Mathematica® 5.0 Workbook Library

One Dimensional Motion with Constant Acceleration (pdf)

This is a very simple example for starters. The display is an animated graph of x versus t together with a stroboscopic combination of the frames. The parameters that can be varied are x0, v0, and the acceleration.

Two Dimensional Trajectory with Gravity (pdf)

The two dimensional trajectory of an object launched in a uniform gravitational field with velocity (vx0, vy0) is found. The landing point and maximum height are calculated and the solution is plotted.

Shooting Barney (or Curious George) (pdf)

This is intended as backup for the actual demonstration. A bullet is fired with velocity (vx1,vy1) from the origin at t=0. Barney is also dropped from (x2,y2) at t=0, and they collide provided vy1/vx1 = y2/x2. The result are an animated x-y plot and a stroboscopic view of the experiment.

Rotational Motion (pdf)

This is a simple demonstration of the equation for circular rotation of an object. The equations are differentiated to give the velocity and acceleration of the object, and then animated and stroboscopic plots or the motion are shown with a circle overlaid.

Drag Force (pdf)

This demonstration is copied directly from Mathematica for Physics. The motion of a projectile under the influence of both gravity and a viscous drag force -bv is studied. The terminal velocity is found, and plots are made of the position and velocity of the projectile for various values of the drag force constant b.

Coriolis Force (pdf)

Both lab frame and rotating frame animated and stroboscopic plots are made of a ball being thrown at a particular velocity and angle by a subject in the rotating frame. By changing v and the angle one can observe the Coriolis effect in the rotating frame while in the lab the trajectory is always a straight line.

One Dimensional Collision (pdf)

Two objects with m1, v1 and m2, v2 collide elastically in one dimension. The solution is derived, and animated and stroboscopic plots are displayed for x versus t.

Three Dimensional Collision (pdf)

Two objects of m1 and m2 (protons) repel each other with a inverse square force (Coulomb scattering). The solution is found numerically and is stroboscopically plotted in two dimensions. The parameters are set so that m2 is at rest and m1 is incident with velocity x1' with impact parameter y1. More glancing collisions can be observed by increasing the impact parameter, and CMS collisions can be created by setting x1' = -x2'.

Gyroscopic Nutation (pdf)

The nutation of a gyroscope precessing in the horizontal plane is solved for by integration. The parameters are ωh and ωv, the components of the angular velocity in the horizontal plane and in the vertical, and both are initially set to zero. The motion is displayed in animated and stroboscopic plots. The gyroscope falls, then gradually picks up precessional velocity, then rises until it is again at rest. By setting ωh instead equal to ωp, the intrinsic precessional angular velocity, the nutation vanishes. Note that the intrinsic nutational angular velocity ωn is set to four times ωp, so that the gyro completes four nutational cycles for one precession.

Simple Harmonic Motion (pdf)

The equation for an object of mass m moving in one dimension with a Hooke's Law force -kx is solved. The solution for x0 non-zero and v0 zero is graphed and also plotted stroboscopically. The two are superposed.

Damped Harmonic Motion (pdf)

The equation for harmonic motion or an object with a viscous damping force is solved and graphed as a function of t. The "envelopes" of the motion given by the exponential decay term are also plotted and superposed on the full solution.

The Pendulum (pdf)

The motion of a simple pendulum is solved two ways: analytically in the small angle approximation and numerically for any angular amplitude. Graphs of the two solutions are superposed showing the longer period of the exact solution (the restoring force is smaller at large angles because of the sinusoid). The initial amplitude can be changed showing how the difference vanishes at small amplitudes.

Elliptical Orbits in a Gravitational Field (constant L) (pdf)

Starting with the radial effective potential and combining this with the conservation of angular momentum, the orbit of a planet in a gravitational field is calculated. A circular orbit with radius r0 is shown and compared with an elliptical orbit having the same angular momentum L, but with its apogee radius given by r0 +dr.

Elliptical Orbits in a Gravitational Field (constant perigee) (pdf)

This is similar to the previous demonstration, but the circular orbit is compared to an elliptical one having its perigee radius equal to r0. The elliptical orbit is defined by its angular velocity at the perigee ω0 + dω, where ω0 is the angular velocity of the circular orbit of radius r0.

Simple Harmonic Motion (pdf)

The equation for an object of mass m moving in one dimension with a Hooke's Law force -kx is solved. The solution for x0 non-zero and v0 zero is graphed and also plotted stroboscopically. The two are superposed..

The Pendulum (pdf)

The motion of a simple pendulum is solved two ways: analytically in the small angle approximation and numerically for any angular amplitude. Graphs of the two solutions are superposed showing the longer period of the exact solution (the restoring force is smaller at large angles because of the sinusoid). The initial amplitude can be changed showing how the difference vanishes at small amplitudes.

Damped Harmonic Motion (pdf)

The equation for harmonic motion or an object with a viscous damping force is solved and graphed as a function of t. The "envelopes" of the motion given by the exponential decay term are also plotted and superposed on the full solution.

Coupled Pendula (pdf)

The small angle approximation is used to solve the equations of motion for two pendula coupled by a spring. The position of each pendulum as a function of time is plotted and a movie of their motion is displayed.

Wave Propagation - Cosine Shape (pdf)

A Cosine shaped wave is propagated and plotted stroboscopically.

Wave Propagation - Gaussian Shape (pdf)

A Gaussian shaped wave is propagated and plotted stroboscopically.

3D Wave Plots - Cosine Shape (pdf)

Three dimensional plots are made of a Cosine shaped wave: x and t are the horizontal axes and the wave amplitude is plotted on the vertical axis.

Spring-Mass Transmission Line (pdf)

The equations for a short spring-mass transmission line are solved and the positions of the masses are plotted stroboscopically. Only four masses are used to minimize execution time.

Wave Reflection I, II, III (pdf I), (pdf II), (pdf III)

In these notebooks a Gaussian wave is propaged to the right and reflected at a boundary. This is simulated using a left propagating virtual wave with the same shape. A stroboscopic movie is made in each case. In the first case only the physical region is shown. In the second one the region behind the boundary containing the virtual wave is also shown. In the third notebook the amplitude of the wave as well as its time derivative (the vertical velocity) are plotted.

Standing Wave from Reflected Traveling Wave (pdf)

We define two Cosine waves with k=(n π)/L and velocity cw: one wave traveling in the +x direction, and a reflected (inverted) wave originating at x=2L and traveling in the -x direction. Standing waves result when n is an integer. Note: the number of anti-nodes in the resulting standing wave is given by n.

Fourier Transforms (pdf)

The Fourier transforms are calculated and plotted for: a square wave of width T, a Gaussian wave of width τ, and cosine and sine waves.

Wave Propagation and Dispersion (pdf)

We define a Gaussian wave packet of initial width a propagating in the +x direction with v=1. Assuming a quadratic dispersion relation ω(k), the width of the travelling wave packet grows as a function of time. The motion of the packet is plotted stroboscopically.

2D Standing Wave Plot (pdf)

A 3D plot is made of a two dimensional standing wave with rectangular boundaries. A stroboscopic movie is made of the wave motion.

3D Spherical Wave Plot (pdf)

A 2D density plot is made of a periodic 3D spherical wave (with a 1/r falloff of the amplitude). A stroboscopic movie is made of the wave motion.

Spherical Wave from Moving Source (pdf)

The periodic spherical wavefronts from a moving source are summed in a color intensity plot. Wave velocity is c and source velocity is v.  The wavefronts have Gaussian shapes with 1/r falloff.  The values v=1, c=1.5 illustrate the Doppler effect. Try v=c=1.5 for a shock wave, and v=2, c=1.5 for a bow wave. A stroboscopic movie is made of the wave motion.

Huygen's Principle: Many Sources in a Row -> Plane Wave (pdf)

We stroboscopically display the motion of single sperical wavefronts from sources arrayed along y-axis. The wave velocity is c, and the wavefronts have Gaussian shapes. The spacing of the sources is dy, and they are delayed relative to their neighbor by dt. Note that the angle made by the "planar wavefront" with the y axis is then given by Sin (θ) = c dt / dy. Also note that there are symmetrical waves going left and right.

Two Source Interference (pdf)

A stroboscopic density plot is made of the sum of periodic spherical wavefronts being emitted by two sources. The sources are separated by a distance d and the wavelength is set to d/4.

Multiple Source Interference Movie (pdf)

We add together N+1 periodic spherical wave sources spread out evenly over a distance d along the y axis. A density plot movie is made of the propagating (and interfering) wave fronts.

Multiple Source Interference Intensity Projection (pdf)

We add together N+1 periodic spherical wave sources spread out evenly over a distance d along the y axis. The time-averaged wave intensity at a fixed distance from the sources is plotted as a function of y.

Multiple Sources --> Fraunhofer Diffraction (pdf)

Calculate and plot the intensity vs x = π (Slit Width) Sin θ / λ for different numbers of sources spaced evenly across the slit: 2 (blue), 3(green), 4(red), ∞(black). Note how the Fraunhofer diffraction pattern (with small lobes instead of secondary maxima) emerges as the number of sources increases.