Forces & free-body diagrams
The idea
A free-body diagram is the most valuable habit you can build in mechanics: isolate ONE object, draw every force acting on it as a labeled arrow from a dot, and ignore everything the object exerts on the rest of the world. You already know how to spot pushes, pulls, gravity, and support forces; the diagram organizes them so Newton's second law can be applied one axis at a time.
The working method is to choose axes that fit the motion, split any tilted force into components using sine and cosine, then write the second law separately for each axis. Along a direction with no acceleration the forces must sum to zero; along the direction of acceleration they sum to ma. Common players are weight (mg, always straight down), the normal force (perpendicular to the surface), tension (along the rope, pulling), and friction (along the surface).
The most common error is treating the normal force as automatically equal to the weight. It is not a law — it is whatever value makes the perpendicular forces balance. A rope pulling upward at an angle, a hand pressing down, or a tilted surface all change the normal force. Always solve for it from the perpendicular equation instead of assuming mg.
Worked example
A 5.0 kg box sits on a frictionless horizontal floor. A rope pulls it with a tension of 40 N at 30° above the horizontal. Find the box's acceleration and the normal force from the floor.
- Draw the free-body diagram: weight mg = 5.0 × 9.8 = 49 N straight down, the normal force N straight up, and the 40 N tension tilted 30° above the horizontal.
- Resolve the tension into components: horizontal part 40 × cos 30° = 40 × 0.866 ≈ 34.6 N, vertical part 40 × sin 30° = 40 × 0.500 = 20 N upward.
- Apply the second law horizontally, where all the acceleration happens: 34.6 = 5.0 × a, so a ≈ 6.9 m/s² in the direction of the pull.
- Apply it vertically, where acceleration is zero: N + 20 − 49 = 0, so N = 29 N. The rope's upward component carries part of the weight, so the floor pushes up with less than mg.
- Sanity-check: if the rope were horizontal the normal force would be the full 49 N, and if the rope pulled hard enough straight up the box would leave the floor entirely — 29 N sits sensibly between those extremes.
Answer. The box accelerates at about 6.9 m/s² horizontally, and the floor pushes up with a normal force of 29 N.
Check your understanding
- Why does pulling a sled with an angled rope reduce the normal force, and how does that help once friction enters the picture?
- How do you decide which forces belong on a free-body diagram and which forces are exerted by the object on something else?
- What would change in the two equations if the same box were being dragged up a 30° ramp instead of across a floor?
- Where do students most often go wrong when splitting a force into components, and how can you check your sine and cosine choices?
Build the foundations first
Forces & free-body diagrams builds on these concepts. If any feel shaky, start there.