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Forces and Free-Body Diagrams

Introduction

Forces are everywhere: the ground pushes up on your shoes, gravity pulls you toward Earth, a rope can pull a sled, and friction can oppose sliding. In this aiMOOC you will learn how to identify these interactions, represent them with vectors, combine them into a net force, and use free-body diagrams to predict motion. The course is designed for Grades 9–10 and assumes basic algebra and familiarity with speed and acceleration.

A force is a push or pull caused by an interaction. Force is measured in newtons, symbol N. Because a force has both magnitude and direction, it is a vector quantity. A free-body diagram turns a real situation into a simple model so that you can focus on the external forces acting on one chosen object.

Use the image above as an overview: Newton's laws connect forces to motion. As you work through the course, keep asking two questions: What object am I analyzing? and What external forces act on that object?

The video above introduces force types and free-body diagrams. Watch for the difference between a real object and the simplified representation used in a diagram.

Learning Goals

By the end of this aiMOOC, you should be able to identify common forces, draw clear free-body diagrams, calculate one-dimensional net force, apply Newton's first and second laws, distinguish balanced from unbalanced forces, and explain why Newton's third-law pairs act on different objects. You should also be able to transfer these ideas to unfamiliar situations such as ramps, elevators, sports, vehicles, and simple engineering systems.

Forces as Vectors

A vector is represented by an arrow. The arrow points in the direction of the force, and its length can be used to show relative magnitude. For example, a 30 N force to the right and a 10 N force to the left do not cancel. If right is positive, the net horizontal force is 30 N - 10 N = 20 N to the right.

The net force is the vector sum of all external forces acting on the chosen object. A zero net force does not necessarily mean that the object is at rest. It may be at rest, or it may move with constant velocity. A nonzero net force means the object accelerates.

Forces can be grouped as contact forces and non-contact forces. Contact forces require physical interaction at a surface or through an object such as a rope. Examples include the normal force, friction, tension, drag, and an applied push or pull. Gravity is a familiar non-contact force.

The image shows a body on a horizontal surface. The downward weight and upward normal force can be equal in magnitude when there is no vertical acceleration, but the normal force is not automatically equal to weight in every situation.

Newton's Laws and Force Analysis

Newton's First Law

Newton's first law states that an object remains at rest or continues with constant velocity unless a nonzero net external force acts on it. This is the idea of inertia. In a free-body diagram, balanced forces correspond to zero net force.

For example, a book resting on a level table has a downward gravitational force and an upward normal force from the table. If no other vertical forces act and the book has no vertical acceleration, those two forces are equal in magnitude.

Newton's Second Law

Newton's second law relates net force, mass, and acceleration:

${\vec{F}}_{n e t} = m \vec{a}$

The acceleration points in the direction of the net force. For the same mass, a larger net force produces a larger acceleration. For the same net force, a larger mass produces a smaller acceleration.

Suppose a 10 kg sled is pushed with 40 N to the right while friction is 15 N to the left. The net force is 25 N to the right. The acceleration is 25 N divided by 10 kg, or 2.5 m/s² to the right.

This video develops Newton's second law and explains how net force and mass determine acceleration.

Newton's Third Law

Newton's third law says that if object A exerts a force on object B, then object B exerts an equal-magnitude force in the opposite direction on object A. The two forces form an interaction pair, but they act on different objects. Therefore, you normally place only one member of that pair on a free-body diagram for a single object.

Do not confuse a third-law pair with two balanced forces on one object. The normal force on a book and the book's weight may be equal and opposite, but both act on the book. They are not a Newton's third-law pair.

What a Free-Body Diagram Shows

A free-body diagram, often abbreviated FBD, isolates one object or system and shows every relevant external force acting on it. The object is often represented by a dot or simple box. Each external force is drawn as a labeled arrow starting at or near the object.

A good diagram makes your system boundary clear. If you choose a cart as the system, show forces exerted on the cart by Earth, the floor, a rope, a person, or air when those forces matter. Do not draw the cart's velocity as a force. Do not draw the force that the cart exerts on another object as though it acted on the cart.

The pushed refrigerator example shows why a free-body diagram is useful: vertical forces can balance while horizontal forces produce acceleration.

This video demonstrates how to construct free-body diagrams on level surfaces and inclines.

A Reliable Drawing Procedure

Choose the object. State exactly what body or system you are analyzing.
Sketch the object simply. Use a dot or box unless shape matters.
Identify interactions. Ask what touches the object and what non-contact interactions act on it.
Draw one arrow for each external force. Point each arrow in the direction that force acts on the chosen object.
Label each force. Use clear labels such as weight, normal force, tension, friction, drag, or applied force.
Choose axes and add forces. Use a sign convention, sum forces by direction, and apply Newton's laws.

Arrow length may represent relative force size when you know or can compare magnitudes. If the force sizes are unknown, the diagram can still be useful as long as the directions and force types are correct.

This worked-example video from The Physics Classroom shows several situations and emphasizes force type, direction, and relative arrow size.

Common Forces You Should Recognize

Weight

Weight is the gravitational force exerted by Earth on an object near Earth's surface. Its magnitude is approximately $W = m g$ , where $g$ is about 9.8 m/s². In the usual near-Earth model, weight points vertically downward toward Earth's center.

Normal Force

The normal force is a contact force exerted by a surface on an object. It points perpendicular to the contact surface. On a horizontal table it is vertical; on an inclined plane it is tilted with the surface normal. Its magnitude depends on the situation and is not always equal to the object's weight.

Friction

Friction acts parallel to a contact surface and opposes relative sliding or the tendency to slide. Static friction adjusts up to a maximum value as needed to prevent slipping. Kinetic friction acts when surfaces slide relative to one another.

The diagram highlights applied force and friction. In a complete free-body diagram for a block on a horizontal surface, you would also consider weight and the normal force.

Tension and Other Pulling Forces

Tension is the pulling force transmitted by a taut rope, string, or cable. On the object you are analyzing, tension points along the rope and away from the object. In an ideal massless rope over an ideal pulley, the tension can be modeled as having the same magnitude along one continuous rope, but real systems may require a more detailed model.

A pulley system can require more than one free-body diagram because different parts of the system experience different external forces.

Drag and Air Resistance

Drag is a resistive force from a fluid such as air or water. It acts opposite the object's motion relative to the fluid. Drag often increases with speed, so it may start small and become large enough to balance weight during a fall.

This free-body diagram isolates a falling body affected by gravity and air resistance. If the two forces become equal in magnitude, the net force becomes zero and the object continues at constant velocity.

Balanced and Unbalanced Forces

Balanced forces produce zero net force. A stationary lamp hanging from a single vertical cable can have tension upward equal to weight downward. A hockey puck moving at constant velocity on nearly frictionless ice can also have zero net horizontal force even though it is moving.

Unbalanced forces produce acceleration. Acceleration means any change in velocity, including speeding up, slowing down, or changing direction. The direction of acceleration follows the net force, not necessarily the current direction of motion.

A useful question is: Which arrows cancel by components, and which do not? In two dimensions, analyze horizontal and vertical components separately.

From a Diagram to Equations

Once the free-body diagram is correct, choose positive directions and write force sums. In one dimension:

$\sum F_{x} = m a_{x}$

For vertical motion you can write:

$\sum F_{y} = m a_{y}$

Signs come from your chosen coordinate directions. A leftward force is not inherently negative; it is negative only if you chose right as positive.

Example: Book on a Table

A 2.0 kg book rests on a level table. Its weight is approximately 2.0 kg × 9.8 m/s² = 19.6 N downward. Because the book has no vertical acceleration and no other vertical forces act, the table's normal force is 19.6 N upward. The net vertical force is zero.

Example: Pushed Box

A 6.0 kg box is pushed horizontally with 24 N to the right while kinetic friction is 9 N to the left. The net horizontal force is 15 N to the right. Newton's second law gives an acceleration of 15 N divided by 6.0 kg = 2.5 m/s² to the right. The weight and normal force can balance vertically if there is no vertical acceleration.

Example: Falling Object

A falling object has weight 12 N downward and air resistance 5 N upward. The net force is 7 N downward, so the object accelerates downward. If air resistance later grows to 12 N upward, the net force becomes zero. The object can then continue falling at constant velocity.

Inclined Planes

On an inclined plane, the normal force is perpendicular to the surface and friction is parallel to the surface. Weight still points vertically downward. Choosing axes parallel and perpendicular to the slope often makes the equations easier.

For learners who know trigonometry, the weight can be resolved into a component parallel to the slope and a component perpendicular to it. If the incline angle is $θ$ , those components are commonly written as $m g \sin θ$ along the slope and $m g \cos θ$ perpendicular to the slope. These components are not extra forces; together they are an alternative representation of the single weight force.

Common Mistakes and How to Fix Them

Mistake: drawing motion as a force. Velocity and acceleration are not force arrows. Draw only actual external interactions.

Mistake: drawing both members of a third-law pair on one object's diagram. A free-body diagram includes only forces acting on the chosen object.

Mistake: assuming normal force always equals weight. They are equal only in specific situations, such as a level surface with no vertical acceleration and no other vertical force components.

Mistake: assuming zero net force means zero velocity. Zero net force means zero acceleration, so velocity is constant and may be nonzero.

Mistake: choosing friction from the direction of motion alone. Friction opposes relative slipping or the tendency to slip at the contact surface.

Mistake: resolving a force into components and also keeping the original force in the same force sum. Use either the original vector or its components for a chosen calculation, not both.

Virtual Investigation

Use the PhET Forces and Motion: Basics simulation to explore net force, motion, friction, and acceleration. Turn force values on, vary the applied force and mass, and record how the acceleration changes. Before each trial, predict the free-body diagram and the direction of the net force.

A useful investigation table can include the chosen object, mass, applied force, friction force, net force, predicted acceleration direction, observed acceleration, and a short explanation. Compare at least one balanced case with one unbalanced case.

Interactive Tasks

Quiz: Test Your Knowledge

What does a free-body diagram show for a chosen object? (External forces acting on the object) (!All objects in the surrounding scene) (!The object's path through space) (!Only forces that cause motion)

Which statement describes a force correctly? (A force has magnitude and direction) (!A force has magnitude but no direction) (!A force is measured in kilograms) (!A force is the same as velocity)

A box has 30 N to the right and 10 N to the left. What is the net horizontal force? (20 N to the right) (!40 N to the right) (!20 N to the left) (!Zero net force)

What does zero net force imply? (The object's acceleration is zero) (!The object must be at rest) (!The object has no velocity) (!No forces act on the object)

Which direction does the normal force act? (Perpendicular to the contact surface) (!Always vertically upward) (!Parallel to the contact surface) (!Always opposite the velocity)

Which force acts along a taut rope? (Tension) (!Weight) (!Normal force) (!Drag)

According to Newton's second law, what determines acceleration? (Net force and mass) (!Velocity and time only) (!Weight and distance only) (!Normal force only)

Why are Newton's third-law forces not usually drawn on the same free-body diagram? (They act on different objects) (!They always cancel before drawing) (!They have different magnitudes) (!They point in the same direction)

A moving object has zero net force. What happens to its velocity? (It remains constant) (!It immediately becomes zero) (!It must increase) (!It must reverse)

Which arrow should not appear as a force arrow on a standard free-body diagram? (Velocity) (!Weight) (!Friction) (!Tension)

Memory Game

Weight	Gravitational force acting on an object
Normal force	Surface force perpendicular to the contact surface
Friction	Contact force that opposes relative slipping
Tension	Pull transmitted by a taut rope or cable
Net force	Vector sum of all external forces on the chosen object
Equilibrium	State in which the net force is zero
Drag	Resistive force from motion relative to a fluid
Acceleration	Rate of change of velocity

Drag and Drop

Match the correct terms.	Topic
Points perpendicular to a surface	Normal force
Points toward Earth's center	Weight
Acts along a taut rope	Tension
Opposes relative slipping	Friction
Equals the vector sum of external forces	Net force

...

Crossword Puzzle

Newton	What is the SI unit of force?
Tension	What pulling force is transmitted by a taut rope?
Friction	What contact force opposes relative slipping?
Gravity	What interaction gives an object its weight near Earth?
Resultant	What word can describe the single vector equivalent to several forces?
Equilibrium	What state has zero net force?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Force Hunt: Photograph or sketch four everyday situations and label at least two forces acting on one chosen object in each scene.
Schoolbag Diagram: Draw a free-body diagram for a schoolbag resting on the floor and write two sentences explaining why the forces are balanced.
Arrow Scale Challenge: Create three simple force diagrams that use arrow length to compare magnitudes, then ask a classmate to identify the net-force direction.
Balanced Forces Comic: Produce a four-panel comic showing an object at rest or moving at constant velocity and explain how zero net force is possible.

Standard

Friction Experiment: Slide the same object across two safe surfaces, compare the force needed to keep it moving at roughly constant speed, and explain the result with free-body diagrams.
Motion Video Analysis: Record a short video of a toy cart being pushed, choose three moments, and draw a free-body diagram for the cart at each moment.
Physics Interview: Interview a mechanic, bicycle technician, coach, or engineer about one situation where forces must be considered, then translate one example into a free-body diagram.
Virtual Force Lab: Use the PhET force simulation to test how changing mass and net force affects acceleration, then present your data and one evidence-based conclusion.

Advanced

Incline Investigation: Build or use a safe adjustable ramp, test an object at several angles, and explain with free-body diagrams how the normal force and tendency to slide change.
Acceleration Study: Use a phone sensor or classroom motion sensor to measure a safe cart or elevator-like motion and compare the observed acceleration with your predicted net-force direction.
Engineering Safety Case: Analyze the forces on a cyclist, passenger, package, or playground user in a sudden stop and propose one design feature that manages those forces more safely.
Teach Free-Body Diagrams: Produce a two-to-four-minute teaching video that corrects three common free-body-diagram mistakes and includes at least two original worked examples.

Learning Assessment

System Boundary Analysis: Given a person standing in an accelerating elevator, define the system, draw the free-body diagram, and justify which external forces are included and excluded.
Error Analysis: Correct a diagram that contains velocity as a force, an incorrect normal-force direction, and both members of a Newton's third-law pair, explaining every correction.
Net Force Transfer: Compare two objects with different masses but the same net force and explain, using Newton's second law, which object accelerates more and why.
Evidence from Motion: From a description of an object moving at constant velocity, infer what must be true about the net force and propose at least two different force combinations that fit the evidence.
Ramp Reasoning: Draw and analyze a block on an incline, then explain why weight remains vertical while the normal force rotates with the surface.
Experiment Design: Design a safe investigation that tests how friction affects acceleration, identify variables to control, state the measurements needed, and predict the free-body diagrams for two contrasting trials.

Evidence of Learning

Knowledge: You can define force, net force, equilibrium, weight, normal force, friction, tension, and drag, and you can connect these ideas to Newton's laws.

Skills: You can choose a system, identify external interactions, draw and label free-body diagrams, use vector directions consistently, calculate one-dimensional net force, and apply $F_{n e t} = m a$ to simple problems.

Products: Strong evidence includes accurate diagrams, annotated photographs, experiment records, simulation data, short explanations, worked calculations, interview summaries, and explanatory videos.

Reasoning: You can explain why zero net force does not require zero velocity, why normal force is not always equal to weight, and why Newton's third-law partners belong to different objects.

Transfer: You can apply force analysis to unfamiliar situations such as ramps, vehicles, sports, lifts, pulleys, falling objects, and simple engineering designs.

OERs on the Topic

For further study, use the OpenStax Physics section on force, the OpenStax guide to drawing free-body diagrams, and the PhET Forces and Motion: Basics simulation. These resources support practice with force identification, net force, Newton's laws, and diagram construction.

Linked Learning Areas

Forces and Free-Body Diagrams

The topic connects physics with mathematics through vectors, signs, algebra, and proportional reasoning; with engineering through force analysis and design; with sports science through motion and contact forces; and with technology through sensors, simulations, and data collection.