Center of Mass Motion

Scenario Type

System View

Mass m₁

Mass m₂

Mass m₃

× Center of Mass

COM Trajectory

Momentum Analysis

Total Momentum P 0 kg·m/s

COM Velocity v_cm 0 m/s

Total Mass M 0 kg

Position Coordinates

Force Analysis

Gravity F_g 0 N

COM Acceleration a_cm 0 m/s²

Internal Forces Cancel Out

Parameters

Particle Masses

Mass m₁: 2.0 kg Mass m₂: 1.5 kg Mass m₃: 1.0 kg

Initial Conditions

Initial Velocity v₀: 15 m/s Launch Angle θ: 45° Initial Height h₀: 0 m

Explosion Parameters

Explosion Force: 50 N Explosion Time: 1.0 s Duration: 0.2 s

Environment

Gravity g: 9.8 m/s² Air Resistance k: 0

Display Options

Show Particle Trails Show COM Marker Show Motion Decomposition Show Velocity Vectors Auto Scale View

Animation Controls

Animation Speed: 1x

Quick Presets

Center of Mass Equations

COM Position r_cm = (Σmᵢrᵢ)/(Σmᵢ)

COM Velocity v_cm v_cm = (Σmᵢvᵢ)/(Σmᵢ)

Total Momentum P P = M·v_cm = Σmᵢvᵢ

External Force F_ext = M·a_cm

Internal Forces ΣF_int = 0 (no effect on COM)

Motion Decomposition rᵢ = r_cm + r'_i

What is Center of Mass Motion?

The center of mass (COM) of a system of particles is the weighted average position of all the mass in the system. It's a crucial concept in mechanics because the motion of the center of mass follows simple laws regardless of the complexity of internal forces or motions within the system.

Center of Mass Definition

The center of mass position is calculated by weighting each particle's position by its mass and dividing by the total mass. For a continuous object, this becomes an integral. The COM is the balance point where the system would be in equilibrium if supported there.

Projectile Motion of Systems

When a multi-particle system is launched as a projectile, every individual particle may follow complex paths due to internal motions, rotations, or deformations. However, the COM always follows a simple parabolic trajectory.

Explosions and Internal Forces

When a system explodes, internal forces push fragments apart in different directions. However, these forces are equal and opposite (Newton's 3rd Law), so the total momentum remains unchanged.

Sand Pendulum

A sand pendulum (or any pendulum with a leaking container) demonstrates COM motion with changing mass. As sand leaks out, the pendulum's mass decreases, but interestingly, the pendulum's period remains nearly constant if the sand leaks slowly.

Motion Decomposition

Any particle's motion can be decomposed into COM motion plus motion relative to the COM. This is powerful because COM motion follows simple laws while relative motion describes internal dynamics.

Real-World Applications

Sports and athletics, ballistics and firearms, robotics and space, vehicle dynamics, structural engineering.

Historical Context

The concept of center of mass has roots in ancient Greek mathematics and physics. Archimedes used the concept of center of gravity in his work on levers and buoyancy.