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.