Relevant material from MIT's introductory courses to support students as they study and educators as they teach the AP Physics curriculum.

The EJS Coupled Oscillators and Normal Modes model displays the motion of coupled oscillators, two masses connected by three springs. The initial position of the two masses, the spring constant of the three springs, the damping coefficient for each mass, and the driving force and driving force frequency for the left mass can be changed via text boxes.

The EJS Damped Driven Harmonic Oscillator Phasor model displays the motion of damped driven harmonic oscillator. The resulting differential equation can be extended into the complex plane, and the resulting complex solution is displayed with the real part of this solution being the position of the oscillator. The natural frequency of the oscillator, the damping coefficient, and the driving force and driving frequency can be changed via textboxes.

Study of ordinary differential equations, including modeling of physical problems and interpretation of their solutions. Standard solution methods for single first-order equations, including graphical and numerical methods. Higher-order forced linear equations with constant coefficients. Complex numbers and exponentials. Matrix methods for first-order linear systems with constant coefficients. Non-linear autonomous systems; phase plane analysis. Fourier series; Laplace transforms.

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

The EJS Oscillations and Lissajous Figures model displays the motion of a superposition of two perpendicular harmonic oscillators. The simulation shows the result of the superposition. The amplitude and frequency of the oscillators can be changed via textboxes.

This Java archive contains a collection of simple Easy Java Simulations (EJS) programs for the teaching of computer-based modeling. The materials and text of this resource appeared in an article of the same name in The Physics Teacher [Phys. Teach. 76, No. 45, pp. 474-480 (2007)].

This unit is intended to develop your understanding of Newtonian mechanics in relation to oscillating systems. In addition to a basic grounding in calculus, this unit assumes that you have some understanding of how to solve second-order linear constant-coefficient differential equations; how to take the dot product of two vectors; of solving statics problems; and of applying Newton's second law to mechanical problems.

This activity is a lab where students make measurements of a mass on a spring and work through appropriate calculations dealing with simple harmonic motion.

The EJS Pendulum on an Accelerating Train model displays the model of a pendulum on an accelerating train. The problem assumes that the pendulum rod is rigid and massless and of length, L = 2, and the pendulum bob is of mass, m = 1. The initial pendulum angle and the initial velocity of the train can be changed via textboxes, and the acceleration of the train can be changed via a slider.

The EJS Platform on Two Rotating Cylinders model displays the model of a platform resting on two equal cylinders are rotating with opposite angular velocities. There is kinetic friction between each cylinder and the platform. The separation between the cylinders and the coefficient of kinetic friction can be changed via textboxes.

The EJS Quartic Oscillator model displays the motion of a bead moving without friction along a horizontal rod, while tied to two symmetric springs. Both the motion of the masses and the phase space plot are shown in the simulation. The natural length of the springs can be changed via textboxes.

The EJS Spinning Hoop model displays the model of a bead moving along a hoop which is spinning about its vertical diameter with constant angular velocity ?. Friction is negligible. The simulation displays the motion of the bead as well as a plot of angle vs. time for the bead. The angular velocity of the hoop and the starting angle of the bead can be changed via sliders.

The EJS Spring Pendulum model displays the model of a hollow mass that moves along a rigid rod that is also connected to a spring. The mass, therefore, undergoes a combination of spring and pendulum oscillations. The initial position and velocities, as well as the spring constant can be changed via textboxes.

The EJS Strange Harmonic Oscillator model displays the motion of two masses connected by a massless rigid rod and the masses may move without friction along two perpendicular rails in a horizontal table. The energy of the oscillator system can be changed via a slider.