Unit 2 · Topic 2.2

2.2 Forces and Free-Body Diagrams

A force is a push or pull from an interaction between two objects. A free-body diagram shows every force on one object or system as a separate arrow, and it's the step that turns a physical situation into equations. Getting the diagram right is often worth points by itself.

Key terms

force
contact force
free-body diagram
normal force
tension
weight

What a force is

A force is a vector that comes from an interaction: something pushes or pulls on something else. Every force has an agent (the object exerting it), so you should always be able to say "the force of ___ on ___". An object can't exert a net force on itself.

Forces are measured in newtons (N), where 1 N = 1 kg·m/s². Some forces act at a distance, like gravity. Contact forces need touching: the normal force, friction, tension and spring forces. On the atomic scale, contact forces are electric forces between the atoms of the two surfaces.

The forces you'll use

Weight (gravitational force), $F_g = mg$ : Earth's pull, straight down toward Earth's center.
Normal force, $F_N$ : a surface pushing on an object, perpendicular to the surface. It's whatever size is needed to stop the object from sinking into the surface, so it is not always equal to mg.
Friction, $F_f$ : a surface's push parallel to the surface, opposing sliding or attempted sliding (2.7).
Tension, $F_T$ : a string or rope pulling along its length, away from the object. Ropes can pull but never push.
Spring force, $F_s$ : a spring pushing or pulling back toward its relaxed length (2.8).
Applied force: a push or pull from a person or other object, in whatever direction the problem says.

Drawing a free-body diagram

A free-body diagram (FBD) isolates one object or system. Draw it as a dot, then draw each force as an arrow starting at the dot and pointing in the force's direction. Label each one clearly, like $F_N$ , $F_g$ or $F_{T1}$ .

On the AP exam, draw each whole force as one arrow. Don't draw components on the FBD, and don't add a "net force" or "centripetal force" arrow; those aren't separate forces. If two forces point the same way, draw them side by side so they don't overlap. Make arrow lengths roughly show relative sizes when you can tell (for a block at rest on a table, $F_N$ and $F_g$ should look equal).

Only include forces acting on the object. The force the object exerts on something else belongs on that other object's diagram.

From the diagram to equations

After the diagram, choose axes. If you know the direction of acceleration, put one axis along it. On an incline, tilt your axes so x runs along the slope, because the acceleration is along the slope. Then only gravity needs splitting into components: $mg\sin\theta$ along the slope and $mg\cos\theta$ into it, where θ is the incline angle.

Write Newton's second law separately for each axis: $\sum F_x = ma_x$ and $\sum F_y = ma_y$ . Components go in these equations even though they don't go on the diagram.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
A block resting on an incline
A 4.0 kg block sits at rest on a 30° incline. Draw (describe) its free-body diagram and find the normal force and the friction force. Use g = 9.8 m/s².
Show the solutionHide the solution
1. Step 1: FBD: three arrows from a dot. Weight, $F_g = mg = 39.2$ N, straight down. Normal force perpendicular to the incline, away from it. Static friction along the incline, pointing up the slope, because without friction the block would slide down.
2. Step 2: Tilt the axes: x down the slope, y perpendicular to it. The block is at rest, so the net force is zero along both axes.
3. Step 3: Perpendicular: $F_N = mg\cos 30^\circ = (39.2)(0.866) \approx 33.9$ N.
4. Step 4: Along the slope: $F_f = mg\sin 30^\circ = (39.2)(0.5) = 19.6$ N, up the slope.
Answer: $F_N \approx 33.9$ N; friction 19.6 N up the incline.
Example 2Calculator allowed
Pulling at an angle (classic trap)
You pull a 10 kg box across a level floor with a rope at 30° above the horizontal. The tension is 50 N. Find the normal force on the box. Use g = 9.8 m/s².
Show the solutionHide the solution
1. Step 1: FBD: weight 98 N down, normal force up, tension 50 N at 30° above horizontal, and friction backward if the floor isn't smooth.
2. Step 2: The box doesn't accelerate vertically, so $\sum F_y = 0$ : $F_N + F_T\sin 30^\circ - mg = 0$ .
3. Step 3: $F_N = 98 - (50)(0.5) = 73$ N.
4. Step 4: The trap is writing $F_N = mg$ . The rope lifts part of the box's weight, so the floor pushes up less. That also means less friction.
Answer: $F_N = 73$ N, less than the box's 98 N weight.

Common mistakes

Assuming the normal force always equals mg. It doesn't on inclines, with angled pulls, or in accelerating elevators.
Drawing components or a "net force" arrow on the FBD. Draw each real force once, as a whole arrow.
Including forces the object exerts on other things. An FBD only shows forces on the chosen object.
Mixing up the incline components. Along the slope it's $mg\sin\theta$ , into the slope $mg\cos\theta$ ; check with θ = 0, where the slope component should vanish.

On the exam

Nearly every free-response question about forces asks for a free-body diagram. Points are usually lost for extra forces, missing forces or arrows that start away from the dot.
A derivation is graded on starting from a correct form of Newton's second law, built from your diagram. Write $\sum F = ma$ for the axis you chose before substituting.

Connected topics

Videos

Forces and free-body diagrams | AP Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Topic 2.2 - Forces and Free Body Diagrams
Lessons With LondotWatch on YouTube (opens in a new tab)
Free-Body Diagram Tips Every AP Physics Student Needs
Flipping PhysicsWatch on YouTube (opens in a new tab)
AP Physics 1 - Unit 2 - Lesson 2 - Drawing FBDs
Allen Tsao The STEM CoachWatch on YouTube (opens in a new tab)
High School Physics - Free Body Diagrams
Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)
Free-Body Diagrams
Bozeman ScienceWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 2.2 Forces and Free-Body Diagrams. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A block sits at rest on a rough ramp. Which list includes every force exerted on the block, and nothing else?

Question 2 of 4Calculator allowed

A ball is thrown upward and is still rising. Ignoring air resistance, which forces act on the ball?

Question 3 of 4Calculator allowed

A 5.0 kg block slides down a frictionless ramp tilted 30° above the horizontal. What is the normal force on the block? Use g = 10 m/s².

Question 4 of 4Calculator allowed

A crate is pushed across a floor. Which force should NOT be drawn on the crate's free-body diagram?

0 of 4 answered