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Forces &
Free-Body Diagrams

Master the art of identifying, decomposing, and representing forces acting on objects to solve any mechanics problem.

AP Physics 1

What Is a Force?

A force is a push or pull that acts on an object due to its interaction with another object. Forces are vectors: they have both magnitude (measured in Newtons, N) and direction.

Contact Forces
Require physical contact: normal, friction, tension, applied, spring
Field Forces
Act at a distance: gravity, electric, magnetic

The Common Forces in Mechanics

Gravity \(\vec{F}_g\)
Always downward. Magnitude: \( F_g = mg \)
Normal \(\vec{F}_N\)
Perpendicular to the contact surface
Friction \(\vec{f}\)
Opposes motion. Magnitude: \( f \le \mu F_N \)
Tension \(\vec{T}\)
Pulls along a rope, away from the object

Why Free-Body Diagrams Matter

A free-body diagram (FBD) isolates a single object and shows every force acting on it as an arrow. FBDs are the critical first step in applying Newton's Second Law.

Object \(F_g\) \(F_N\) \(F_a\) \(f\)

Newton's Second Law

The net force on an object equals its mass times its acceleration. This is the foundation for solving every force problem:

The free-body diagram lets you identify every force, resolve it into components, and sum them along each axis. The result: two equations you can solve simultaneously.

From FBD to Equations

\(mg\) \(F_N\) Component Equations x: \(\sum F_x = ma_x\) y: \(\sum F_y = ma_y\)

Mass vs. Weight

Mass Weight
Symbol \(m\) \(F_g\) or \(W\)
Unit kg N (Newtons)
Type Scalar Vector (downward)
Depends on \(g\)? No Yes: \(F_g = mg\)
Changes with location? No Yes (Moon, Mars, etc.)

How to Draw a Free-Body Diagram

1
Identify the object. Isolate it from its surroundings. Draw it as a simple dot or box.
2
List all interactions. What is touching the object? Is there gravity? Air resistance?
3
Draw force arrows. Each arrow starts on the object, points in the force's direction. Longer arrows mean larger forces.
4
Label every force. Use standard symbols: \(\vec{F}_g\), \(\vec{F}_N\), \(\vec{f}\), \(\vec{T}\), \(\vec{F}_a\).
5
Choose coordinate axes. Align one axis with the direction of acceleration when possible.

Example 1: Book on a Table

A 2 kg book sits at rest on a table. Draw its free-body diagram and find the normal force.

Book \(F_g = 19.6\) N \(F_N = 19.6\) N +y

Example 2: Pulling a Box

A 10 kg box is pulled across a floor by an applied force of 50 N at an angle of \(30^\circ\) above the horizontal. The coefficient of kinetic friction is \( \mu_k = 0.3 \). Find the acceleration.

\(mg = 98\) N \(F_N\) 50 N \(30^\circ\) \(f\)

Example 3: Inclined Plane

A 5 kg block slides down a frictionless ramp inclined at \(37^\circ\). Find the acceleration and the normal force.

\(37^\circ\) \(mg\) \(F_N\) \(mg\sin 37^\circ\) \(mg\cos 37^\circ\) \(37^\circ\) Tip: Decompose \(\vec{F}_g\) into components parallel (\(\parallel\)) and perpendicular (\(\perp\)) to the ramp.

Static Equilibrium

An object is in static equilibrium when it is at rest and the net force on it is zero. This means every force is balanced by other forces.

Dynamic Equilibrium
Moving at constant velocity: \(\vec{F}_{\text{net}} = 0\), but \(\vec{v} \neq 0\)
Static Equilibrium
At rest: \(\vec{F}_{\text{net}} = 0\) and \(\vec{v} = 0\)

Newton's Third Law and FBDs

For every force, there is an equal and opposite reaction force. But reaction forces act on different objects, so they never appear on the same FBD.

A B F on A by B F on B by A = Equal magnitude, opposite direction, on DIFFERENT objects

Tension in Ropes and Strings

\(m\) \(T\) \(mg\) FBD of box: \(T = mg\) (at rest, \(a=0\))

Common Mistakes

✕ Including Reaction Forces
Drawing forces the object exerts on other objects on its own FBD. Only include forces acting on the object.
✕ Forgetting Components
Applying forces at angles without decomposing into \(x\) and \(y\) components using \(\sin\) and \(\cos\).
✕ Inventing Forces
Adding a "force of motion" or \(ma\) as a force on the diagram. \(ma\) is the result, not a force.

Force Summary Table

Force Symbol Direction Formula
Gravity \(F_g\) Downward \(mg\)
Normal \(F_N\) \(\perp\) to surface From \( \sum F_y = ma_y \)
Friction \(f\) Opposes motion \(\mu F_N\)
Tension \(T\) Along rope, pulls From \( \sum F = ma \)
Applied \(F_a\) As specified Given

Question 1 of 5

A 3 kg object is at rest on a horizontal surface. What is the normal force acting on it? (Use \( g = 9.8 \, \text{m/s}^2 \))

Question 2 of 5

Which of the following should never appear as a force arrow on a free-body diagram?

Question 3 of 5

A block slides down a frictionless incline at angle \( \theta \). What is the block's acceleration along the incline?

Question 4 of 5

A 10 kg box is pushed with a horizontal force of 60 N across a floor with \( \mu_k = 0.4 \). What is the net force on the box?

Question 5 of 5

Two objects interact via a rope. Object A pulls object B to the right with a tension of 25 N. According to Newton's Third Law, what force does B exert on A?