Free-Body Diagrams
A four-step recipe that makes force problems almost mechanical.
Lesson at a glanceShow the key points
- Step 1: choose the system; all its parts must share the same acceleration.
- Step 2: list only the forces exerted ON the system BY other bodies; never forces the system exerts.
- Step 3: draw the free-body diagram with every force as an arrow from one point.
- Step 4: take an axis along the (likely) acceleration, write ΣFₓ = maₓ and ΣF_y = ma_y, and solve.
- For a block pulled at angle θ on a smooth table: T cos θ = Ma and N = Mg − T sin θ.
Most mistakes in force problems happen before any algebra: a force left out, a force counted twice, a force that acts on the wrong body. A short recipe prevents all three.
Newton’s laws are about a particle. Real objects are huge collections of particles, but the laws still work for an extended body as long as every part of it has the same acceleration. With that in mind, here is a four-step algorithm for turning any situation into equations.
Step 1: decide the system
Choose what you will apply the laws to: a block, two blocks stacked together, two blocks joined by a taut string, a piece of rope. The only rule is that every part of the system must share the same acceleration.
For example, if block B rides on block A without slipping, and A is tied by a taut string to block C, then A, B and C all move together: (A), (B), (A + B), (A + B + C) are all fine systems. But a block hanging vertically and a block sliding horizontally, though joined by a string over a pulley, move the same distance in different directions, so their accelerations differ and they cannot be lumped together (at least not yet).
Step 2: list the forces on the system
Now, and only now, list the forces acting on the system by bodies outside it. Forces the system exerts on other things do not belong in the list. Asking “which forces act here?” before choosing the system has no answer, as the widget shows.
| Exerted by | Force | Direction | Nature |
|---|---|---|---|
| Earth | weight of the person, W | down | gravitational |
| Floor | normal force N₁ | up | electromagnetic |
| Box | box presses on the hands, N₂ | down | electromagnetic |
Step 3: draw the free-body diagram
Represent the system by a single point in a fresh diagram and draw every listed force as an arrow starting from that point, with its direction and (if known) its size. This picture is the free-body diagram. Most problems involve forces in one plane, which you draw in the plane of the page.
Step 4: choose axes and write the equations
Pick the x-axis along the direction in which the system accelerates (or is likely to), and the y-axis perpendicular to it. Then resolve every force and write
If the system is in equilibrium, any perpendicular pair of axes will do. If you need more equations, return to step 1 with a different system. Accelerations of connected bodies are often related by kinematics (a taut string gives equal magnitudes), so use the same symbol for them. Then solve.
A 4.0 kg block on a smooth horizontal table is pulled by a string at 30° above the horizontal and accelerates at 1.5 m/s². Find the tension in the string and the normal force from the table. (g = 9.8 m/s²)
An equilibrium example
A 5.0 kg sign hangs at rest from two cables, one at 30° to the horizontal and the other at 60° to the horizontal on the opposite side. Find both tensions.
- 11 · Choose a system whose parts share one acceleration.
- 22 · List only forces ON it BY other bodies.
- 33 · Draw them from a single point: the free-body diagram.
- 44 · Take x along the acceleration; write ΣFₓ = ma, ΣF_y = 0; add systems or kinematic links as needed.
Check yourself
Keep exploring
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