Newton's laws

Newton's laws are a compact method for turning forces into predictions about motion. The main skill is not memorizing three sentences: it is choosing the object, identifying the forces that act on it, and adding those forces correctly.

Three laws, one framework

First law: if the net external force is zero, velocity stays constant. Rest is only the special case v = 0.

Second law: a non-zero net external force changes velocity by producing acceleration.

ΣF = ma

Force and acceleration are vectors. The acceleration points in the direction of the net force, not necessarily in the direction of motion.

Third law: when object A exerts a force on object B, B exerts an equal-magnitude, opposite-direction force on A.

A force diagram is an accounting tool. Draw only external forces acting on the chosen object; do not add “motion” or “net force” as extra arrows.

First law: zero net force does not mean zero motion

An object does not need a continuing net force to keep moving at constant velocity. A puck gliding on nearly frictionless ice can continue moving even after the push has ended. What a net force changes is velocity: its magnitude, its direction, or both.

This statement assumes an inertial reference frame—a frame that is not itself accelerating. In an accelerating car, for example, apparent effects appear that are not ordinary interaction forces.

Second law: add forces before using F = ma

The symbol F in the familiar expression is the net external force, not one selected force. If 20 N acts right and 8 N acts left, the horizontal net force is 12 N right. A 4 kg object then accelerates at 3 m s⁻² right.

Mass measures inertia: for the same net force, a larger mass has a smaller acceleration. Weight is different from mass; near Earth's surface, weight is the gravitational force mg.

Third law: interaction forces belong to different objects

Third-law pairs never cancel on one free-body diagram. The two forces are equal and opposite, but they act on different objects.

A common mistake is to say that the forces on a moving object “cancel because of Newton's third law.” Third-law partners do not act on the same object. To decide whether one object's forces cancel, use that object's free-body diagram.

A reliable force-problem method

Choose the object or system you are analyzing.
Draw every external force acting on it.
Choose axes and resolve angled forces into components when needed.
Write ΣF_x = ma_x and ΣF_y = ma_y.
Use the signs and units to check the result.

Do not assume the normal force always equals mg. It does so only in particular situations, such as a horizontal surface with no vertical acceleration and no other vertical force.

Common forces have specific physical origins: weight comes from gravity, a normal force acts perpendicular to a contact surface, friction acts along a contact surface, and tension pulls along a rope or cable. Naming the interaction helps prevent invented forces. If an object is moving right, for example, there is not automatically a rightward force; the net force could be zero or even point left.

Worked examples

1. A horizontal push

A 5.0 kg cart is pushed to the right with 18 N while friction is 3.0 N to the left. Find its acceleration.

Solution

Choose right as positive. The net horizontal force is 18 − 3 = 15 N.

a = ΣF/m = 15/5.0 = 3.0 m s⁻² to the right.

2. An elevator at constant speed

A 70 kg passenger moves upward at constant speed. What is the net force on the passenger, and what is the floor's normal force?

Solution

Constant velocity means a = 0, so the net force is 0 N. Therefore N = mg = 70 × 9.81 ≈ 687 N.

3. Action and reaction

A person pushes a wall with 120 N. What force does the wall exert on the person?

Solution

Newton's third law gives an equal and opposite interaction force: the wall pushes the person with 120 N in the opposite direction. These two forces act on different bodies.