Newton laws | PhysicsRank

An empty trolley and a loaded trolley respond differently to the same push. Before reaching for F = ma, decide which trolley you are studying and which interactions act on it. A force diagram is the part of the calculation where the physical story becomes precise.

We will work in an approximately inertial laboratory frame: a frame in which an isolated body moves with constant velocity. Everyday ground-based experiments can often use this approximation. A frame accelerating with a lift needs additional inertial-force terms, so we will observe the lift from the ground instead.

What you will learn to do

Distinguish a force from a velocity or an acceleration.
Draw a free-body diagram containing only forces on the chosen body.
Translate the diagram into signed equations along chosen axes.
Distinguish balanced forces from a third-law pair.
Choose between a single-body and a combined-system calculation.

Useful before this chapterVectors and their components →Velocity and acceleration →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Does an object need a nonzero net force to keep moving at constant velocity?

Compare your reasoning

No, in an inertial frame constant velocity means zero acceleration and hence zero net force. Forces can still be present and balanced. The law concerns their vector sum.

Revisit the explanation →

01 / From interactions to equations

Motion needs no continuing push

Slide a puck along an ideal horizontal surface with no friction or air resistance. Once your hand lets go, the puck keeps moving at constant velocity. A continuing force is needed to change its velocity, not to maintain it. Everyday objects slow down because resisting interactions remain after the push ends.

Newton’s first law describes rest and uniform straight-line motion on equal terms. Both have zero acceleration. A body can therefore be moving while its net force is zero. Conversely, it can have zero velocity at one instant while a nonzero net force changes its motion, as at the top of a vertical throw.

\sum\vec F=\vec 0\quad\Longrightarrow\quad\vec a=\vec 0\quad\Longrightarrow\quad\vec v\text{ is constant}

Constant velocity includes the special case v = 0. It does not mean every individual force is absent.

STOP & THINK

A puck travels right at constant velocity on a horizontal surface. Which must be true in an inertial frame?

There is a net force to the right
The net force is zero
There are no forces of any kind

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: Motion needs no continuing push
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
\sum\vec F=\vec 0\quad\Longrightarrow\quad\vec a=\vec 0\quad\Longrightarrow\quad\vec v\text{ is constant}
Meaning: Constant velocity includes the special case v = 0. It does not mean every individual force is absent.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

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02 / From interactions to equations

One diagram, one chosen body

A force is an interaction between bodies. Earth pulls on the book; the table pushes on it; a hand may push it sideways. Before drawing arrows, name the body you are analysing. Then ask what outside objects interact with that body. Draw only their forces on it.

For a small body modelled as a particle, draw the force arrows from a representative point. A more detailed extended-body model may also need the point of application to calculate torque. We begin with translation, for which the vector sum of forces controls the centre-of-mass acceleration.

A book resting on a horizontal table
Interaction	Force on the book	Direction
Earth–book	Gravitational force mg	Downward
Table–book	Normal contact force N	Upward
No hand contact	No hand force	No arrow to draw

The upward normal force is not automatically mg. Here it equals mg because the book has zero vertical acceleration and these are its only vertical forces. With a hand pressing down, an upward pull, or a vertically accelerating support, the normal force changes.

Do not draw “ma” as an extra force. It is the result of adding the actual forces. Likewise, velocity is a property of motion, not another interaction. A separate velocity arrow can be useful, but it must not be included in the force sum.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: One diagram, one chosen body
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / From interactions to equations

Turn the picture into a signed equation

For a body of constant mass in an inertial frame, Newton’s second law states that its net external force is mass times acceleration. The net force and acceleration point in the same direction. Velocity need not point that way: an object may be slowing down or turning.

\sum\vec F=m\vec a;\qquad \sum F_x=ma_x,\quad\sum F_y=ma_y

Choose axes, resolve each force onto them, and add signed components. Each component equation describes the same body.

Explain this equation

Read it in words

The vector sum of the forces on one object equals its mass multiplied by its acceleration. The same balance can be written separately along each coordinate axis.

Symbols and units

$\sum\vec F$: Net external force on the chosen object; add all force vectors on that object.Newtons (N).
$m$: Mass of the object in this constant-mass model.Kilograms (kg).
$\vec a$: Acceleration: rate of change of velocity, including direction.Metres per second squared (m/s²).
$F_x,\ F_y,\ a_x,\ a_y$: Signed components along the chosen x and y axes.Force components in N; acceleration components in m/s².

Signs and operations

Σ means add; arrows mean vectors. A force opposite the positive axis contributes a negative component. Equal-and-opposite third-law partners act on different objects and do not both belong on one object’s diagram.

When it applies

Use an inertial frame, nonrelativistic speeds and constant mass. Identify the body before listing forces. A moving object can have zero acceleration if its velocity is constant.

Make it concrete

For a 2 kg cart pulled 9 N right and resisted by 3 N left, choose right positive: net force is +6 N and acceleration is +3 m/s². Its current direction of motion need not be rightward.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): \sum\vec F=m\vec a;\qquad \sum F_x=ma_x,\quad\sum F_y=ma_y
Read it as: The vector sum of the forces on one object equals its mass multiplied by its acceleration. The same balance can be written separately along each coordinate axis.
Symbols:
\sum\vec F: Net external force on the chosen object; add all force vectors on that object.; Newtons (N).
m: Mass of the object in this constant-mass model.; Kilograms (kg).
\vec a: Acceleration: rate of change of velocity, including direction.; Metres per second squared (m/s²).
F_x,\ F_y,\ a_x,\ a_y: Signed components along the chosen x and y axes.; Force components in N; acceleration components in m/s².
Signs and operations: Σ means add; arrows mean vectors. A force opposite the positive axis contributes a negative component. Equal-and-opposite third-law partners act on different objects and do not both belong on one object’s diagram.
Conditions: Use an inertial frame, nonrelativistic speeds and constant mass. Identify the body before listing forces. A moving object can have zero acceleration if its velocity is constant.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

WORKED EXAMPLE

Two horizontal forces on one cart

A 3 kg cart has an 8 N force to the right and a 2 N resisting force to the left. Its vertical forces balance. Find its acceleration.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

The cart above is currently moving left. What does the rightward acceleration initially do?

Makes it move right immediately
Reduces its leftward speed
Increases its leftward speed

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: Turn the picture into a signed equation
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
\sum\vec F=m\vec a;\qquad \sum F_x=ma_x,\quad\sum F_y=ma_y
Meaning: Choose axes, resolve each force onto them, and add signed components. Each component equation describes the same body.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / From interactions to equations

A support force comes from the motion constraint

WORKED EXAMPLE

A scale inside an accelerating lift

A 4 kg object rests on a scale in a lift accelerating upward at 2 m/s². Use g = 10 m/s². What normal force does the scale exert?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Forces on the 2 kg block in the angled-pull example. Draw the applied pull once. The dashed lines show its components; they are not additional forces. Vertical balance sets N = 15 N, while the horizontal component causes acceleration.

WORKED EXAMPLE

Pulling upward and sideways

A 2 kg block on a smooth horizontal floor is pulled by a 10 N force at 30° above the horizontal. Use g = 10 m/s². Find its horizontal acceleration and the normal force while contact is maintained.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: A support force comes from the motion constraint
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / From interactions to equations

Equal and opposite forces can act on different bodies

If body A exerts a force on body B, body B exerts an equal and opposite force on body A at the same time. This is Newton’s third law. The two forces belong to two different free-body diagrams. They do not cancel when you calculate the force on either body individually.

For the book and table, the table pushes upward on the book and the book pushes downward on the table. That is a third-law pair. The book’s weight and the table’s upward force both act on the book, so they are not a third-law pair—even when their magnitudes happen to match.

The third-law partner of Earth’s gravitational force on the book is the book’s gravitational force on Earth. Equal forces do not imply equal accelerations: the masses are extremely different. If you choose the combined Earth–book system, the two gravitational forces are internal and cancel in the system’s total force balance.

\vec F_{A\to B}=-\vec F_{B\to A}

The subscripts identify the recipient. Changing the recipient changes the body whose motion is being analysed.

STOP & THINK

A hand pushes a cart. Which force is the third-law partner of the hand’s force on the cart?

The floor’s friction force on the cart
The cart’s force on the hand
The cart’s weight

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: Equal and opposite forces can act on different bodies
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
\vec F_{A\to B}=-\vec F_{B\to A}
Meaning: The subscripts identify the recipient. Changing the recipient changes the body whose motion is being analysed.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / From interactions to equations

Choose a system to reveal the unknown

Horizontal forces only. The separate diagrams show the 9 N contact pair acting on different carts. For the combined system the pair is internal; its resultant cancels, leaving the 15 N external push. Weight and normal forces balance vertically on each cart.

WORKED EXAMPLE

Two touching carts move together

Two carts of masses 2 kg and 3 kg touch on a smooth horizontal track. A horizontal 15 N force pushes the 2 kg cart toward the 3 kg cart. Find their acceleration and the contact force between them.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

System choice removes some unknowns but cannot answer every question at once. The combined-system equation finds acceleration without knowing contact force. It also hides that force because the force is internal. To recover it, return to a subsystem.

Before leaving a solution, list the assumptions that made it possible: smooth track, maintained contact, horizontal applied force, constant masses, negligible air resistance and an inertial frame. Changing one of these can change the equations. In the next chapter, we add friction by determining how much is required and whether the contact can provide it.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: From interactions to equations
Focus: Choose a system to reveal the unknown
Learning goals:
- Distinguish a force from a velocity or an acceleration.
- Draw a free-body diagram containing only forces on the chosen body.
- Translate the diagram into signed equations along chosen axes.
- Distinguish balanced forces from a third-law pair.
- Choose between a single-body and a combined-system calculation.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Name the body before drawing forces.
Net force determines acceleration, not velocity.
Find a normal force from the relevant component equation.
Third-law partners act on different bodies.
Use a combined system for overall acceleration and a subsystem for an internal contact force.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A 5 kg body has a 14 N force rightward and a 4 N force leftward. What is its acceleration to the right?

Answer in m/s².

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SELF CHECK 2

A lift travels upward at constant velocity. A supported object inside it has only weight and normal force acting. Which is correct?

N > mg because it moves upward
N = mg because its acceleration is zero
N = 0 because velocity is constant

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

A 3 kg object is supported in a lift accelerating downward at 2 m/s². Use g = 10 m/s². Find the normal force.

Answer in N.

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SELF CHECK 4

For two touching carts treated as a combined system, why do their mutual contact forces disappear from the total horizontal force?

Neither cart feels a contact force
They are an equal-and-opposite internal pair
Contact forces are always negligible

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

A scale during downward accelerationIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

A book rests on a table. Why are the upward normal force and downward weight not a third-law pair?

Compare your reasoning

Both act on the book. Third-law partners act on different bodies: table on book pairs with book on table; Earth on book pairs with book on Earth.

Revisit the explanation →

A few days later: change the situation

A lift accelerates upward with a person standing inside. Explain why a scale reading exceeds the person’s weight.

Compare your reasoning

The upward contact force must exceed the downward gravitational force to give upward net force. The scale measures that contact force. The gravitational weight has not increased.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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