AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/2/2-2)
Unit 2 · Topic 2.2
2.2 Forces and Free-Body Diagrams
A force is a push or pull that one object exerts on another. A free-body diagram shows every force the environment exerts on one object or system, and it's the starting point for nearly every force problem. The AP exam has specific rules for drawing these diagrams.
Key terms
- force
- contact force
- free-body diagram
- normal force
- tension
- weight
What a force is
Every force is an interaction between two objects, so every force has an object that exerts it and an object it's exerted on. Say it fully: "the force exerted on the box by the floor." If you can't name what's exerting a force, it isn't a real force.
Forces are vectors, measured in newtons (1 N = 1 kg·m/s²). An object can't exert a net force on itself; you can't lift yourself by pulling up on your own belt.
Contact forces happen when objects touch: the normal force, friction, tension and spring forces. At the atomic level they all come from electric forces between the atoms of the surfaces. The only force that acts at a distance in AP Physics 1 is gravity.
| Force | Symbol | Direction | Exerted by |
|---|---|---|---|
| Gravitational (weight) | F_g | toward the center of Earth (down) | Earth |
| Normal | F_N | perpendicular to the surface, away from it | a surface |
| Tension | F_T | along the string, away from the object | a string, rope or cable |
| Friction | F_f | parallel to the surface, opposing sliding or a tendency to slide | a surface |
| Spring | F_s | toward the spring's relaxed length | a spring |
Drawing a free-body diagram
A free-body diagram shows one object or system as a dot, which represents its center of mass. Each force the environment exerts on it is a straight arrow that starts on the dot and points in the force's direction. Label every arrow (F_g, F_N, F_T and so on).
The exam's rules: draw each force as its own arrow; don't draw components; if two forces point the same way, draw them side by side, not on top of each other; and make bigger forces longer when you know their relative sizes.
Leave off anything that isn't a force on the object: no velocity arrows, no "ma" arrow, no "centripetal force" arrow and no forces the object exerts on something else.
When your system is several objects that move together, such as two stacked boxes, draw one dot for the whole system and include only the forces from outside it. Forces between the boxes are internal and don't appear.
A quick check: if the object isn't accelerating, the arrows should balance, so the up arrows together should be as long as the down arrows. If it's accelerating, the arrows in the direction of acceleration should win.
From diagram to equations
Next you turn the diagram into equations, one for each direction. Choose axes so that one axis points along the acceleration. On an incline, tilt the axes so x runs along the slope and y is perpendicular to it. Then only gravity has to be split into components: mg sin θ along the slope and mg cos θ into the slope, where θ is the incline angle.
The components belong in your equations, not on the diagram. If a question asks for a separate sketch of components, it will say so.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Box pulled by an angled rope
A 10 kg box is pulled across a rough floor by a rope at 30° above horizontal with a tension of 50 N. The box doesn't leave the floor. Describe its free-body diagram and find the normal force. Use g = 9.8 m/s².
Show the solutionHide the solution
- Step 1: Forces on the box: gravity (down, from Earth), the normal force (up, from the floor), tension (up and to the right at 30°, from the rope) and kinetic friction (to the left, opposing the sliding, from the floor). That's four arrows from one dot.
- Step 2: The box doesn't accelerate vertically, so the vertical forces balance: F_N + F_T sin 30° − mg = 0.
- Step 3: F_N = mg − F_T sin 30° = (10)(9.8) − (50)(0.50) = 98 − 25 = 73 N.
- Step 4: The rope lifts part of the box's weight, so the normal force is less than mg.
Answer: Four forces: gravity, normal, tension and friction; F_N = 73 N
- Example 2Calculator allowed
Block resting on an incline (classic trap)
A 2.0 kg block rests on a 30° incline without sliding. Describe its free-body diagram, then find the components of gravity along and perpendicular to the incline. Use g = 9.8 m/s².
Show the solutionHide the solution
- Step 1: Three forces: gravity straight down (not perpendicular to the incline), the normal force perpendicular to the incline's surface, and static friction pointing up the slope, since the block tends to slide down.
- Step 2: Tilt your axes to match the incline. Gravity (mg = 19.6 N) is the only force not along an axis.
- Step 3: Along the incline: mg sin 30° = 19.6 × 0.50 = 9.8 N, pointing down the slope. Perpendicular: mg cos 30° = 19.6 × 0.866 ≈ 17 N, pointing into the surface.
- Step 4: The trap is drawing gravity perpendicular to the incline or adding extra arrows for mg sin θ and mg cos θ. Those components go in the equations only.
Answer: Gravity, normal force and static friction; components 9.8 N along the slope and about 17 N into the surface
- Example 3Calculator allowed
The force that isn't there
A ball has just left your hand and is rising straight up. Ignoring air resistance, what forces act on it?
Show the solutionHide the solution
- Step 1: List every object touching the ball: nothing. Your hand let go, so it no longer pushes.
- Step 2: Gravity, exerted by Earth, acts at a distance.
- Step 3: There's no upward "force of the throw." The ball keeps moving up because it already has upward velocity, not because a force pushes it.
Answer: Only gravity, pointing down
Common mistakes
- Drawing a force that has no source, like a "force of motion" or "force of the throw." Every force needs an object exerting it.
- Drawing components on the free-body diagram. On the AP exam, each force is one arrow; components go in the equations.
- Assuming the normal force always equals mg. It equals whatever keeps the object from moving into the surface, which changes on inclines, with angled pulls and in elevators.
- Including a centripetal force or an "ma" arrow as if it were a separate force.
On the exam
- Free-response questions very often begin with "draw a free-body diagram." Readers check that every force is present, correctly directed, labeled, starts on the dot, and that nothing extra is drawn.
- Many questions then ask you to write a Newton's second law equation "consistent with your diagram." Make sure every force in your equation appears on your diagram, and the other way around.
Connected topics
Videos
Check yourself
4 questions on 2.2 Forces and Free-Body Diagrams. Pick an answer to see if you got it, and why.
A book rests on a horizontal table. Which list includes every force exerted on the book, and nothing else?
A ball has left a player's hand and is moving up and to the right. Air resistance is negligible. Which forces act on the ball?
A student pulls a sled across rough, level snow with a rope angled above the horizontal. The sled slides at constant speed. How many forces should appear on the sled's free-body diagram?
At the atomic level, what is the origin of contact forces such as the normal force and friction?
0 of 4 answered