Forces and Free-Body Diagrams

Lean on a wall and the wall pushes back on your hands; let go of a pencil and Earth pulls it down. Every force has a source, some other object doing the pushing or pulling. A free-body diagram is the drawing that lists those forces for one chosen object, and every Newton's-law equation you write in this course is read off one.

A force is an interaction between two objects

A force is a push or a pull that one object exerts on another. It is a vector: it has a magnitude, measured in newtons ($1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$), and a direction. A $2.0\ \text{N}$ push to the left and a $2.0\ \text{N}$ push to the right do opposite things.

Because a force is an interaction, you can always name it in the form "force on A by B". Before you draw an arrow, name the object that exerts it. If you can't name one, the force doesn't exist. "The force of the throw" on a ball that has left your hand fails this test: nothing is touching the ball anymore, so nothing is throwing it.

The same idea explains why an object can't exert a net force on itself. Sitting in a stalled car and pushing hard on the dashboard does not move the car: your hands push the dashboard forward, the dashboard pushes your hands backward, and the seat back pushes you forward to keep you in place. Every push between parts of the car-plus-you system is matched by an equal push the other way (Topic 2.3 makes this precise). To change the motion of a system, something outside it has to act on it. A car accelerates because the road pushes forward on its tires, not because the engine pushes on the car; the engine is part of the car.

Contact forces are electric forces seen from far away

Forces come in two kinds. Long-range forces act across empty space; in mechanics the one you meet is gravity (Topic 2.6). Contact forces act only when two objects touch: normal forces, friction, tension, spring forces and air resistance.

Contact forces aren't a separate kind of physics. When the atoms at two surfaces get very close, the electrons around them repel each other. A book resting on a table squeezes the table's surface atoms slightly closer together, like compressing trillions of tiny stiff springs, and those atoms push back. The sum of all those microscopic electric forces is what you call the normal force. This picture explains a fact students often miss: the normal force has no fixed size. Press down on the book and the surface compresses a bit more and pushes back harder. The normal force is whatever it needs to be to keep the book from sinking into the table.

A useful search procedure: an object feels gravity, plus up to two forces (a normal force and friction) from each surface it touches, plus one force from each rope, spring or hand attached to it, plus drag if it moves through air or a liquid fast enough for that to matter. Nothing else.

Drawing a free-body diagram

A free-body diagram shows every force the environment exerts on one object or system, and nothing else. Draw it this way:

1. Pick the object or system and draw it as a dot. The dot stands for the center of mass (Topic 2.1): the model treats the object as if all its mass sat at that point.
2. Find every force on it with the search procedure above, naming the object that exerts each one.
3. Draw each force as one straight arrow that starts at the dot and points in the direction of the force. Label it with a symbol, and preferably with what exerts it.
4. Make the lengths honest when you know how the sizes compare. If the forces balance, opposite arrows should be the same length.

When an AP free-response question asks for a free-body diagram, it asks for the forces, not their components: each force is its own straight arrow starting on the dot and pointing away from it. Splitting a force into components on the diagram draws the same force twice. Two forces that point the same way would lie on top of each other, so draw them side by side, both starting at the dot, so a reader can see that there are two.

Equally important is what stays off the diagram: forces that the object exerts on other things, the net force, $m\vec{a}$, and the velocity. They are not forces on the object by something else, so they fail the "on A by B" test.

From the diagram to equations: choosing axes

Newton's laws (Topics 2.4 and 2.5) are applied one axis at a time, so you need axes. Any pair of perpendicular axes gives correct answers, but one choice makes the algebra easy: put one axis along the direction of the acceleration. Then the acceleration has only one component, and the other axis gets the equation "forces balance".

On an incline, an object that slides or is pulled moves along the surface, so tilt the axes: $x$ parallel to the incline and $y$ perpendicular to it. In the crate's diagram above, the normal force and the tension then lie along the axes, and only gravity has to be split. The $-y$ axis is perpendicular to the incline and $\vec{F}_g$ is perpendicular to the ground. The incline is tilted $\theta$ from the ground, so those two perpendicular lines are tilted $\theta$ from each other, and

Check it at the extremes: with $\theta = 0$ the ground is flat and nothing pulls the crate along it ($\sin 0 = 0$); with $\theta = 90^\circ$ the "incline" is a wall and gravity is entirely along it.

These components go into your equations, not onto the diagram. For the crate in the figure, take $m = 12\ \text{kg}$ and $g = 9.8\ \text{m/s}^2$. It is at rest, so the forces along each axis balance:

The normal force is less than the crate's weight, $mg = 118\ \text{N}$, because the incline supports only the part of gravity that presses into it.

Worked examples

Worked exampleStacked blocks

Block A, of mass $2.0\ \text{kg}$, rests on top of block B, of mass $5.0\ \text{kg}$, which sits on a frictionless horizontal table. A horizontal force $\vec{F}$ to the right is applied to block B. The blocks move together, without slipping, with an acceleration of $1.5\ \text{m/s}^2$ to the right. Take $g = 9.8\ \text{m/s}^2$.

(a) On separate dots, draw and label the forces (not components) exerted on block A and on block B.

(b) Justify the direction of the friction force exerted on block A.

(c) Calculate the magnitude of the friction force on block A and the magnitude of $\vec{F}$.

Show solution

(a) Go object by object and ask what touches each one.

Block A touches only block B, so it feels gravity, a normal force from B, and friction from B. Block B touches the table (a normal force only, since the table is frictionless), touches block A (a normal force and friction), and has the applied force $\vec{F}$, plus gravity. The force A exerts on B is a normal force pushing down on B's top surface, not "the weight of A": A's weight is a force on A by Earth, so it belongs only on A's diagram.

(b) Block A accelerates to the right, so a net force to the right must act on it. Its vertical forces must balance, since it has no vertical acceleration. The only horizontal force on A is friction from B, so that friction must point to the right. It is static friction, because the surfaces don't slide relative to each other.

(c) With $x$ to the right (the direction of acceleration), Newton's second law (Topic 2.5) for block A alone gives

$$f_s = m_A a = (2.0\ \text{kg})(1.5\ \text{m/s}^2) = 3.0\ \text{N}.$$

For $\vec{F}$, choose the two blocks together as the system. The friction forces between A and B are now internal, and the only external horizontal force is $\vec{F}$:

$$F = (m_A + m_B)a = (7.0\ \text{kg})(1.5\ \text{m/s}^2) = 10.5\ \text{N}.$$

Check with block B alone: $F - f_s = 10.5 - 3.0 = 7.5\ \text{N} = (5.0)(1.5)$, as it must be. Choosing a different system for each unknown is the move that keeps every equation to one unknown.

Where students lose points