AP Physics C: Mechanics — Unit 4 Study Guide

Center of Mass (CoM)

The point representing the average position of the matter in a system or object.

Discrete Systems of Particles

Systems made of individual, separate particles where CoM is calculated using mass-weighted average.

Mathematical Formulation for CoM in 1D

x{cm} = (m1x1 + m2x2 + … + mnxn) / (m1 + m2 + … + mn) = (Σ mixi) / M.

Total Mass (M)

The sum of all individual masses in a system: M = Σ m_i.

CoM Calculation in 3D

x{cm}, y{cm}, z_{cm} are calculated independently using respective coordinates.

Position Vector of CoM

r{cm} = (1/M) Σ mi r_i.

Velocity of CoM

v{cm} = (1/M) Σ mi v_i, related to total momentum.

Acceleration of CoM

a{cm} = (1/M) Σ mi a_i.

Newton's Second Law for Systems

F{net, external} = M a{cm}, where F includes only external forces.

Conservation of Momentum

If F{net, external} = 0, then v{cm} remains constant.

Linear Density (λ)

Mass per unit length: dm = λ dx.

Surface Density (σ)

Mass per unit area: dm = σ dA.

Volume Density (ρ)

Mass per unit volume: dm = ρ dV.

Center of Mass of Uniform Objects

Located at the geometric center (centroid) in uniform density cases.

Center of Mass of Non-Uniform Objects

Requires integration due to varying density.

Infinitesimal Mass Element (dm)

Used in integration; represents a small mass defined in terms of density.

Internal Forces

Do not affect the motion of the Center of Mass of a system.

External Forces

Forces acting on the system from the outside that affect the motion of CoM.

Result of pushing at CoM

The object will move in a straight line without rotating.

Result of pushing away from CoM

The object will rotate around the CoM while translating.

Common Mistake: Mixing $m$ and $M$

Forgetting the total mass M when using integration to find CoM.

Coordinate System Neglect

Overlooking that coordinates can be negative, affecting CoM calculations.

Uniform vs. Non-Uniform Density

Do not assume CoM is at L/2 unless uniform density is specified.

Worked Example: Three-Mass System

Illustrated how to find CoM for a system with distinct mass points.

Worked Example: Non-Uniform Rod

Calculated CoM using variable linear density.

Memory Aid: The Weighted Balance

Think of CoM like calculating a grade point average for weighted sums.