AP Physics 1 Unit 2: Force and Translational Dynamics Notes

Introduction

Translational dynamics connects environmental interactions to changes in a chosen system’s center-of-mass velocity.

Choose the system before drawing arrows or writing equations. Forces between parts of that system are internal; forces exerted by the environment are external. A free-body diagram contains only forces acting on the isolated object or system, each labeled by interaction, agent, and receiver.

Newton’s laws relate the external force sum to acceleration in an inertial frame. The direction of motion does not determine the net-force direction. Gravity, friction, springs, contact forces, and tension are specific interactions that must satisfy their own models and conditions.

System and center of mass

define what is being modeled

Interaction map and free-body diagram

identify external force vectors

Newton’s laws

connect the force sum to acceleration

Force models

apply gravity, friction, spring, and contact conditions

Radial and tangential components

describe curved motion without inventing a force

Current exam guidance: Aligned to the current 2026–27 AP Physics 1 framework, with algebra-based modeling, experimental reasoning, and explicit assumptions.

2.1

Systems and Center of Mass

How can an extended collection be replaced by one point when analyzing translation?

Short answer

The center of mass is the mass-weighted average position, and its acceleration is controlled by the net external force on the whole chosen system.

Know firstVector components · Weighted averages · System boundaries

The system boundary decides what counts as external

A system is the object or collection chosen for analysis. An interaction between two parts inside that boundary is internal; an interaction crossing the boundary is external. Internal third-law pairs cancel in the force sum for the complete system, so only external forces change its center-of-mass motion. That does not mean internal forces vanish physically—they can change separations, rotations, and stored energy.

For particles on one axis, $x_{CM}=\sum m_ix_i/\sum m_i$ ; use the same weighted-average rule separately for $y_{CM}$ . The center of mass lies closer to larger masses and can occupy empty space, as at the center of a uniform ring. Geometric symmetry identifies the center of mass only when the mass distribution shares that symmetry. Translating every particle by the same amount translates the center of mass equally, while multiplying every mass by the same factor does not move it.

Physics model

Center of mass follows the weighted mass distribution

Change one mass and watch the balance point shift toward the greater concentration of mass.

Right mass3 kg

Right mass: 3 kg
Center of mass: 0.6 m

left massright masscenter of mass

x_{cm}=\frac{\sum m_ix_i}{\sum m_i}

Read this figure without the graphic

Change one mass and watch the balance point shift toward the greater concentration of mass. The table below states the model assumptions and relationships without relying on color or motion.

Quantity	Meaning
Left mass	2 kg at −3 m
Right mass	Selected mass at +3 m
Center of mass	Mass-weighted position

Locate a three-particle center of mass

System: Three particles treated as one composite system.
Frame: Inertial laboratory frame with a horizontal $x$ axis.
Assumptions: Particles are point masses and their positions are simultaneous.
Givens: $(m_i,x_i)=(1\ \mathrm{kg},0\ \mathrm{m}),(2\ \mathrm{kg},3\ \mathrm{m}),(3\ \mathrm{kg},5\ \mathrm{m})$ .

Compute the total mass: $M=1+2+3=6\ \mathrm{kg}$ .
Compute the first moment: $\sum m_ix_i=1(0)+2(3)+3(5)=21\ \mathrm{kg\,m}$ .
Divide first moment by total mass.
Check that the result lies between the smallest and largest particle positions.

Result

x_{CM}=\frac{21}{6}=3.5\ \mathrm{m}

Units: The numerator has kg·m; division by kg leaves metres.

Boundary cases

Equal masses reduce the expression to the ordinary average position.
A ring’s center of mass can lie where there is no material.

The three-kilogram particle at five metres contributes the largest weight, placing the center of mass at three point five metres.

Choose and audit a center-of-mass model

Draw the system boundary.
List masses and simultaneous coordinates.
Compute $\sum m_ix_i$ and divide by total mass for each axis.
Confirm the result is consistent with symmetry and lies within the coordinate range for positive masses.
Classify each interaction as internal or external to the selected system.

Keep these three ideas

Center of mass is a mass-weighted position.
System boundaries determine internal and external interactions.
Only net external force changes whole-system center-of-mass motion.

Force and Translational Dynamics