Why do the pendulums sometimes pass through each other without colliding?

The model defines a collision only when the relative velocity of the bobs along the line connecting their centers is negative (they are moving toward each other) and they are within a very small separation distance. If they are moving apart or their motion is primarily perpendicular to that line, no collision is triggered. This mimics a 'normal direction only' impact rule, a key simplification for isolating 1D collision physics in a 2D system.

Is the total energy of the system perfectly conserved?

In the idealized model, yes. Between collisions, energy is conserved as the pendulums swing (neglecting air resistance and pivot friction). During the perfectly elastic collisions, kinetic energy is also conserved. However, small numerical errors from the integration algorithm may cause very slight drift over extremely long simulations.

How is this different from Newton's Cradle?

Newton's Cradle typically involves 1D collisions along a straight line. This simulator extends the concept into two dimensions, where the collision direction (the normal) changes with the pendulums' positions. It demonstrates that the 1D collision rules still apply, but only along that instantaneous line of impact, while tangential motion is unaffected.

What does θ¨ represent and why is it important?

θ¨ (theta double-dot) is the angular acceleration of a pendulum bob. It is governed by −(g/L) sin θ, linking the restoring force of gravity to the bob's angular displacement. Monitoring θ¨ helps understand how the pendulum's motion changes between collisions and how a collision provides an impulsive torque, instantly changing θ¨ and thus the future swing.

Pendulum Collision

A pendulum collision simulator explores the dynamics of two spherical bobs suspended as simple pendulums. The core phenomenon is a two-dimensional collision where the bobs interact only when their connecting line aligns with the direction of motion at the moment of impact, approximating a one-dimensional elastic collision along that normal. This condition is enforced by the model: the collision algorithm calculates the relative velocity along the center-to-center line and applies the standard 1D elastic collision formulas for equal masses, conserving both momentum and kinetic energy in that specific direction. Between collisions, each pendulum evolves under the influence of gravity, governed by the torque equation τ = Iα, leading to the nonlinear differential equation θ¨ = −(g/L) sin θ. The simulator numerically integrates this equation, allowing students to observe how the complex, coupled motion emerges from the interplay of simple harmonic oscillation (for small angles) and impulsive interactions. Key learnings include the distinction between 1D and 2D collision rules, the conservation of linear momentum and mechanical energy in an idealized elastic system, and how the transfer of kinetic energy during a collision dramatically alters the subsequent oscillatory phase and amplitude of each pendulum. Simplifications include point-mass bobs, massless rigid rods, frictionless pivots, and perfectly elastic, instantaneous collisions occurring only when specific geometric and kinematic conditions are met.

Who it's for: Undergraduate physics students studying classical mechanics, particularly courses covering collisions, conservation laws, and oscillatory systems.

Key terms

Elastic Collision
Conservation of Momentum
Simple Pendulum
Angular Acceleration (θ¨)
Normal Direction
Kinetic Energy
Numerical Integration
Coupling

Live graphs

How it works

Two equal-length pendulums hang from nearby pivots. While the bobs are in contact we treat them like smooth spheres: resolve velocities into components along the line of centers (normal) and tangential to that line. The normal components undergo a one-dimensional collision with restitution e (e = 1 is elastic); tangential components are unchanged, so no rotational dynamics are modeled. Between hits each bob is a simple planar pendulum. For e = 1 total kinetic energy is conserved at each impact; horizontal momentum is not exactly conserved in the full 2D problem because the pivots exert forces, but the graphs track the sum of horizontal bob momenta as a qualitative check.

Key equations

v₁n′, v₂n′ same as 1D · v_t unchanged · ω = (v·t̂)/L

Pendulum: θ¨ = −(g/L) sin θ between collisions

Frequently asked questions

Why do the pendulums sometimes pass through each other without colliding?: The model defines a collision only when the relative velocity of the bobs along the line connecting their centers is negative (they are moving toward each other) and they are within a very small separation distance. If they are moving apart or their motion is primarily perpendicular to that line, no collision is triggered. This mimics a 'normal direction only' impact rule, a key simplification for isolating 1D collision physics in a 2D system.
Is the total energy of the system perfectly conserved?: In the idealized model, yes. Between collisions, energy is conserved as the pendulums swing (neglecting air resistance and pivot friction). During the perfectly elastic collisions, kinetic energy is also conserved. However, small numerical errors from the integration algorithm may cause very slight drift over extremely long simulations.
How is this different from Newton's Cradle?: Newton's Cradle typically involves 1D collisions along a straight line. This simulator extends the concept into two dimensions, where the collision direction (the normal) changes with the pendulums' positions. It demonstrates that the 1D collision rules still apply, but only along that instantaneous line of impact, while tangential motion is unaffected.
What does θ¨ represent and why is it important?: θ¨ (theta double-dot) is the angular acceleration of a pendulum bob. It is governed by −(g/L) sin θ, linking the restoring force of gravity to the bob's angular displacement. Monitoring θ¨ helps understand how the pendulum's motion changes between collisions and how a collision provides an impulsive torque, instantly changing θ¨ and thus the future swing.