AP Physics 1 · Topic 2.2

Topic 2.2: Forces and Free-Body Diagrams

Unit 2: Force and Translational Dynamics18-23% of the multiple-choice section

A force is an interaction between two objects, so every force on your object comes from something else you can name. A free-body diagram collects those forces as straight arrows drawn from a single dot standing for the object's center of mass, and shows nothing the object exerts on anything else.

AP Physics: Unit 2 (topics 2.2 Forces and Free-Body Diagrams). AP Physics 1 Unit 2, Topic 2.2, covering learning objective 2.2.A (describe a force as an interaction between two objects or systems) and 2.2.B (describe the forces exerted on an object or system using a free-body diagram). The topic's boundary statement fixes how forces must be drawn on AP exams: individual straight arrows originating on the dot, whole forces rather than components, and same-direction forces side by side rather than overlapping. The CED lists suggested skills 1.A, 2.B, 2.C, and 3.C here, and weights Unit 2 at 18 to 23 percent of the multiple-choice section.

What Topic 2.2 requires

Topic 2.2 has two learning objectives. Objective 2.2.A asks you to describe a force as an interaction between two objects or systems. Objective 2.2.B asks you to describe the forces exerted on an object or system using a free-body diagram. The first is a definition you can be asked to defend in words; the second is the most reused skill in the course.

This page works through the CED statements for the topic. For the drawing routine itself, step by step, with a force inventory table and the errors that break diagrams, use the how to draw a free-body diagram guide and build a few setups in the free-body diagram builder.

The CED's suggested skills here are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Unit 2 carries 18 to 23 percent of the multiple-choice section.

A force is an interaction, not a possession

Forces are vector quantities that describe the interactions between objects or systems. The CED then sharpens that into a rule you can apply arrow by arrow: a force exerted on an object or system is always due to the interaction of that object with another object or system.

Objects do not have forces. They take part in interactions, and each interaction hands one force to each participant. That gives you a two-question test for any arrow you are tempted to draw. What is the other object? What kind of interaction is it? If you cannot answer both, the arrow is not a force and does not belong on the diagram.

The habit that makes the test automatic is naming every force with both nouns. Not the normal force, but the force on the box by the floor. Not tension, but the force on the crate by the rope. The symbols you actually write are shorter, $F_N$ for a normal force and $F_T$ for a tension, but keep the long form in your head, because the long form is what Topic 2.3 turns into paired forces. Speed, momentum, and inertia are properties rather than interactions, so none of them ever gets an arrow.

Nothing exerts a net force on itself

An object or system cannot exert a net force on itself. The CED gives that its own essential knowledge item, 2.2.A.1.ii, and it rules out a whole family of wrong answers.

You cannot lift yourself by pulling up on your own belt. A car engine does not push the car forward; it turns the wheels, and the road pushes the car forward through friction at the tires. A rocket does not push on itself either; it pushes exhaust gas backward and the gas pushes the rocket forward. In every case, whatever changes the motion of the system comes from outside the system.

Notice the exact wording: no net force on itself. Parts of a system certainly do push and pull on other parts. The string between two blocks pulls on both of them, and your muscles pull on your bones. Those internal forces come in canceling pairs, so they can change the shape of a system or the motion of its parts without changing the motion of its center of mass. Which forces count as internal depends on where you drew the boundary, and that is Topic 2.1.

Contact forces, and the one force at a distance

Contact forces describe the interaction of an object or system touching another object or system and are macroscopic effects of interatomic electric forces. The normal force, friction, tension, an applied push, a spring force: underneath, all of them are the same electric interaction between the atoms of two surfaces, averaged into one arrow.

Two consequences are worth carrying. Contact forces exist only while the surfaces touch, so the instant a thrown ball leaves your hand the hand force stops, and no leftover push travels with the ball. And contact forces have no formula of their own to memorize; you solve for them from the rest of the diagram, which is why how to find normal force spends most of its time on the other forces.

The exception is interaction at a distance. In AP Physics 1 that is limited to gravitational forces, as the CED's Topic 2.3 boundary statement states, adding that AP Physics 2 brings in gravitational, electric, and magnetic forces. So on any AP Physics 1 free-body diagram exactly one arrow can be there without something touching the object, and it is the gravitational force.

What goes on the diagram, and what never does

Free-body diagrams are useful tools for visualizing forces being exerted on a single object or system and for determining the equations that represent a physical situation. The two working words in that sentence are single and equations.

Single means one object per diagram. Essential knowledge 2.2.B.2 says the diagram shows each of the forces exerted on the object by the environment, so anything your object exerts on something else belongs on a different diagram. A two-block problem gets two diagrams, or one diagram of the combined system, never a mixture of the two.

Equations means the picture is not the destination. Each arrow becomes a signed term in a Newton's second law equation, one equation per axis, which is why the CED pairs this objective with skill 2.B. A diagram you cannot turn into equations has not finished its job.

Velocity and acceleration are not forces and never get arrows. Neither does the product $ma$ , which is what the arrows already on the page add up to rather than one more of them. The free-body diagram builder checks your inventory against the setup while the habit sets.

The drawing rules the CED writes down

The Topic 2.2 boundary statement is unusually specific about the picture, and it is worth reading closely because it says what a diagram has to look like. AP Physics 1 only expects students to depict the forces exerted on objects, not the force components on free-body diagrams. On the AP Physics exams, individual forces represented on a free-body diagram must be drawn as individual straight arrows, originating on the dot and pointing in the direction of the force. Individual forces that are in the same direction must be drawn side by side, not overlapping.

Four rules fall out of that paragraph.

Whole forces only. Resolve $F_g$ into components in the algebra beside the diagram, not on it.
Straight arrows, starting on the dot. No curves, no arrows floating next to the object, no chains laid tip to tail.
Same direction means side by side. Two downward forces get two visible parallel arrows, not one arrow hiding another.
The dot is the center of mass. Forces are represented as vectors originating from the representation of the center of mass, such as a dot, and a system is treated as though all of its mass is located there. That is Topic 2.1 doing its job.

Point one axis along the acceleration

Essential knowledge 2.2.B.4 hands you the decision that makes the algebra short. A coordinate system with one axis parallel to the direction of acceleration of the object or system simplifies the translation from free-body diagram to algebraic representation. The CED supplies its own example: in a free-body diagram of an object on an inclined plane, it is useful to set one axis parallel to the surface of the incline.

Here is the payoff. Line an axis up with the acceleration and the perpendicular axis has zero acceleration, so its equation reads these forces sum to zero and hands you an unknown immediately, usually the normal force. Keep standard horizontal and vertical axes on a ramp instead and both equations carry an acceleration with two nonzero components, which doubles the work for no gain.

The cost is that gravity is no longer along an axis, so gravity becomes the force you resolve. On a ramp at angle $\theta$ the component along the surface is $mg\sin\theta$ and the component into the surface is $mg\cos\theta$ . Drag the angle in the inclined plane simulator until those two stop swapping places in your head.

How Topic 2.2 is tested, and what to do next

The four suggested skills name the formats worth rehearsing. Skill 1.A is the diagram itself, drawn to the boundary-statement rules above. Skill 2.B is the calculation the diagram sets up. Skill 2.C is the comparison version: one object, two scenarios, which normal force is larger and why. Skill 3.C is the written justification, where a correct diagram is the evidence you cite.

The CED's own sample activity for this topic is a good drill. It asks students to consider an accelerating two-object system from everyday life, such as a person pushing a shopping cart or a car pulling a trailer, then to draw the forces on one object, then on the other, and then the external forces exerted on the two-object system. Running that on three or four setups is the fastest way to make the single-object rule automatic.

Next, take the diagram into Topic 2.3, which fixes what the arrows on the other object's diagram are allowed to be, and drill the arithmetic in how to find net force.

Two objects, two diagrams: a book resting on a crate

A 1.2 kg book rests on top of a 2.8 kg crate, which rests on a level floor. Nothing else touches either object and nothing accelerates. Draw the free-body diagram for each object and find every force magnitude.

Take up as the positive direction and hold it for both diagrams. Neither object accelerates, so on each diagram the vertical forces sum to zero.
Book diagram: two arrows leave the dot. The gravitational force on the book by Earth points down, and the normal force on the book by the crate points up. Nothing else touches the book. $F_g = mg = (1.2 \, \mathrm{kg})(9.8 \, \mathrm{m/s^2}) = 11.76 \, \mathrm{N}$ downward.
Vertical equation for the book: $F_{N,\text{crate on book}} - 11.76 \, \mathrm{N} = 0$ , so the crate pushes up on the book with 11.76 N, which is 12 N to two significant figures.
Crate diagram: three arrows leave the dot. The gravitational force on the crate by Earth, $(2.8 \, \mathrm{kg})(9.8 \, \mathrm{m/s^2}) = 27.44 \, \mathrm{N}$ downward; the normal force on the crate by the book, 11.76 N downward; and the normal force on the crate by the floor, upward. The book force points down here because the book presses on whatever supports it, a relationship Topic 2.3 names.
Draw those two downward arrows side by side rather than one on top of the other, and label them separately. That is the boundary-statement rule for forces in the same direction.
Vertical equation for the crate: $F_{N,\text{floor on crate}} = 27.44 \, \mathrm{N} + 11.76 \, \mathrm{N} = 39.20 \, \mathrm{N}$ , which is 39 N to two significant figures.
Check against the combined system. Treat book and crate as one 4.0 kg object and the two book-crate forces become internal: the floor supports $(4.0 \, \mathrm{kg})(9.8 \, \mathrm{m/s^2}) = 39.2 \, \mathrm{N}$ . Same number, one diagram instead of two.

Crate on book, 12 N up. Book on crate, 12 N down. Floor on crate, 39 N up, plus the two weights of 12 N and 27 N. Five arrows across two diagrams, and no arrow appears on both, because each diagram shows only what the environment does to that one object.

Choosing axes on a 20 degree ramp

A 3.5 kg block slides on a frictionless ramp inclined at 20 degrees above the horizontal. Draw the free-body diagram, choose axes, and find the normal force and the block's acceleration.

Two forces act, so two arrows leave the dot: the gravitational force on the block by Earth, straight down, and the normal force on the block by the ramp, perpendicular to the ramp surface. Frictionless means no third arrow, and nothing pushes the block along the surface.
Choose axes by essential knowledge 2.2.B.4. The acceleration lies along the surface, so set the $x$ axis parallel to the ramp with the positive direction down the slope, and the $y$ axis perpendicular to the surface.
Resolve gravity in the algebra, off the diagram. $F_g = mg = (3.5 \, \mathrm{kg})(9.8 \, \mathrm{m/s^2}) = 34.3 \, \mathrm{N}$ . Along the ramp, $F_g \sin 20^\circ = (34.3 \, \mathrm{N})(0.3420) = 11.73 \, \mathrm{N}$ . Into the ramp, $F_g \cos 20^\circ = (34.3 \, \mathrm{N})(0.9397) = 32.23 \, \mathrm{N}$ .
Perpendicular axis: the block stays on the surface, so the acceleration there is zero and those forces sum to zero. $F_N = 32.23 \, \mathrm{N}$ , or 32 N to two significant figures. It is smaller than the 34.3 N weight, as it always is on a plain ramp.
Along the ramp: the only force component is the 11.73 N pull down the slope, so $a = F_{net}/m = (11.73 \, \mathrm{N})/(3.5 \, \mathrm{kg}) = 3.35 \, \mathrm{m/s^2}$ , which is 3.4 m/s squared down the slope.
Check symbolically: $a = g\sin\theta = (9.8 \, \mathrm{m/s^2})(0.3420) = 3.35 \, \mathrm{m/s^2}$ . The mass cancels, so every block on this frictionless ramp gets the same acceleration.

$F_N = 32 \, \mathrm{N}$ and $a = 3.4 \, \mathrm{m/s^2}$ down the slope. Tilting the axes turned the perpendicular equation into an instant value for $F_N$ and left one unknown on the other axis. Horizontal and vertical axes describe the same physics and cost twice the algebra.

Frequently asked questions

What is a force in AP Physics 1?

A vector quantity that describes an interaction between two objects or systems. Every force on your object is exerted by some other object you can name, which is why forces are best written with both nouns: the force on the box by the floor, rather than just the normal force. Speed, mass, and inertia are properties of an object, not interactions, so they are never forces.

Can an object exert a force on itself?

Not a net one. The CED states that an object or system cannot exert a net force on itself. Parts of a system do push and pull on each other, but those internal forces come in canceling pairs, so they sum to zero for the system and cannot change the motion of its center of mass. Anything that changes a system's motion is external to it.

Which forces belong on a free-body diagram?

Only the forces exerted on the chosen object by its environment. Forces your object exerts on other things go on those other objects' diagrams. Velocity and acceleration are not forces, and the product of mass and acceleration is not a separate arrow. One object, one diagram, and every arrow needs a source you can name.

How does the AP exam expect free-body diagram arrows to be drawn?

The Topic 2.2 boundary statement is specific. Individual forces must be drawn as individual straight arrows, originating on the dot and pointing in the direction of the force, and forces in the same direction must be drawn side by side rather than overlapping. Components of a force are not drawn on the diagram at all, only the whole forces.

Which way should the axes point?

Put one axis parallel to the direction of the acceleration. The CED says that choice simplifies the translation from diagram to algebra, and offers the inclined plane as its example: set one axis parallel to the ramp surface. The perpendicular axis then has zero acceleration, so its forces sum to zero and the normal force drops out in one line.