11.3.1.2: Explorations

Exploration 1: Spring and Pendulum Motion

The animations depict the motion of a mass on a spring and a pendulum, respectively. Restart.

1. For Animation 1, plot the period of the motion vs. amplitude. Drag the ball from equilibrium, varying the amplitude from \(1\text{ m}\) to \(10\text{ m}\) in \(1\text{-m}\) steps.
2. For Animation 2, plot the period of the motion vs. amplitude. Drag the ball from equilibrium, varying the amplitude from \(0.1\) radian to \(1.0\) radian in \(0.1\)-radian steps.
3. What can you say about the period's dependence on amplitude for each animation?

Exploration authored by Mario Belloni.
Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

Exploration 2: Pendulum Motion and Energy

A \(4\text{-kg}\) mass is a pendulum bob and undergoes periodic motion (position is given in meters and time is given in seconds). Also shown are bar graphs representing the kinetic and gravitational potential energies in joules. Restart.

Use the animation and the bar graphs to guide your answers to the following questions.

1. What is the period of the motion?
2. What is the amplitude of the motion (in radians)?
3. From the motion and the bar chart, how do you know energy is conserved?
4. Is this simple harmonic motion?

Exploration 3: Simple Harmonic Motion With and Without Damping

Enter a value for the damping coefficient, the spring constant of the restoring force, or check the "show velocity" box, then press the "set parameters, then drag the ball" button. When you have done this, drag the ball into position and press "play" to run the animation (position is given in meters and time is given in seconds). Restart. When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

1. Find the mass of the ball by using your knowledge of simple harmonic motion.
2. Enable the velocity graph. Does the velocity lead or lag the position graph during simple harmonic motion?
3. How do the frequencies compare if the restoring force is \(-2\ast y\), \(-4\ast y\), and \(-8\ast y\text{ N/m}\)? You may right-click on the graph to create a copy at any time.

Now focus on the damping coefficient and how it affects the motion.

4. Set the restoring force to \(-2\ast y\) and the initial displacement from equilibrium to \(5\text{ m}\). Vary \(b\) from \(0\) to \(2\text{ N}\cdot\text{s/m}\) in steps of \(0.25\text{ N}\cdot\text{s/m}\). What can you say about the frequency of motion as a function of \(b\)?

Exploration 4: Pendulum Motion, Forces, and Phase Space

A \(1\text{-kg}\) pendulum bob is attached to a \(9.8\text{-m}\) massless string to form a pendulum (position is given in meters and time is given in seconds). A graph of angular velocity (\(\text{rad/s}\)) vs. angle (\(\text{rad}\)) is shown. This graph is sometimes called a "phase space" representation of the motion. Restart. In addition,

• the red arrow represents the total force
• the blue arrow represents the force of gravity
• the green arrow represents the velocity

The phase-space representation of motion is just another way to describe an object's motion (like a position vs. time graph). For example, when would the phase-space representation of the motion be circular? Well, \(x\) and \(v\) would have to have the same frequency, be out of phase with each other by \(\pi /2\) radians (or \(90^{\circ}\)), and \(x_{\text{max}}\) and \(v_{\text{max}}\) would have to have the same magnitude. This occurs with simple harmonic motion when \(\omega = 1\text{ rad/s}\).

You must first select the "drag pendulum" button, drag the pendulum bob into place, and then press "play" to begin the animation for a different initial angle.

1. Given the information above and the information depicted in the animation, when does the pendulum motion approximate simple harmonic motion?
2. Determine the maximum angle for approximate simple harmonic motion from the animation.
3. We have considered a special case of simple harmonic motion, \(\omega = 1\text{ rad/s}\). What would the phase-space diagram look like for simple harmonic motion with a general \(\omega\)?

Exploration 5: Driven Motion and Resonance

Enter a value for the magnitude of the driving force and its frequency, the spring constant of the restoring force, or check the "show velocity" box, then press the "set parameters, then drag the ball button. When you have done this, drag the ball into position and press "play" to run the animation (position is given in meters and time is given in seconds). Restart. When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

1. Find the mass of the ball by using your knowledge of simple harmonic motion.
2. Enable the velocity graph. Does the velocity lead or lag the position graph during simple harmonic motion?

Set the restoring force to \(-2\ast y\) and the initial displacement from equilibrium to \(0\text{ m}\). Also set the magnitude of the driving force to \(-1\text{ N}\). Vary the driving frequency between \(0.10\text{ Hz}\) and \(0.20\text{ Hz}\) in \(0.01\text{-Hz}\) steps. You may want each animation to run a while before determining the maximum amplitude.

3. Draw a graph of the maximum amplitude of motion as a function of the frequency.
4. What frequency gives the maximum amplitude?

Note that the mass is not allowed to oscillate past about \(22\text{ m}\).

Exploration 6: Damped and Forced Motion

A mass can be driven by an external force in addition to an internal restoring force and friction. Restart. Specifically, \(F_{\text{net}} = F_{\text{restore}} + F_{\text{friction}} + F_{\text{driving}}\), where the default values are

\[F_{\text{restore}}=-2\ast y,\quad F_{\text{friction}}=-0.2\ast vy\quad\text{and}\quad F_{\text{driving}}=\sin(t)\nonumber\]

You can change these default values as you see fit. Remember to use the proper syntax such as \(-10+0.5\ast t\), \(-10+0.5\ast t\ast t\), and  \(-10+0.5\ast t\wedge 2\). Revisit Exploration 1.3 to refresh your memory.

1. Find the mass. Hint: consider a linear restoring force.
2. Change the restoring force to \(-y-0.1\ast y\ast y\). Is the motion periodic? Is it harmonic? What about \(-y-2.0\)?
3. Design your own force that produces periodic, but not necessarily harmonic, motion.
4. Drive the mass at resonance and explain the behavior of the position graph. How does the behavior change with and without friction?

Drive the system (use a linear restoring force of \(-1\ast y\) and initially no friction) with a function that switches a constant force on and off. This can be achieved with the step function: \(\text{step}(\sin(t/4))\). The step function is zero if the argument is negative and one if the argument is positive. The given function, \(\text{step}(\sin(t/4))\), will therefore produce a square wave with amplitude of one and an angular frequency of one quarter. Note that the total force you should use is \(-1\ast y+\text{step}(\sin(t/4))\). Start the mass in its original position; do not drag it.

5. Draw a graph of the force vs. time superimposed on the position vs. time graph.
6. Why does the system oscillate, stop, and oscillate again?
7. Does this behavior occur at any other frequencies? For example, notice that the function \(\text{step}(\sin(t/4.5))\) produces qualitatively different behavior. Why is this?

Note that the mass is not allowed to oscillate past about \(22\text{ m}\).

Exploration 7: A Chain of Oscillators

Twenty-nine damped harmonic oscillators are driven by an external force, \(\sin(t)\). Each oscillator can be thought of as a mass connected to the floor with a spring. The masses are not connected to each other in any way. One spring has been shown for demonstration purposes. Restart.

The center oscillator, shown in red, is in resonance with the external force. It has a natural frequency of oscillation of \(\omega  = 1\text{ rad/s}\). Oscillators to the left have a spring with a lower spring constant, while those on the right have a larger spring constant. This animation shows how this collection of oscillators responds to the driving force.

The animation starts with all oscillators at rest. The oscillators then begin to move up and down in phase with the driving force during the first few cycles. This motion is, however, transient; and differing amplitudes and phases soon manifest themselves. Since oscillators to the right of the center have a higher resonance frequency, they begin to lead the driving force; while those to the left of center begin to lag. Although the above oscillators are not connected, this phase shift gives the appearance of a traveling wave. After a few hundred oscillations the transient behavior has dissipated; and a resonance curve appears since the amplitude and phase of each oscillator approach their steady state behavior.

1. Find an example of a resonance curve (amplitude vs. frequency) in your textbook. How does the motion of the masses relate to the resonance curve you found in your book? Hint: Look at both amplitude and phase.
2. What effect does the damping coefficient have on the motion of the masses?
3. Assume that the mass of each ball is \(1\text{ kg}\) and that the spring constant for the center spring is \(1\text{ N/m}\). By how much does the spring constant change between neighboring springs?

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.