Free AP Physics 1 Force and Translational Dynamics

AP Physics 1 · Force and Translational Dynamics · free lesson

Forces and Free-Body Diagrams

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The idea

A force is an interaction

You pushed a shopping cart, pulled a door, and felt the ground hold you up. In physics each of these is a force. A force is never something an object has on its own. It is always one object acting on another.

Definition.

A force is a vector that describes an interaction between two objects or systems. Its magnitude is measured in newtons (N), and it has a direction. A force exerted on an object is always due to the interaction of that object with some other object.

This gives you a question to ask about every force: what object is exerting it? If you cannot name an object outside the one you are studying that exerts the force, there is no such force. A "force of motion" that keeps a thrown ball going is a favorite example of a force with no source: nothing is touching the ball once it leaves the hand.

Rule.

An object or system cannot exert a net force on itself. You cannot lift yourself by pulling up on your own belt. The forces that matter for the motion of an object are the ones exerted on it by other objects, the environment.

Contact forces

Many forces act only when two objects touch. A table pushes up on a book, a rope pulls on a crate, a floor drags on a sliding box. These are contact forces. At the scale of atoms, every one of them comes from the electric forces between the atoms of the two surfaces. You do not see those tiny forces; what you see is their sum, the macroscopic effect.

Force	Symbol	Exerted by	Direction
normal	F_N	a surface the object touches	perpendicular to the surface, away from it
tension	F_T	a rope, string or cable	along the rope, pulling on the object
friction	F_f	a surface the object touches	along the surface
applied push or pull	F_A	a person or other object in contact	the way the push or pull goes

In this course only one force acts without touching: gravity. The gravitational force on an object, written $F_g$ , is exerted by Earth (or another large body) on the object and points toward Earth's center, which near the ground is straight down.

Free-body diagrams

A free-body diagram is a picture of one object, or one system, with the forces exerted on it. It leaves out everything else. Drawing one is the first step in turning a situation into equations, and you will draw many.

Definition.

A free-body diagram shows the forces exerted on a single object or system by its environment. The object is drawn as a dot at its center of mass. Each force is one straight arrow that starts on the dot and points in the direction of the force. Longer arrows are larger forces.

Because a system acts as though all its mass were at its center of mass, even a large object is drawn as a dot. The rules for the exam, which this page follows:

Draw each force as its own arrow, starting on the dot.
Draw forces that point the same way side by side, not on top of each other.
Do not draw force components. Draw each force whole.
Draw only forces exerted on the object, never forces the object exerts on something else.
Label each arrow with the force and, when it helps, the object exerting it.

A book on a table

A book lies on a table. Two objects touch or pull on it: Earth, which pulls with $15\,\mathrm{N}$ , and the table, which pushes up with $15\,\mathrm{N}$ . The scale is one square for $5\,\mathrm{N}$ .

The dot is the book. The normal force is an arrow pointing up, $3.00$ squares long. The gravitational force points down, $3.00$ squares long. Each label names the force and the object that exerts it.

A crate pulled along the floor

Now a crate is pulled to the right along a floor by a rope that slants $30$ degrees above the floor. Four forces are exerted on it: Earth pulls down with $49\,\mathrm{N}$ , the floor pushes up with $29\,\mathrm{N}$ and drags back with $10\,\mathrm{N}$ of friction, and the rope pulls with $40\,\mathrm{N}$ along the rope. The floor gives two arrows, the normal force, perpendicular to the floor, and friction, along it. The rope's force is one arrow at its angle, not split into a horizontal and a vertical part. The scale is one square for $10\,\mathrm{N}$ .

The arrows are $2.90$ squares (normal), $4.90$ (gravity), $4.00$ (rope) and $1.00$ (friction). The rope's arrow is drawn at its angle of $30$ degrees.

Note.

Only forces on the crate. The crate also pulls on the rope and pushes down on the floor, but those forces act on the rope and the floor, not on the crate. They do not belong in this diagram. The next topic is about those paired forces.

What is not a force

A ball is thrown straight up and is now rising, after it left the hand. What forces are exerted on it? Ask the question for each candidate: which object exerts it? Earth pulls on the ball, so the gravitational force is there. The hand is no longer touching the ball, and no object exerts an "upward force of the throw". The air is ignored here. The ball's velocity is a vector, but it describes the ball's motion, not an interaction, so it is not drawn as a force.

Only one arrow is on the free-body diagram. The dashed gray arrow is the ball's velocity, drawn off to the side so nobody mistakes it for a force.

Choosing the object, and a convenient axis

You choose the object or system whose free-body diagram you draw, and the choice decides which arrows appear. Two blocks are stacked on a table, a 10 N block on top of a 20 N block. In the first diagram the system is the top block alone: the bottom block exerts a normal force on it. In the second diagram the system is both blocks together: that contact is now inside the system, so it does not appear, and the table's normal force and Earth's pull on both blocks do.

Both diagrams use the same scale, one unit of length for 10 N. The second diagram has no arrow for the contact between the blocks, because both blocks are inside the system.

An axis along the motion

When you turn a free-body diagram into equations, you set up coordinate axes. The axes are yours to choose. It simplifies the algebra to put one axis in the direction the object accelerates, or, for an object on an incline, parallel to the surface. The forces are still drawn whole, at their true directions; the axes are drawn beside the diagram, tilted with the surface.

A block rests on a ramp inclined at 30 degrees. Earth pulls on it with $20\,\mathrm{N}$ straight down. The ramp pushes on it with $17\,\mathrm{N}$ perpendicular to the surface, and with a frictional force of $10\,\mathrm{N}$ up along the surface. The scale is $5\,\mathrm{N}$ per square.

The three forces are drawn whole. The black pair of axes at the lower left is tilted to lie along the ramp: $x$ along the surface and $y$ perpendicular to it. With that choice two of the three forces lie along an axis, which makes the algebra shorter later on.

Rule.

Drawing a free-body diagram. Pick the object or system. Draw a dot for its center of mass. For each object that touches it, draw the contact force or forces that object exerts. Add the gravitational force. Draw each force whole, starting on the dot, and name it. Check that every arrow is a force on this object, and that none is the object's motion.

Worked examples

Five examples of the kinds the AP exam asks on this topic: deciding whether something is a force and who exerts it, drawing a free-body diagram to scale, drawing one for a more complex object, changing the system, and justifying a claim.

Example.

Example 1. Is it a force on the ball?

A soccer ball is flying through the air after being kicked. A student lists three things "acting on the ball": the pull of Earth, the kick that sent it off, and the air's drag. Which belong on the ball's free-body diagram, if drag is ignored?

Solution

For each candidate, name the object that exerts it, and check that it is acting on the ball now.

Candidate	Exerted by	On the diagram?
the pull of Earth	Earth	yes: gravity acts on the ball, with no touching needed
the kick	the foot	no: the foot is no longer in contact, so it exerts no force on the ball now
the air's drag	the air	not when drag is ignored

Result.

Only the gravitational force of Earth is on the diagram. The kick was a force while the foot touched the ball. Once contact ends, no object exerts it, and a force that has no object exerting it is not a force.

Example.

Example 2. A hanging lamp

A lamp hangs from the ceiling by a chain. Earth pulls down on the lamp with $24.5\,\mathrm{N}$ and the chain pulls up with $24.5\,\mathrm{N}$ . Draw the free-body diagram of the lamp, with one square for $5\,\mathrm{N}$ .

Solution

The lamp is the system, a dot. Two objects exert forces on it: Earth, with no contact, and the chain, which touches it. The arrow lengths are the forces divided by the scale.

\begin{aligned} L_g &= \frac{F_g}{k} \\ &= \frac{24.5\,\mathrm{N}}{5\,\mathrm{N}} \\ &= 4.90 \\ L_T &= \frac{F_T}{k} \\ &= \frac{24.5\,\mathrm{N}}{5\,\mathrm{N}} \\ &= 4.90 \end{aligned}

Result.

An upward arrow of $4.90$ squares labeled with the chain, and a downward arrow of $4.90$ squares labeled with Earth, both starting on the dot. Nothing else touches the lamp, so nothing else is drawn.

Example.

Example 3. A box pushed along the floor

A person pushes a box to the right along a level floor. The box is pushed with an applied force of $60\,\mathrm{N}$ to the right. Earth pulls on it with $98\,\mathrm{N}$ , the floor pushes up with $98\,\mathrm{N}$ , and friction from the floor pulls back with $35\,\mathrm{N}$ . Draw the free-body diagram, one square for $20\,\mathrm{N}$ , and say which forces are contact forces.

Solution

\begin{aligned} s_g &= \frac{F_g}{k} \\ &= \frac{98\,\mathrm{N}}{20\,\mathrm{N}} \\ &= 4.90 \\ s_N &= \frac{F_N}{k} \\ &= \frac{98\,\mathrm{N}}{20\,\mathrm{N}} \\ &= 4.90 \\ s_A &= \frac{F_A}{k} \\ &= \frac{60\,\mathrm{N}}{20\,\mathrm{N}} \\ &= 3.00 \\ s_f &= \frac{F_f}{k} \\ &= \frac{35\,\mathrm{N}}{20\,\mathrm{N}} \\ &= 1.75 \end{aligned}

Result.

Four arrows start on the dot: up $4.90$ squares (normal force), down $4.90$ squares (gravity), right $3.00$ squares (the push) and left $1.75$ squares (friction). The normal force, the push and friction are contact forces: the floor and the person touch the box. Gravity is the only force that acts without contact.

Example.

Example 4. What changes if the system changes?

A suitcase sits on a cart. First the system is the suitcase alone, then the suitcase and the cart together (the cart sits on the floor). Which forces are on each free-body diagram?

Solution

A force belongs on the diagram if an object outside the system exerts it.

System	Forces on it	Left out
suitcase	Earth's pull; the push of the cart up on the suitcase	the floor's push on the cart: the floor does not touch the suitcase
suitcase and cart	Earth's pull on both; the push of the floor up on the cart	the push between suitcase and cart: now inside the system

Result.

Both diagrams have two forces. They are not the same two: the suitcase's second force comes from the cart, and the pair's second force comes from the floor. The push between the suitcase and the cart is external to the suitcase and internal to the pair, so it is on the first diagram and not on the second.

Example.

Example 5. Justifying a claim

A block slides down a ramp that is inclined at 30 degrees. A student draws its free-body diagram with three forces: gravity ( $30\,\mathrm{N}$ , straight down), the ramp's normal force ( $26\,\mathrm{N}$ ) and friction ( $10\,\mathrm{N}$ ). She then adds a fourth arrow, "the force of motion", pointing down the ramp. She also sets up an $x$ axis horizontal and a $y$ axis vertical. Is the fourth arrow right? Is her choice of axes the best?

Solution

The block moves down the ramp. But for the fourth arrow ask which object exerts it. Nothing does: the ramp's contact is already in the normal force and in friction, and Earth's pull is already drawn. Motion is not an interaction.

The axes are allowed, but the better choice has one axis along the ramp. The block accelerates along the ramp, and then friction points along one axis and the normal force along the other.

Result.

The fourth arrow is wrong: no object exerts "the force of motion", so the diagram has three forces. Horizontal and vertical axes are a legal choice, but axes along and perpendicular to the ramp make two of the three forces lie along an axis, which makes the later algebra shorter.

Practice

Answer AP-style questions on forces and free-body diagrams. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.

Practise forces and free-body diagrams with new numbers