AP Physics C: Mechanics · Topic 2.2

Topic 2.2: Forces and Free-Body Diagrams

Unit 2: Force and Translational Dynamics20-25% of the multiple-choice section

A force is an interaction between two objects, so every arrow on your diagram must come from something you can name. The AP Physics C treatment of free-body diagrams is the same as the AP Physics 1 treatment. What differs is what the exam asks for next: a symbolic expression, not a number.

AP Physics: Unit 2 (topics 2.2 Forces and Free-Body Diagrams). AP Physics C: Mechanics Unit 2, Topic 2.2. Two learning objectives. 2.2.A, describe a force as an interaction between two objects or systems, supported by 2.2.A.1 (forces are vector quantities describing interactions), 2.2.A.1.i (a force on an object is always due to interaction with another object or system), 2.2.A.1.ii (an object or system cannot exert a net force on itself) and 2.2.A.2 (contact forces are macroscopic effects of interatomic electric forces). 2.2.B, describe the forces exerted on an object or system using a free-body diagram, supported by 2.2.B.1 (diagrams visualize forces on a single object and determine the equations that represent a physical situation), 2.2.B.2 (the diagram shows forces exerted by the environment), 2.2.B.3 (forces are vectors originating from a representation of the center of mass, such as a dot; a system is treated as though all its mass is at the center of mass) and 2.2.B.4 (one axis parallel to the acceleration simplifies the translation to algebra, with the inclined plane as the example). Boundary statement, in full: AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect 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. This topic's content is IDENTICAL to AP Physics 1 Topic 2.2: same eight statements, same suggested skills (1.A, 2.C, 3.B, 3.C), and a boundary statement whose only difference is which courses it names. No calculus is involved. The C exam differs in expecting symbolic answers: skill 2.A carries 25 to 30 percent of the multiple-choice section, and Science Practice 1 is not assessed on multiple choice at all (20 to 35 percent on free response). The CED's sample free-response question 2 aligns to 2.2.B and restates the arrow rules in its prompt, adding that arrow lengths should reflect relative magnitudes.

This topic is the same in both courses, and that is worth saying plainly

Topic 2.2 is one of the few places in AP Physics C: Mechanics where calculus changes nothing. Compare the two frameworks statement by statement and they line up: 2.2.A.1, 2.2.A.1.i, 2.2.A.1.ii, 2.2.A.2, 2.2.B.1, 2.2.B.2, 2.2.B.3 and 2.2.B.4 appear in AP Physics 1 with the same numbers and the same content. Both courses print a boundary statement, and the two are word for word the same except that the AP Physics 1 version opens "AP Physics 1 only expects students" and the AP Physics C version opens "AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect students".

There is no derivative and no integral in Topic 2.2. Do not go looking for one.

What does change is what the exam does with the diagram once you have drawn it. Skill 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway, carries 25 to 30 percent of the AP Physics C: Mechanics multiple-choice section, the largest band the CED lists for any single skill. The free-response section states its expectation directly: the Mathematical Routines question says students will be expected to symbolically derive relationships between variables, as well as calculate numerical values, and the Translation Between Representations question expects them to create a visual representation that describes a scenario and then derive equations that are mathematically relevant to it.

So the diagram is not the answer on this exam. The diagram is the first line of a derivation, and the answer is usually an expression in $m$ , $\theta$ , $\mu$ and $g$ with no numbers in it at all. Practise finishing in symbols.

What the CED requires of Topic 2.2

Two learning objectives, with suggested skills 1.A, 2.C, 3.B and 3.C.

Objective 2.2.A: describe a force as an interaction between two objects or systems.

2.2.A.1: forces are vector quantities that describe the interactions between objects or systems.
2.2.A.1.i: a force exerted on an object or system is always due to the interaction of that object or system with another object or system.
2.2.A.1.ii: an object or system cannot exert a net force on itself.
2.2.A.2: contact forces describe the interaction of an object or system touching another object or system and are macroscopic effects of interatomic electric forces.

Objective 2.2.B: describe the forces exerted on an object or system using a free-body diagram.

2.2.B.1: 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.
2.2.B.2: the free-body diagram of an object or system shows each of the forces exerted on the object or system by the environment.
2.2.B.3: forces exerted on an object or system are represented as vectors originating from the representation of the center of mass, such as a dot. A system is treated as though all of its mass is located at the center of mass.
2.2.B.4: 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. For 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.

Read 2.2.B.1 again: free-body diagrams are for determining the equations that represent a physical situation. The CED says the point of the drawing is the algebra it produces, which is exactly why the C exam rewards the derivation and not the picture.

The boundary statement, quoted whole

Topic 2.2 carries an unusually procedural boundary statement, and it is worth following to the letter because the exam scores against it:

"AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect 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, and each one is a point somebody has lost.

Forces, not components. If gravity acts on a block on a ramp, draw one arrow straight down. Do not draw $mg\sin\theta$ and $mg\cos\theta$ instead of it, and do not draw them in addition to it. The components belong in your algebra, off to the side.
Straight arrows. Not curved, not double-headed, not a squiggle for a spring.
Originating on the dot, pointing in the direction of the force. The tail sits on the dot. An arrow drawn pointing into the dot is a different force from the one you meant.
Same-direction forces side by side, not overlapping. Two downward forces get two visibly separate arrows. A grader cannot count arrows that sit on top of each other.

The CED's sample free-response question 2 repeats the instruction in the prompt itself: it asks students to draw and label the other forces exerted on a car and says each force must be represented by a distinct arrow starting on, and pointing away from, the dot, and that the lengths of the arrows should reflect the relative magnitudes of the forces. Relative arrow length is scored. If the normal force must exceed the gravitational force in your scenario, draw it longer.

Statement 2.2.B.4 then tells you to tilt the axes to line one up with the acceleration. Notice the two rules coexisting: you tilt the coordinate system, and you still do not draw the components on the diagram.

A force is an interaction, so name the other object

Statement 2.2.A.1.i is the sentence to carry out of this topic: a force exerted on an object is always due to the interaction of that object with another object or system. Every arrow you draw needs a second object behind it. If you cannot name that object, the arrow does not belong.

That test kills the standard invented forces.

There is no forward force on a thrown ball once it leaves the hand. The hand is gone, so its force is gone.
There is no separate centripetal force on a car in a turn. Something real supplies it, and on a flat road that something is friction from the road. Note the AP Physics C: Mechanics framework never uses the phrase "centripetal force" anywhere: every occurrence of the word "centripetal" in it is followed by "acceleration", and where a force is meant, statement 2.10.A.2.ii writes "the net force producing centripetal acceleration".
There is no force of inertia and no force of motion. Statement 2.4.A.3 covers what happens without a net force, and it is not that the object needs one to keep going.

Statement 2.2.A.1.ii adds the other half: an object or system cannot exert a net force on itself. A car's engine turns the wheels, the wheels push on the road, and it is the road that pushes the car forward. That is why 2.2.B.2 says the diagram shows the forces exerted on the object by the environment.

Statement 2.2.A.2 defines contact forces as the macroscopic effects of interatomic electric forces. Normal force, tension and friction are all the same underlying interaction seen at different scales. In Unit 2 the only non-contact force you will meet is gravity; the electric and magnetic ones belong to AP Physics C: Electricity and Magnetism, which reprints this same boundary statement.

Tilting the axes, and why it is a decision and not a habit

Statement 2.2.B.4 says a coordinate system with one axis parallel to the direction of acceleration simplifies the translation from free-body diagram to algebraic representation, and gives the inclined plane as its example.

The logic is worth stating rather than memorising. Newton's second law is a vector statement, and you turn it into scalar equations by projecting onto axes. If one axis lies along the acceleration, the other axis has zero acceleration, so its equation reads "sum of components equals zero" and one unknown drops out for free.

So pick the axes from the acceleration, not from the picture.

Block sliding on a ramp: tilt, because the acceleration is along the surface. Then the perpendicular equation gives you the normal force immediately.
Block in a circle on a flat road: keep the axes horizontal and vertical, because the acceleration points horizontally toward the center. The vertical equation gives you the normal force.
Block on a banked road: keep the axes horizontal and vertical, not tilted with the bank. The acceleration is horizontal, toward the center of the circle, even though the surface is not. This is the case where copying the ramp habit costs you the question.

Once the axes are chosen, declare the positive direction and hold it to the end of the problem. This site uses down the slope as positive for incline problems. Any consistent choice is fine; a choice that flips halfway through a derivation is not.

Site guides that walk the mechanics of the drawing itself: how to draw a free-body diagram, how to find the net force, how to find the normal force, how to find tension and the inclined plane walkthrough. There is also a free-body diagram builder if you want to try the arrow rules on a live diagram.

Traps this topic sets

Drawing components instead of forces. The boundary statement forbids it and it costs a drawing point. Draw the gravitational force straight down, then resolve it in your algebra.

Adding a force with no second object. Covers the phantom forward force, the phantom centripetal force and the phantom force of inertia. Run the 2.2.A.1.i test on every arrow.

Overlapping arrows. If two forces point the same way, draw them beside each other. This is explicit in the boundary statement.

Assuming the normal force equals $mg$ . It does only when the surface is horizontal and nothing else has a vertical component. Statement 2.7.A.2.ii defines it as the perpendicular component of the force exerted on an object by the surface with which it is in contact, directed away from the surface, which is a definition and not a formula. Always get it from the perpendicular equation.

Putting third-law partners on the same diagram. A free-body diagram is for one object. The partner of a force in your diagram acts on the other object and belongs on that object's diagram. Topic 2.3 is where this is developed.

Forgetting that a system's diagram hides its internal forces. Statement 2.2.B.3 says a system is treated as though all of its mass is located at the center of mass. Choose two blocks as one system and the contact force between them vanishes from the diagram, which is exactly why the system choice is worth making deliberately.

If you want the algebra-based version of this topic

The two treatments are the same, so the honest recommendation depends only on which course you are in. If you are taking AP Physics 1, read AP Physics 1 Topic 2.2: Forces and Free-Body Diagrams, which covers the same eight statements and the same drawing rules at a slower pace with algebra-based examples. If you are taking AP Physics C: Mechanics, stay here for the symbolic framing and the exam context.

	AP Physics 1 Topic 2.2	AP Physics C Topic 2.2
Objectives	2.2.A, 2.2.B	2.2.A, 2.2.B
Essential-knowledge statements	8	8
Boundary statement	same text, named for AP Physics 1	same text, named for both C courses
Suggested skills	1.A, 2.C, 3.B, 3.C	1.A, 2.C, 3.B, 3.C
Typical expected answer	a number	a symbolic expression
Calculus involved	none	none

Even the suggested skills match. This is not a topic where the calculus-based course is harder; it is a topic where the calculus-based exam asks a differently shaped question about the same content.

For the vocabulary, the free-body diagram glossary entry and the net force glossary entry are one-paragraph definitions, and scalar versus vector covers why forces need directions in the first place.

How Topic 2.2 is tested

Unit 2 is weighted at 20 to 25 percent of the multiple-choice section across about 15 to 25 class periods, the highest minimum weighting of any unit in AP Physics C: Mechanics (only Unit 3, at 15 to 25 percent, reaches the same 25 percent ceiling). Topic 2.2 supplies the representation that most of the rest of the unit runs on.

The suggested skills are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.C, compare physical quantities between two or more scenarios or at different times and/or locations within a single scenario; 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

Skill 1.A is not assessed on the multiple-choice section at all. The CED's exam-weighting table marks Science Practice 1 as not applicable for multiple choice and 20 to 35 percent for free response. So the drawing itself is a free-response skill, and Topic 2.2 shows up in the multiple-choice section through 2.C, 3.B and 3.C instead: comparing scenarios and justifying claims from a diagram someone else drew.

The CED's sample free-response question 2, the Translation Between Representations question worth 12 points, aligns to seven learning objectives and 2.2.B is one of them. Part A is a drawing task with the arrow rules restated in the prompt. The unit opener's "Preparing for the AP Exam" note ties Unit 2 to question four, the Qualitative/Quantitative Translation, and warns that students exposed primarily to numerical problem solving often struggle with that question because it requires them to express a conceptual understanding of course content and representations.

The CED also notes, under the Qualitative/Quantitative Translation description, that students may not be directly assessed on their ability to create diagrams to answer that question, and that a student who draws a free-body diagram will earn points for the explanation and conclusions the diagram indicates rather than for creating the diagram itself. Draw it anyway; just do not expect it to score on its own.

One of Unit 2's six optional sample instructional activities is on this material: pairs of students produce free-body diagrams, trade them, and suggest situations where the forces on an object would be described by the diagram they were given.

From diagram to symbolic acceleration: a block pushed horizontally into a ramp

A block of mass $m = 3.5$ kg sits on a ramp inclined at $\theta = 32^\circ$ . A horizontal force of magnitude $P = 12$ N pushes the block toward the hill. The coefficient of kinetic friction between block and ramp is $\mu_k = 0.18$ , and the block is sliding down the slope. Derive an expression for the acceleration and then evaluate it, using $g = 9.8\ \mathrm{m/s^2}$ .

Draw the diagram first, following the boundary statement. Four arrows, all starting on the dot: the gravitational force straight down, the normal force perpendicular to the ramp surface and away from it, the applied force $P$ horizontal, and kinetic friction along the surface. No components on the diagram.
Choose axes by 2.2.B.4: one axis along the surface, because the acceleration is along the surface. Declare the positive direction as down the slope, and hold it to the end.
Resolve each force in your algebra, off the diagram. Along the slope: gravity contributes $+mg\sin\theta$ , the horizontal push contributes $-P\cos\theta$ because it points partly up the slope, and friction opposes the downhill sliding so it contributes $-\mu_k F_N$ .
Perpendicular to the slope there is no acceleration, so those components sum to zero: $F_N - mg\cos\theta - P\sin\theta = 0$ , giving $F_N = mg\cos\theta + P\sin\theta$ . The horizontal push presses the block into the ramp, so it increases the normal force.
Combine, still in symbols. $ma = mg\sin\theta - P\cos\theta - \mu_k\left(mg\cos\theta + P\sin\theta\right)$ , so $a = g\left(\sin\theta - \mu_k\cos\theta\right) - \dfrac{P}{m}\left(\cos\theta + \mu_k\sin\theta\right)$ . This is the answer the C exam wants; the number is a bonus.
Check the expression before substituting. Set $P = 0$ and it collapses to $a = g(\sin\theta - \mu_k\cos\theta)$ , the familiar sliding-block result. Increase $P$ and $a$ decreases, which is right because the push is uphill and it also increases friction.
Now the numbers. $\sin 32^\circ = 0.5299$ , $\cos 32^\circ = 0.8480$ . First bracket: $0.5299 - 0.18(0.8480) = 0.5299 - 0.1526 = 0.3773$ . Times $g$ : $9.8(0.3773) = 3.697\ \mathrm{m/s^2}$ .
Second bracket: $0.8480 + 0.18(0.5299) = 0.8480 + 0.0954 = 0.9434$ . Times $P/m = 12/3.5 = 3.4286$ : $3.4286(0.9434) = 3.235\ \mathrm{m/s^2}$ .
Subtract: $a = 3.697 - 3.235 = 0.462\ \mathrm{m/s^2}$ , so $0.46\ \mathrm{m/s^2}$ down the slope in the declared convention.
Cross-check by forces. $F_N = 34.3(0.8480) + 12(0.5299) = 29.09 + 6.36 = 35.45$ N, friction $= 0.18(35.45) = 6.38$ N, and the driving term $mg\sin\theta - P\cos\theta = 18.18 - 10.18 = 8.00$ N. Then $a = (8.00 - 6.38)/3.5 = 0.46\ \mathrm{m/s^2}$ , which matches.

$a = g(\sin\theta - \mu_k\cos\theta) - \dfrac{P}{m}(\cos\theta + \mu_k\sin\theta) = 0.46\ \mathrm{m/s^2}$ , directed down the slope.

A rope pulling at an angle to the ramp surface

An $8.0$ kg crate is dragged up a $25^\circ$ ramp by a rope. The rope makes an angle of $15^\circ$ with the ramp surface, and its tension is $55$ N. The coefficient of kinetic friction is $\mu_k = 0.20$ . Find the crate's acceleration, with $g = 9.8\ \mathrm{m/s^2}$ .

Five arrows on the dot: gravity down, normal force perpendicular to the ramp, tension along the rope, and kinetic friction along the surface. Nothing is drawn in components.
Declare the convention: positive down the slope, as everywhere on this site. The crate moves up the slope, so kinetic friction acts down the slope, which is the positive direction here. Expect a negative answer; that will mean the acceleration points up the slope.
Perpendicular equation. The rope pulls partly away from the surface, so it reduces the normal force: $F_N + T\sin\phi - mg\cos\theta = 0$ , giving $F_N = mg\cos\theta - T\sin\phi$ , with $\phi = 15^\circ$ .
Numbers for the normal force: $mg\cos\theta = 78.4(0.9063) = 71.05$ N and $T\sin\phi = 55(0.2588) = 14.24$ N, so $F_N = 56.82$ N. Compare that with $mg\cos\theta = 71.05$ N if the rope had been parallel to the surface; lifting the rope has taken 14.2 N off the normal force and therefore off the friction.
Along-slope equation with down-slope positive: $ma = mg\sin\theta - T\cos\phi + \mu_k F_N$ .
Numbers: $mg\sin\theta = 78.4(0.4226) = 33.13$ N, $T\cos\phi = 55(0.9659) = 53.13$ N, and $\mu_k F_N = 0.20(56.82) = 11.36$ N.
Sum: $33.13 - 53.13 + 11.36 = -8.63$ N, so $a = -8.63/8.0 = -1.08\ \mathrm{m/s^2}$ .
Read the sign against the convention you declared. Negative means up the slope, so the crate accelerates up the ramp at $1.08\ \mathrm{m/s^2}$ . Reporting it as $+1.08$ without saying you changed convention is the error this step exists to prevent.
Worth noticing: raising the rope angle helps twice over on friction (less normal force, less friction) but wastes tension (less along-slope component). There is an optimum angle, and finding it by setting a derivative to zero is exactly the kind of extension a Mathematical Routines free-response question can ask for.

$a = -1.08\ \mathrm{m/s^2}$ in the down-slope-positive convention, that is, $1.08\ \mathrm{m/s^2}$ directed up the slope. The angled rope reduces the normal force to 56.8 N from the 71.1 N it would be with the rope parallel to the surface.

Frequently asked questions

Are free-body diagrams different in AP Physics C than in AP Physics 1?

No. The essential knowledge statements 2.2.A.1 through 2.2.B.4 are the same in both course and exam descriptions, and the boundary statement is word for word the same except that the AP Physics C version names both AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism instead of AP Physics 1. Both courses list the same suggested skills for this topic: 1.A, 2.C, 3.B and 3.C. What differs is the expected answer. Skill 2.A, derive a symbolic expression, carries 25 to 30 percent of the AP Physics C: Mechanics multiple-choice section, so a C question is more likely to end in an expression than a number.

Can you draw force components on an AP Physics free-body diagram?

No. The boundary statement under Topic 2.2 reads that AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect students to depict the forces exerted on objects, not the force components on free-body diagrams. It continues that 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, and that individual forces that are in the same direction must be drawn side by side, not overlapping. Resolve into components in your algebra beside the diagram, not on it.

Should you tilt your axes for an inclined plane problem?

Yes, and essential knowledge 2.2.B.4 says why: 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, and it gives the inclined plane as its example. The rule is to pick the axes from the acceleration, not from the picture. A block sliding on a ramp accelerates along the surface, so tilt. A car on a banked curve accelerates horizontally toward the center of the circle, so keep the axes horizontal and vertical even though the surface is tilted.

What counts as a real force on a free-body diagram?

Anything you can attribute to a second object. Essential knowledge 2.2.A.1.i of the AP Physics C: Mechanics framework says a force exerted on an object or system is always due to the interaction of that object or system with another object or system, and 2.2.A.1.ii adds that an object or system cannot exert a net force on itself. So if you cannot name the object doing the pushing or pulling, the arrow does not belong. That test removes the imagined forward force on a thrown ball, the imagined force of inertia, and any separate arrow labelled centripetal force.

Why does the AP Physics C framework never say centripetal force?

Because centripetal acceleration is caused by ordinary forces rather than being a force of its own. Every occurrence of the word centripetal in the AP Physics C: Mechanics course and exam description is followed by the word acceleration. Where a force is meant, the framework writes it out: essential knowledge 2.10.A.2.ii refers to the net force producing centripetal acceleration of an object traveling in a circle on a banked surface, and 2.10.A.2 says centripetal acceleration can result from a single force, more than one force, or components of forces. Drawing an extra arrow labelled centripetal force double counts a force already on the diagram.

Does the normal force always equal mg?

No, and assuming so is an expensive habit in this unit. Essential knowledge 2.7.A.2.ii defines the normal force as the perpendicular component of the force exerted on an object by the surface with which it is in contact, directed away from the surface. That is a definition, not a formula. On a horizontal surface with nothing else acting vertically, it happens to equal mg. Tilt the surface and it becomes mg cos theta. Push down at an angle and it grows; pull up at an angle and it shrinks; accelerate the surface vertically and it changes again. Always get it from the perpendicular equation.

Do free-body diagrams earn points on the AP Physics C exam?

On the free-response section, sometimes directly. The CED's sample question 2, Translation Between Representations, opens with a drawing task and restates the arrow rules in the prompt, adding that the lengths of the arrows should reflect the relative magnitudes of the forces. On the Qualitative/Quantitative Translation question the CED says students may not be directly assessed on their ability to create diagrams, and that a student who draws a free-body diagram will earn points for the explanation and conclusions the diagram indicates rather than for creating the diagram itself. Science Practice 1 is marked not applicable for the multiple-choice section entirely.