yes, a phase space diagram can be used to predict the future behavior of a spring in simple harmonic motion. By analyzing the shape and trajectory of the ellipse, we can determine the amplitude, frequency, and period of the motion, and make predictions about the future behavior of the system.

## 5. How does the energy of a spring in simple harmonic motion relate to its phase space diagram?

The total energy of a spring in simple harmonic motion remains constant. This means that as the spring moves along its trajectory in the phase space diagram, it will always stay on the same ellipse. This relationship between energy and the phase space diagram allows us to better understand the conservation of energy in the system.

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