Forces and Free-Body Diagrams: Study Guide (Physics 1 2.2)

This guide is for the algebra-based Physics 1 course. In Topic 2.1 you learned to choose a system. Now you will describe the forces exerted on it, and draw them in a way that turns straight into equations.

A force is an interaction

A force is a push or a pull that one object exerts on another. That second object is the key idea: every force is exerted on something by something else. If you cannot name the object exerting a force, the force is not real.

Get into the habit of naming each force in full:

• the force on the book by Earth (gravity),
• the force on the book by the table (the normal force),
• the force on the sled by the rope (tension).

Forces are vectors. Each has a size, measured in newtons (1 N = 1 kg·m/s²), and a direction. The same object can interact with several others at once, so several forces can act on it together.

No object can push itself

An object or system cannot exert a net force on itself. Picture a child sitting inside a large cardboard box on the floor, pushing hard on the front wall. The box does not slide forward. While she pushes the wall forward, the box pushes her backward, and she presses back on the floor of the box through her seat. Every one of these interactions is between parts inside the child–box system. Only an interaction with something outside the system, such as a friend pulling the box, can change how the system as a whole moves. Topic 2.3 shows exactly why internal forces always cancel.

Contact forces and long-range forces

Contact forces act only while two objects touch. Normal forces, friction, tension and the push of your hand are all contact forces. They look simple, but at the scale of atoms they are electric forces. When a book rests on a table, the atoms at the two surfaces are squeezed slightly closer. The charged particles in them repel, and the combined effect of enormous numbers of these tiny repulsions is the upward normal force.

Long-range forces act across a distance, with no touching needed. In this course the only one you will use is gravity. Earth pulls on a falling apple while nothing touches it. Magnetic and electric attractions also act at a distance, but they are outside this course.

Forces you will meet in Unit 2

Force	Usual symbol	Exerted by	Direction	Developed in
Gravitational force (weight)	F_g	Earth (or another mass)	towards Earth’s centre: “down”	Topic 2.6; near Earth, F_g = mg
Normal force	F_N	a surface	perpendicular to the surface, away from it	Topic 2.7
Friction	F_f	a surface	parallel to the surface, opposing sliding (or the tendency to slide)	Topic 2.7
Tension	F_T	a string, rope or cable	along the string, pulling away from the object	Topic 2.3
Spring force	F_s	a spring	along the spring	Topic 2.8
Applied force	F_A	a hand, a person, a motor’s contact	as described	—

Two directions in this table are worth learning now. The normal force is perpendicular to the surface, which is only vertical when the surface is level. Tension always pulls: a rope cannot push.

What a free-body diagram shows

A free-body diagram is a picture of the forces exerted on one chosen object or system by its environment. It lets you see every force at once, and it gives you the terms for your equations.

The course uses these conventions, and exam answers are expected to follow them:

1. Represent the object or system by a dot. The dot stands for its center of mass, where all its mass can be imagined to be (Topic 2.1).
2. Draw each force as its own straight arrow, starting on the dot and pointing in the direction the force acts.
3. Label every arrow, for example “F_N, floor on crate” or just “F_N” if the meaning is clear.
4. Make relative lengths sensible. Bigger forces get longer arrows.
5. If two forces point the same way, draw them side by side, not on top of each other. For example, two people pushing a crate in the same direction give two parallel arrows next to each other.
6. Draw whole forces only. Do not draw components on the free-body diagram. Work out components separately.

Leave off anything that is not a force on the system: velocity, acceleration, the net force, and any force the system exerts on other objects.

A method that never misses a force

1. Choose the system and imagine its boundary.
2. Draw the dot.
3. Add the long-range force: Earth’s gravity, downward.
4. Go round the boundary. Every place where something touches the system can give a contact force: a surface (normal force and maybe friction), a string (tension), a hand (applied force).
5. Check each arrow: “on what, by what?” If there is no “by what”, delete it.

Worked example 1: a sled pulled by a rope

Question. A 12 kg sled is pulled across level snow by a rope at 30° above the horizontal. The rope’s tension is 50 N, and the sled slides to the right at a steady speed. (a) Draw the free-body diagram. (b) Find the size of the gravitational force. (c) Take +x to the right and +y up, write the rope’s force in components, and use the fact that the forces balance (Topic 2.4 explains why, for a steady velocity) to find the normal force and the friction force.

(a) Things that interact with the sled: Earth (long-range), the snow (touching it), the rope (attached). So there are four forces:

• F_g, Earth on sled, straight down;
• F_N, snow on sled, straight up (perpendicular to the level snow);
• F_f, snow on sled, to the left (opposing the sliding);
• F_T, rope on sled, 30° above the horizontal to the right.

(b) F_g = mg = (12 kg)(9.8 N/kg) = 118 N, downward.

(c) Components of the rope’s force, with the 30° measured from +x:

• T_x = (50 N) cos 30° = +43 N
• T_y = (50 N) sin 30° = +25 N

y-direction (no vertical acceleration, so the forces balance): F_N + T_y − F_g = 0, so F_N = 118 N − 25 N = 93 N.

x-direction (steady velocity, so the forces balance): T_x − F_f = 0, so F_f = 43 N.

Interpretation. The normal force is less than the weight. The rope is helping to hold the sled up, so the snow does not need to push as hard. “Normal force equals mg” is not a rule; it is what happens only in special cases.

Choosing axes

Any pair of perpendicular axes gives correct answers, but one choice makes the algebra easiest: put one axis along the direction of the acceleration (or of the motion, if the object might accelerate along it). Then the other axis has zero acceleration, and its forces simply balance.

On a ramp, that means tilting the axes: one axis parallel to the ramp surface and the other perpendicular to it. Then the normal force and friction each lie along an axis. Only gravity has to be split into components.

With the ramp’s angle θ and the axes tilted:

• component of F_g along the ramp, pointing down the slope: mg sin θ
• component of F_g into the ramp, perpendicular to it: mg cos θ

A quick check: on a level surface (θ = 0) there is no down-slope part (sin 0 = 0), and the whole weight pushes into the surface (cos 0 = 1).

Worked example 2: a block resting on a ramp

Question. A 2.0 kg block rests without moving on a rough ramp tilted at 25° to the horizontal. (a) Draw the free-body diagram and choose axes. (b) Find the components of the gravitational force along your axes. (c) The block is at rest, so the forces along each axis balance. Find the normal force and the friction force.

(a) Three forces: F_g (Earth on block, vertically down), F_N (ramp on block, perpendicular to the ramp surface, away from it), and F_f (ramp on block, up the slope, because the block tends to slide down). Take +x up the slope, parallel to the surface, and +y perpendicular to the surface, away from it.

(b) F_g = (2.0 kg)(9.8 N/kg) = 19.6 N.

• Along the ramp (down the slope, so −x): mg sin 25° = 19.6 × 0.423 = 8.3 N
• Perpendicular to the ramp (into it, so −y): mg cos 25° = 19.6 × 0.906 = 17.8 N

(c) y-direction: F_N − 17.8 N = 0, so F_N = 17.8 N. x-direction: F_f − 8.3 N = 0, so F_f = 8.3 N, up the slope.

Check. Rebuild F_g from its two components: √(8.28² + 17.76²) = 19.6 N. With horizontal and vertical axes instead, you would have to split both F_N and F_f into components, which is twice the work for the same answer.

Worked example 3: the system changes the diagram

Question. On a smooth floor (negligible friction), a hand pushes box A (3.0 kg) to the right, and box A pushes box B (1.0 kg) ahead of it. List the forces on (a) box A, (b) box B and (c) the system of both boxes.

(a) Box A: F_g, Earth on A (29.4 N, down); F_N, floor on A (up); F_A, hand on A (right); and the force of B on A (left, a contact force because B pushes back). Four forces.

(b) Box B: F_g, Earth on B (9.8 N, down); F_N, floor on B (up); the force of A on B (right). Three forces. The hand does not touch B, so it exerts no force on B.

(c) Both boxes: F_g, Earth on the system (39.2 N, down); the floor’s normal force on the system (up; you may draw the floor’s push on each box as two side-by-side arrows or as one combined arrow); F_A, hand on the system (right). Three forces. The pushes between A and B are internal to this system, so they are not drawn.

Interpretation. The physical situation never changed; only the system did. That is why you must always say which system a free-body diagram is for.

Limiting cases worth knowing

• Free fall. A ball in flight, with air resistance negligible, has one force on it: gravity. Nothing else touches it.
• A lamp hanging at rest from a cord. Two forces: gravity down and tension up.
• A smooth surface. “Smooth” or “frictionless” means you draw no friction arrow.
• A slack string. Tension is zero, so draw no tension arrow.
• A level surface. The normal force is vertical; on a ramp it tilts with the surface.

Common misconceptions

• “A moving object needs a force in its direction of motion.” No. After a ball leaves your hand, your hand is no longer touching it. Do not draw a “force of the throw” (Limiting cases).
• Drawing forces the system exerts. The book pushes down on the table, but that force acts on the table, not the book. It does not go on the book’s diagram.
• “The normal force always equals mg.” Not when a rope pulls upward (Worked example 1) or the surface is tilted (Worked example 2).
• “The normal force is always vertical.” It is perpendicular to the surface.
• “Gravity on a ramp points into the ramp.” Gravity always points straight down. Only its component points into the ramp.
• Drawing components or a net-force arrow on the diagram. Components go in a separate sketch; the net force is a result, not a force exerted by anything.
• Overlapping arrows. Forces in the same direction are drawn side by side.
• “I can push myself forward.” Internal forces cannot change the motion of the system as a whole (the child in the box).

Where this leads

Every force on a free-body diagram has a partner: if the floor pushes up on the crate, the crate pushes down on the floor. Topic 2.3 (Newton’s Third Law) makes this exact. Read the Topic 2.3 study guide next, and revisit the Topic 2.1 study guide on choosing systems if you need to. Try the practice questions now, then use the revision notes and the checklist to consolidate. You can also return to the course roadmap.