04Forces and Newton's Laws
Apply Newton's three laws of motion to analyze forces, draw free-body diagrams, calculate net force, and solve friction problems.
Newton's laws explain why objects move the way they do. Every engineering structure, vehicle, and sports play depends on these principles — they are the core of classical mechanics.
How do forces acting on an object determine whether it stays at rest, moves at constant velocity, or accelerates?
Lesson Overview
A force is a push or pull — a vector quantity with both magnitude and direction, measured in newtons (N). Forces can be grouped as contact forces (friction, tension, normal force — require physical touch) or field forces (gravity, magnetism — act at a distance). Isaac Newton's three laws of motion describe exactly how forces change — or fail to change — the motion of objects. Mastering these laws lets you predict acceleration, solve for unknown tensions, and analyze everything from a sliding crate to an Atwood machine.
Newton's Three Laws
1st Law — Law of Inertia
An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted upon by a net external force. Inertia is the tendency of an object to resist changes in its state of motion. When ΣF = 0, the object is in equilibrium.
2nd Law — ΣF = ma
The net force on an object equals its mass times its acceleration. The direction of acceleration is always the same as the direction of the net force. A larger net force produces greater acceleration; a larger mass resists acceleration. Free-body diagrams (FBDs) are the essential tool for identifying all forces before applying this law.
3rd Law — Action-Reaction Pairs
For every action force, there is an equal and opposite reaction force. These forces act on different objects, so they never cancel each other. Example: you push the floor down (action); the floor pushes you up (reaction). The two forces are equal in magnitude and opposite in direction.
Key Equations
Newton's 2nd Law
ΣF = ma
Weight
W = mg (g = 9.8 m/s²)
Kinetic friction
fk = μk · N
Max static friction
fs_max = μs · N
Solve for acceleration
a = ΣF / m
Worked Examples
A 10 kg box on a flat surface is pushed with an applied force of 40 N. Kinetic friction acts backward with a magnitude of 10 N. Find the acceleration of the box.
Draw a free-body diagram: Fapp = 40 N (right), fk = 10 N (left), N and W cancel vertically.
Net force: ΣF = Fapp − fk = 40 N − 10 N = 30 N
Apply Newton's 2nd Law: a = ΣF / m = 30 N / 10 kg
a = 3 m/s² (in the direction of the applied force)
A 5 kg book rests on a horizontal table. Find the normal force exerted by the table on the book.
The book is in equilibrium (a = 0), so ΣF = 0 in every direction.
Vertically: N − W = 0 → N = W = mg
N = (5 kg)(9.8 m/s²) = 49 N
An 8 kg crate is pushed across a floor at constant velocity. The coefficient of kinetic friction is μk = 0.30. Find the applied force needed to maintain constant velocity.
Constant velocity means a = 0, so ΣF = 0 (Newton's 1st Law).
Normal force: N = mg = (8 kg)(9.8 m/s²) = 78.4 N
Kinetic friction: fk = μk · N = 0.30 × 78.4 N = 23.5 N
Since a = 0: Fapp = fk = 23.5 N
Two blocks are connected by a light rope on a frictionless horizontal surface. Block 1 (m₁ = 4 kg) is behind block 2 (m₂ = 6 kg). An external force F = 30 N pulls block 2 forward. Find (a) the acceleration of the system and (b) the tension in the rope.
Treat the system as one object: total mass = m₁ + m₂ = 4 + 6 = 10 kg.
(a) a = F / (m₁ + m₂) = 30 N / 10 kg = 3 m/s²
(b) Isolate block 1: the only horizontal force on it is tension T.
T = m₁ · a = 4 kg × 3 m/s² = 12 N
Identify the Newton's 3rd Law action-reaction pair when a person pushes against a wall with 80 N of force.
Action: the person exerts an 80 N force on the wall (directed into the wall).
Reaction: the wall exerts an 80 N force on the person (directed away from the wall).
These forces are equal in magnitude, opposite in direction, and act on different objects.
They do NOT cancel because they act on different objects (person ≠ wall).
Guided Problems
A 15 kg sled is pulled across ice with an applied force of 45 N. A friction force of 9 N opposes the motion. What is the sled's acceleration?
Hint: Find the net force first (ΣF = Fapp − fk), then use a = ΣF / m.
A 12 kg object hangs from a rope attached to the ceiling. What is the tension in the rope? (g = 9.8 m/s²)
Hint: The object is in equilibrium. Draw an FBD: tension acts upward, weight acts downward. Set them equal.
A 20 kg box sits on a surface where μs = 0.40. What is the maximum static friction force before the box begins to slide?
Hint: Calculate the normal force (N = mg), then use fs_max = μs · N.
In an Atwood machine, mass m₁ = 3 kg hangs on the left and m₂ = 5 kg hangs on the right over a frictionless pulley. Find the acceleration of the system.
Hint: Net force = (m₂ − m₁)g. Total mass = m₁ + m₂. Use a = ΣF / m_total.
A 6 kg block is pushed to the right with 24 N. A 4 kg block is in contact with the right side of the 6 kg block (no rope). The surface is frictionless. Find the contact force between the two blocks.
Hint: Find the system acceleration first (a = F / m_total), then isolate the 4 kg block: contact force = m₂ · a.
Key Vocabulary
Force
A push or pull on an object; a vector quantity with magnitude and direction, measured in newtons (N).
Example: Gravity pulls a 2 kg ball downward with F = mg = 19.6 N.
Inertia
The tendency of an object to resist any change in its state of motion. Greater mass means greater inertia.
Example: A bowling ball is harder to start moving than a tennis ball.
Net Force
The vector sum of all forces acting on an object (ΣF). Determines whether the object accelerates.
Example: 40 N right + 10 N left = 30 N net force to the right.
Newton's First Law
An object remains at rest or in uniform motion unless a net external force acts on it (law of inertia).
Example: A hockey puck slides at constant velocity on frictionless ice.
Newton's Second Law
The net force on an object equals its mass times its acceleration: ΣF = ma.
Example: ΣF = 30 N, m = 10 kg → a = 3 m/s².
Newton's Third Law
For every action force there is an equal and opposite reaction force acting on a different object.
Example: A rocket expels gas downward; the gas pushes the rocket upward.
Friction
A contact force that opposes relative motion between surfaces. Static friction prevents motion; kinetic friction acts during sliding.
Example: fk = μk · N = 0.3 × 49 N = 14.7 N.
Normal Force
The perpendicular contact force a surface exerts on an object resting on it. On a flat surface, N = mg.
Example: A 5 kg book on a table: N = 49 N upward.
Workbook Check
Interactive Practice — 5 Questions
A 4 kg object experiences a net force of 20 N. What is its acceleration?
A 10 kg box rests on a horizontal surface. What is the normal force on the box? (g = 9.8 m/s²)
Which of the following best describes Newton's First Law?
A 6 kg crate slides on a surface with μk = 0.25. What is the kinetic friction force? (g = 9.8 m/s²)
You push a wall with 50 N. According to Newton's 3rd Law, the wall pushes you with:
Independent Practice
A 12 kg box is pushed across a frictionless floor by a horizontal force of 36 N. Calculate the acceleration of the box.
A 5.0 kg object rests on a surface with μₛ = 0.40. What is the minimum horizontal force needed to start the object moving? (g = 9.8 m/s²)
Two forces act on an object: 40 N east and 30 N north. Find the magnitude and direction of the net force.
A 10 kg block sits on a frictionless incline at 30°. Calculate the acceleration of the block along the incline. (g = 9.8 m/s²)
★ A 20 kg box is pulled up a 25° incline by a rope parallel to the surface. The coefficient of kinetic friction is 0.30. (a) Draw a complete free-body diagram. (b) Calculate the tension in the rope needed to pull the box up at constant velocity. (c) What tension would be needed to accelerate it up the incline at 1.5 m/s²? (g = 9.8 m/s²)
ChallengeCommon Mistakes
Drawing the reaction force on the same object instead of on the other object in a Newton's 3rd law pair
Action-reaction pairs act on DIFFERENT objects. If Earth pulls you down, you pull Earth up — those are the pair
Including velocity or acceleration as forces on a free-body diagram
Only real contact forces (normal, friction, tension) and field forces (gravity) go on an FBD. Velocity and acceleration are NOT forces
Forgetting to resolve forces into components before applying ΣF = ma
For inclined planes or angled forces, break every force into x- and y-components, then apply Newton's 2nd law to each axis separately
Confusing static friction (f_s ≤ μ_s N) with kinetic friction (f_k = μ_k N)
Static friction adjusts up to its maximum value to prevent motion; kinetic friction is constant once sliding begins
Math Tips
FBD checklist: (1) draw the object as a dot, (2) add all forces as arrows from the dot, (3) label each force with its type and magnitude, (4) choose x-y axes aligned with motion or the incline
On an incline at angle θ: the component of gravity along the slope = mg sinθ (causes sliding); perpendicular to slope = mg cosθ (determines normal force N = mg cosθ)
Equilibrium means ΣF = 0 in every direction — not that all forces are zero. An object moving at constant velocity is also in equilibrium (Newton's 1st law)
Atwood machine shortcut: a = (m₁ − m₂)g / (m₁ + m₂). The heavier mass accelerates downward; tension T = 2m₁m₂g / (m₁ + m₂)