This applet shows the response of a damped sinusoidally driven pendulum over a wide range of parameters. You can investigate period doubling and the approach to chaos, and see different attractors.
This applet exhibits two responses for different initial conditions, or different driving forces. You can see how nearby paths converge in the nonchaotic regime, then diverge when chaos sets in, by varying the driving force. You can also see how slightly different initial conditions can lead to different stable final states.
We follow two trajectories having identical parameters except for the initial position. Here we plot the log of the absolute separation as a function of time. The difference amplitude goes as Ke^{λt}sinωt decaying exponentially in the nonchaotic regime, but increasing exponentially when the system is chaotic. The coefficient λ is the Lyapunov exponent. You can measure it by adjusting the position and slope of the red line to align it with the peaks of the curve.
A better way to understand the pendulum's motion is to plot its path in state space (a.k.a. phase space) with x,y coordinates its position φ and velocity ̇φ respectively. For very small driving force, the orbit is close to elliptical (circular in the right units) but on increasing the force it distorts somewhat, then splits into two closely parallel tracks--actually one track (of course) that crosses itself at one point. Further increase in force causes visible splits into 4 tracks, then further doubling leading to chaos, in which the path wanders in a larger region. For some values of the driving force it settles into beautiful patterns, called attractors. These we'll discuss further in the next applet, on the Poincaré section.
To focus on the essential features of the changes in the orbital pattern of the driven damped pendulum as the driving force changes, We can take a Poincaré section: plot the position in state space at some instant in time, then repeat exactly one cycle later, two cycles later, etc. For low driving force, this Poincaré section is simply one dot (after transients), on gradually increasing the force it goes to two, then four, then eight, etc., thus the simplest representation of the doubling sequence. On further increase into the chaotic region, the dots become a curve, then in certain value ranges a strange attractor, a fantastic fractal pattern. You can explore all this with the applet.
This applet describes a single atom gas moving in one dimension. It accelerates or decelerates only through classical collisions with the moving piston on its container. This alone is enough to explain why the gas gets warm when it is compressed and cool when expanded.
Newton's imaginary cannon is on a high mountain, and fires a cannonball above the atmosphere. Depending on speed,it could fall, circle the earth, or fly away depending on how hard it was fired, showing the motion of an cannonball near the Earth and the circling motion of the Moon around the Earth are different aspects of the same thing.
Rutherford scattering from a Thomson Atom and from a Nuclear Atom
Relevant Lecture in Physics 252 Modern Physics.
Watch alpha particles scatter from Thomson's model atom with charge spread evenly through all the atom's volume (as most people believed), and another model with all the charge concentrated in a small central nucleus.These were the two competing theories of atomic positive charge distribution in 1910. The startling experimental discovery of occasional large angle scattering proved there had to be a nucleus. This was the birth of nuclear physics.This applet shows how a chain reaction develops. A nucleus in the middle emits a neutron in a random direction, if the neutron hits another nucleus it is absorbed, the hit nucleus emits two neutrons and dies.
A full chain reaction, involving essentially all the nuclei, is much more likely to occur with a bigger piece of material. Try it!
This applet shows how atomic velocities cause the Brownian Motion of a dust particle. In one panel a small ball jitters. In the next, we see that it jitters because many smaller balls bat it rapidly about. Here's the original Java version.
The classic attempt to detect the Earth's motion through the ether: light waves were thought to be oscillations in the ether, analogous to sound waves through air -- so should have slightly different velocities in different directions, with, against, or across the ether flowing past the Earth. The apparatus is a race course between light going up and down stream, and light going accross and back, which should be a little quicker, a detectable difference.
A clock bouncing light between two mirrors is animated, to show vividly why taking the speed of light the same in all inertial frames leads inevitably to time dilation.
Here is Galileo's own diagram explaining that the parabolic path of a projectile, or a ball rolling off the edge of a table, can be regarded as compounded of horizontal motion at constant speed, plus vertical motion identical to that of a ball falling vertically. We've animated his drawing.
Shoot a cannon (or throw a ball!) to see how high and far the ball flies.
This is very accurate, and will plot multiple trajectories. You can include realistic values for air resistance for cannonballs, ping pong balls, and anything in between.
The waving motion of a string can be understood from Newton's laws if you think about a little bit of it, and the force on it at any instant as the imbalance of the tensions tugging at the two ends.
A sound wave is waves of compression and rarefaction generated by a vibrating object. As the wave travels through the air, it gives the impression of carrying some air along, but actually air stays where it is, vibrating about its rest position.
A wave pulse traveling down a uniform string under tension comes to the end of the string. A fixed end reflects it upside down, a free end reflects without inversion. Both are explained by comparing with pulses colliding on a string twice as long.
A wave pulse traveling down a uniform string under tension comes to a join with a string of different density. The pulse is partially transmitted, partially reflected. The relative densities of the two strings can be set by the slider. Note the phase of the reflected wave as a function of relative string densities. The physics is the same for a wide variety of wave reflection phenomena.
The moving object puts out wave pulses at regular intervals. The cursor is a microphone! Hold it somewhere and listen as the source moves past you.
Synchronized waves emanating from two slits form a pattern with destructive interference in certain directions, where the waves are out of phase.
The slit separation and wavelength can be adjusted to explore how the pattern changes.
We move across the screen point by point to see how the two waves successively augment and cancel one another.
The slit separation and wavelength can be adjusted to explore how the pattern changes.
This animation relates the motion of the basic Carnot engine to the corresponding theoretical cycle in the P,V plane, and a thermometer.
A box has red particles in the left hand half, green in the right. They all begin with the same speed but random direction. Watch as they mix! How long will it take? The vertical lines give the average positions of red and green particles.
Try different numbers of particles and different sizes
A dynamic realization: initially we give all molecules the same speed, collisions rapidly spread them out into Maxwell's predicted pattern.
The yellow curve is the theoretical prediction for Maxwell's distribution in two dimensions. (It begins linear, the three-dimensional distribution begins parabolic.)
A dynamic realization: initially we give all molecules the same speed, but a steady gravitational force pushes them down the pipe. On encountering the walls, they lose their horizontal (along the pipe) velocity. Vertically, they bounce off. The velocity down the pipe approaches a parabolic profile.
Natural motion for a free particle is straight line motion: for the particle to go around the sides of a square, it needs an appropriate impulsive kick to change its direction at each corner. Going from a square to a many-sided polygon, the kicks get more frequent, and in the limit of an infinite number of sides, it follows that uniform circular motion must result from a continuous inward force.
Prove Pythagoras' theorem by moving the triangles around--in the original configuration, the total area inside the red square is equal to that of the four identical triangles plus the area of the central square on the hypotenuse (side = triangle's longest side). Rearrange to see that the total area inside the red square is also that of the four triangles plus the sum of the areas of squares on the other two sides.
The only three-dimensional flat faced solids with every face identical, and every vertex identical.
The path traced by a point on a wheel as the wheel rolls on a flat surface.
The path traced by a point on a wheel as the wheel rolls inside a circle.
simple harmonic motion as a shadow of circular motion
Change the group and phase velocities of interfering sine waves. Relevant Lecture for Physics 252 Modern Physics.
A short movie of the Moon entering the Earth's shadow, and how it appears from Earth. The ancient Greeks used this picture to estimate the distance to the Moon. Their estimate was within ten per cent of the correct answer.
A simple applet showing the relative orbital sizes and periods of the inner planets: Mercury, Venus, Earth and Mars.
A simple applet showing the relative orbital sizes and periods of the outer planets: Jupiter, Saturn, Uranus and Neptune.
This applet is a slightly simplified representation of Ptolemy's model for the motion of the inner planets: this simple model can successfully account for the motion of the planets in the heavens to a very good approximation, but keeps the Earth itself at rest. Ptolemy actually refined the model further with small epicycle corrections, necessary because the planetry orbits are not perfect circles, but (almost circular) ellipses.
Point to notice: Venus and Mercury never get very far from the Sun in the sky. Think about how that works in the modern picture.
An applet demonstrating that the motion of Mars through the heavens agrees exactly with the Copernican (modern) view of the Solar System and the ancient Earth-centered epicyclic model of Ptolemy.
It follows that observing the path of Mars through the stars, (or the path of any other planet, by similar model comparisons) cannot settle which model physically represents the real Solar System.
In the 1260's, al Tusi, an astronomer in Iran, created an ingenious model to represent an observed small linear oscillatory component of planetary motion in terms of combined circular motions, thus extending Ptolemy's model. Here is his model.
An applet demonstrating that the motion of Mars through the heavens agrees exactly with the Copernican (modern) view of the Solar System and the ancient Earth-centered epicyclic model of Ptolemy.
It follows that observing the path of Mars through the stars, (or the path of any other planet, by similar model comparisons) cannot settle which model physically represents the real Solar System.
Kepler theorized that the planets moved on spherical surfaces separated by having Platonic solids just fit between neighboring orbits: see his model here.
This applet plots the orbital motion of a spaceship launched from Earth at a given speed. Find the best orbit to Mars by launching at different speeds and at different times, to meet with Mars as it moves around its orbit.
This applet shows Venus' orbit from different perspectives to understand how the sunlit half of the planet exhibits phasess like the Moon.
This applet plots planetary orbits with a force proportional to r^{n} over a range of values of n. Find out when the orbits are unstable!
This applet adds two moving sine waves over a wide range of wavelengths and frequencies. Try to predict the results--some are surprising!
For a heavily damped oscillator, two different time scales emerge. Discover the physics with this applet.
How does a damped simple harmonic oscillator settle after being displaced, or being given an initial kick? This applet traces its path as a function of time for these two different initial conditions. For low damping, it oscillates with decreasing amplitude. For high damping, it just settles down. The dividing point is called critical damping. Find where it is. This turns out not to be the ideal damping for a car's shocks!
How important are initial conditions in dictating how the oscillator behaves over time? Here we plot the two very different complicated curves traced in response to different beginnings. But when we look at the difference between the two, it is revealed to be much simpler!
How does a simple harmonic oscillator react to an external driving force at a different frequency? This applet traces its path as a function of time, with sliders to adjust driving frequency and amplitude, damping, and initial conditions. Look for resonant behavior, and how it's affected by frequency tuning and damping.
A damped oscillator driven well below its natural frequency lags behind the driving force. This changes as it goes through resonance. You can explore how this happens with this applet, and see how the power transfer varies with frequency, maximized at the resonant frequency, although that is not necessarily where the amplitude response is maximum.
As a preliminary to discussing the driven damped anharmonic oscillator, we explore the motion of an undriven, undamped particle in an anharmonic potential, a simple harmonic oscillator potential with adjustable cubic and quartic terms added.
A simple harmonic oscillator with a repulsive quartic potential term (and damping) added has a resonant frequency which increases with amplitude. Howevever, on driving it at a gradually increasing frequency to keep it resonant, there is a sudden discontinuous drop in amplitude, at a frequency that depends on then damping and the coefficients of the potential terms. The behavior can be explored with this applet.