5.1 Interactions, Force Models, and Free-Body Diagrams

Something Is Missing

A book sits on a table. How many forces act on it?

Most people say one: gravity pulls it down.

But think about that for a moment. If gravity were the only force on the book, Newton's second law says the book would accelerate downward --- through the table, through the floor, straight toward the center of the Earth. The book does not do this. It sits perfectly still.

Something else must be acting on the book. Something you did not name. Something invisible, easy to overlook, but absolutely essential.

The table pushes up.

That upward push --- the normal force --- is just as real as gravity. The book is at rest because these two forces balance perfectly. Miss one, and your prediction of the motion is wrong.

Before you read on: Here is a block sitting on a ramp with a rope attached to it, pulling upward along the ramp.

How many forces act on the block? List every one you can think of before continuing.

[Interactive: Predict-Then-Reveal. Student types or selects forces from a list. After submitting, the full set is revealed one by one --- gravity, the normal force from the ramp surface, friction along the ramp, and tension from the rope --- with a brief note about each. Most students miss at least one. A counter shows: "You identified ___ out of 4 forces."]

If you missed one or two, you are in good company. Students routinely overlook friction, or the normal force, or both. The question is: how do we systematically find every force so we never miss one?

Forces Come from Interactions

Here is the key insight that organizes everything: every force on an object comes from an interaction with another specific object.

Gravity on the book? That is an interaction between the book and the Earth. The normal force? That is an interaction between the book and the table. Friction on the block on the ramp? That is an interaction between the block and the ramp's surface. Tension? That is an interaction between the block and the rope.

No force appears out of nowhere. If you cannot name the other object involved in the interaction, the force does not exist.

This gives you a reliable procedure for finding every force:

1. Choose your object. Decide which single object you are analyzing.
2. Ask: what touches it? Every contact is a potential source of force --- normal forces, friction, tension, applied pushes and pulls.
3. Ask: what acts on it at a distance? For most problems in this course, the only non-contact force is gravity (the interaction between your object and the Earth).
4. For each interaction, name the force. Be specific: "the normal force from the ramp on the block," not just "normal force."

That is it. Contact forces and gravity. If you work through the list carefully, you will not miss any.

Pause and think: Go back to the block on the ramp with a rope. Run through the procedure above. What touches the block? (The ramp surface and the rope.) What acts at a distance? (The Earth, via gravity.) Does this recover all four forces?

The Free-Body Diagram

Knowing which forces act on an object is half the battle. The other half is representing them clearly enough that you can write equations. The tool for this is the free-body diagram --- often abbreviated FBD.

A free-body diagram does three things:

1. Isolates one object. You draw the object by itself, removed from everything around it. No table, no ramp, no rope --- just the object, floating alone.
2. Shows only the forces that act on that object. Each force is drawn as an arrow starting on the object, pointing in the direction the force pushes or pulls.
3. Labels every arrow. Each force gets a name and, when known, a magnitude.

That is the entire idea. Isolate. Draw forces. Label.

[Video: Start with a photograph of a book on a table. The scene fades, and the book is shown alone against a blank background --- isolated. An arrow appears pointing downward, labeled $\vec{F}{\text{gravity}}$ (or equivalently, $m\vec{g}$). Then an arrow appears pointing upward, labeled $\vec{F}$. The two arrows are the same length, reflecting that the book is in equilibrium. A caption reads: "This is a free-body diagram. Two interactions, two arrows, one object."]}}$, or $\vec{N

The free-body diagram is simple, but its power is enormous. It is the translation layer between the physical world --- with all its ramps and ropes and surfaces --- and the mathematical equation $\sum \vec{F} = m\vec{a}$. Every force arrow becomes a term in that sum. No diagram, no reliable equation.

Building a Free-Body Diagram Step by Step

Let's build one together for the block on the ramp with a rope.

Step 1: Isolate the block. Draw it alone. No ramp, no rope, no ground --- just the block.

Step 2: Identify interactions. What touches the block? The ramp surface (two interactions: normal force perpendicular to the surface, and friction along the surface) and the rope (tension along the rope). What acts at a distance? Gravity, pulling straight down.

Step 3: Draw each force as an arrow.

• $\vec{W}$ (weight): straight down, from the center of the block.
• $\vec{N}$ (normal force): perpendicular to the ramp surface, pointing away from the surface.
• $\vec{f}$ (friction): along the ramp surface. The direction depends on whether the block tends to slide up or down --- for now, assume it tends to slide down, so friction points up the ramp.
• $\vec{T}$ (tension): along the rope, up the ramp.

Step 4: Label everything. Each arrow gets its symbol and, if known, its magnitude.

[Interactive: FBD Builder. Students see a realistic image of a block on a ramp with a rope attached. Below it is an isolated block outline on a blank background. Students drag force arrows from a palette onto the block, choosing direction and label. The system gives real-time feedback: - "You're missing an interaction --- what does the ramp surface do to the block besides push it outward?" - "This force doesn't act on the block --- check which object it acts on." - "Good --- that's the normal force, and it's perpendicular to the surface." When all four forces are placed correctly, the completed FBD is highlighted and a short congratulatory note appears.]

The Most Common Mistake

Here is a mistake that appears in nearly every introductory physics class.

A student is asked to draw the free-body diagram for a book resting on a table. The student draws three arrows:

• $\vec{W}$: gravity pulling down. Correct.
• $\vec{N}$: normal force pushing up. Correct.
• $\vec{F}_{\text{book on table}}$: the force of the book pushing down on the table.

That third arrow does not belong.

The force of the book pushing down on the table is a real force --- but it acts on the table, not on the book. It would appear on the table's free-body diagram. On the book's diagram, it has no place.

This error comes from confusing Newton's third-law pairs. Yes, the book pushes down on the table, and the table pushes up on the book. These are an interaction pair. But each force in the pair acts on a different object. The FBD shows forces on one object only.

The rule: if a force is exerted by your object on something else, it does not appear on your object's FBD. Only forces exerted on your object by other things belong.

What changed? Compare these two diagrams:

• Diagram A: FBD of the book --- shows gravity (from Earth on book) and normal force (from table on book). Two arrows.
• Diagram B: FBD of the table --- shows gravity (from Earth on table), normal force (from ground on table), and the weight of the book pressing down (from book on table). Three arrows.

Every force appears on exactly one FBD --- the diagram of the object it acts on. Forces never appear on both sides of an interaction.

Contact Forces and Non-Contact Forces

As you catalog forces, a useful classification emerges. Some forces require physical contact. Others act across empty space.

Type	Requires contact?	Examples
Contact forces	Yes	Normal force, friction, tension, air resistance, applied push/pull
Non-contact forces	No	Gravity, electric force, magnetic force

In this course, the only non-contact force you will encounter regularly is gravity. Every other force arises from direct contact between objects.

This classification is more than a label --- it is a diagnostic tool. If you have identified all the things that touch your object and accounted for gravity, you have found every force. Nothing else can act on it.

Why We Isolate One Object

You might wonder: why not draw all the forces in the entire scene on one big diagram? Why bother isolating a single object?

Because Newton's second law applies to one object at a time:

$$\sum \vec{F}{\text{on object}} = m$$}} \, \vec{a}_{\text{object}

The left side is the sum of forces on a specific object. The right side is that object's mass times that object's acceleration. If you mix forces from different objects onto one diagram, you cannot write this equation. You do not know which mass to use, which acceleration to connect it to, or which forces to sum.

The FBD enforces discipline. It says: pick one object, find every force on it, and then write the equation. If you have multiple objects, draw a separate FBD for each one and write a separate equation for each one. The equations can then be connected through shared forces or constraints --- but each equation starts from a clean, unambiguous diagram.

The FBD as a Reasoning Tool

It is tempting to treat the free-body diagram as busywork --- a picture you draw because the instructor requires it, then ignore while you write equations from memory.

Resist that temptation.

The FBD is not decoration. It is where your physical reasoning happens. Every arrow you draw is a claim: "this interaction exists, and it pushes or pulls in this direction." Every arrow you leave off is also a claim: "this interaction is negligible or absent." The diagram encodes your assumptions before you write a single equation.

When a dynamics problem goes wrong, the error is almost never in the algebra. It is almost always in the diagram. A missing force. A force pointing the wrong direction. A force that belongs on a different object's diagram. Fix the diagram, and the algebra fixes itself.

The FBD is the link between the physical world and the mathematical equation $\sum \vec{F} = m\vec{a}$. No diagram, no reliable equation.

Practice

Layer 1: Concrete

Draw free-body diagrams for each of the following setups. For each force, name the interaction (e.g., "normal force from table on book") and draw the arrow in the correct direction.

(a) A book resting on a table.

(b) A ball hanging from a string attached to the ceiling.

(c) A block sitting on a ramp (no rope, the block is stationary).

Check your answer

(a) **Book on table.** Two forces: weight $\vec{W}$ pointing straight down (Earth on book), and normal force $\vec{N}$ pointing straight up (table on book). The arrows are equal in length since the book is at rest. (b) **Ball hanging from string.** Two forces: weight $\vec{W}$ pointing straight down (Earth on ball), and tension $\vec{T}$ pointing straight up along the string (string on ball). Again, equal in length for a stationary ball. (c) **Block on ramp (stationary, no rope).** Three forces: weight $\vec{W}$ pointing straight down (Earth on block), normal force $\vec{N}$ pointing perpendicular to the ramp surface and away from it (ramp on block), and static friction $\vec{f}_s$ pointing up the ramp along the surface (ramp on block). Friction prevents the block from sliding down. Notice the pattern: for each setup, the number of forces equals the number of interactions (contacts plus gravity).

Layer 2: Pattern

For each scenario below, list every force acting on the object in bold. Then classify each force as either contact or non-contact.

(a) A skydiver falling through the air (before the parachute opens).

(b) A car driving on a flat road at constant velocity.

(c) A magnet stuck to a refrigerator door.

(d) A hockey puck sliding across the ice.

(e) A satellite orbiting Earth (no atmosphere).

Check your answer

(a) **Skydiver:** Weight (non-contact), air resistance/drag (contact). Two forces. (b) **Car at constant velocity:** Weight (non-contact), normal force from road (contact), friction from road providing forward traction (contact), air resistance opposing motion (contact). Four forces. Since the car moves at constant velocity, the net force is zero --- the forward friction from the engine-driven tires balances the backward drag and any rolling resistance. (c) **Magnet on refrigerator:** Weight (non-contact), normal force from door surface (contact), static friction from door pointing upward (contact), and the magnetic attraction to the door (non-contact). Four forces. The magnetic force pulls the magnet toward the door; the normal force prevents it from going through. Friction and weight balance vertically. (d) **Hockey puck sliding on ice:** Weight (non-contact), normal force from ice (contact), kinetic friction from ice opposing the motion (contact). Three forces. Air resistance exists but is very small for a compact, low-speed puck --- you could reasonably include or ignore it depending on the problem. (e) **Satellite in orbit (no atmosphere):** Weight/gravitational force (non-contact). That is the only force. There is no surface to provide a normal force, no air for drag, no rope for tension. Gravity alone curves the satellite's path into an orbit.

Layer 3: Structure

Why do we isolate one object when drawing a free-body diagram, instead of drawing all forces for the entire system on a single picture?

Check your answer

Newton's second law relates the net force *on a specific object* to *that object's* mass and acceleration: $\sum \vec{F} = m\vec{a}$. If you put forces from multiple objects onto one diagram, you cannot write this equation --- you do not know which mass goes with which forces, and you conflate forces that act on different objects. Isolating one object also prevents a common error: including forces that the object exerts on other things. By stripping away everything except the chosen object, the FBD forces you to ask the right question --- "what pushes or pulls on *this* object?" --- and nothing else. When you have multiple interacting objects (e.g., two blocks connected by a rope), you draw a separate FBD for each one. Each diagram produces its own $\sum \vec{F} = m\vec{a}$ equation. The equations are then linked through shared forces (like tension) and kinematic constraints (like both blocks having the same acceleration). This systematic approach scales to arbitrarily complex systems.

Layer 4: Debug

A student draws the following free-body diagram for a block being pushed to the right across a rough table by a hand:

• $\vec{W}$: pointing down (weight)
• $\vec{N}$: pointing up (normal force from table)
• $\vec{F}_{\text{push}}$: pointing to the right (push from hand)
• $\vec{F}_{\text{block on table}}$: pointing down (the block pressing on the table)
• $\vec{f}_k$: pointing to the right (kinetic friction)

There are two errors. Find them.

Check your answer

**Error 1:** $\vec{F}_{\text{block on table}}$ does not belong on the block's FBD. This is the force the block exerts *on the table* --- it acts on the table, not on the block. It would appear on the table's free-body diagram. Including it here double-counts the downward force on the block (gravity already accounts for the block's downward pull). **Error 2:** Kinetic friction $\vec{f}_k$ points in the wrong direction. Friction opposes the direction of sliding. If the block is being pushed to the right and sliding to the right, kinetic friction acts to the *left*, not the right. The corrected FBD has four forces: $\vec{W}$ down, $\vec{N}$ up, $\vec{F}_{\text{push}}$ to the right, and $\vec{f}_k$ to the left.

Reflection

Think back over what you have read and explored in this section.

What surprised you about how many forces act on a simple object at rest?

Before this section, you might have said that a book on a table has "just gravity" acting on it. Now you know there are at least two forces, and that if either one were missing, the book's behavior would be dramatically different. Consider: every object you see around you right now --- your phone, your chair, a cup on the desk --- has multiple forces acting on it in a careful balance. The world is full of invisible interactions.

Looking Ahead

You have just built the most important tool in dynamics: the free-body diagram. It may look like a simple sketch, but it is the bridge between the physical setup you can see and the mathematical equation you need to solve. Every problem in the rest of this chapter --- and most problems for the rest of this course --- will begin with a FBD.

In the next section, we turn to the equations themselves. Newton's three laws tell you exactly what to do with the forces you have identified: sum them, set them equal to $m\vec{a}$, and solve. You already know how to find the forces. Now you will learn the rules that govern what those forces do.