Newton's Laws of Motion

Newton's laws describe how forces change the way objects move. The first law deals with what happens when the forces on an object are balanced.

Newton's First Law: if the resultant force on an object is zero, a stationary object stays at rest and a moving object continues at a constant velocity.

The resultant force is the single force that has the same effect as all the forces acting on the object combined. If those forces cancel to zero, the object's velocity does not change: it stays still if it was still, or keeps moving at the same speed in the same direction if it was moving.

Velocity changes (the object speeds up, slows down, or changes direction) only if there is a non-zero resultant force.

A car travelling at a steady velocity on a motorway has a driving force from the engine that exactly balances the resistive forces (friction and air resistance). The resultant force is zero, so the velocity stays constant. When the driver accelerates, the driving force exceeds the resistive forces and the resultant force is no longer zero.

When there is a non-zero resultant force, an object accelerates. Newton's Second Law says how large that acceleration is.

Newton's Second Law: the acceleration of an object is proportional to the resultant force acting on it and inversely proportional to its mass.

$F=ma$

where $F$ is the resultant force in newtons (N), $m$ is the mass in kilograms (kg), and $a$ is the acceleration in metres per second squared (m/s²).

You must recall and apply this equation. $F=ma$ is not given on the equation sheet.

A larger resultant force gives a larger acceleration; a larger mass gives a smaller acceleration for the same force.

Worked example — a resultant force of 600 N acts on a car of mass 1200 kg. Find its acceleration.

$a=\frac{F}{m}=\frac{600}{1200}=0.5{\text{ m/s}}^2$

The car accelerates at 0.5 m/s².

Worked example — what resultant force gives a 0.15 kg ball an acceleration of 40 m/s²?

$F=ma=0.15\times 40=6\text{ N}$

A resultant force of 6 N is needed.

The third law is about pairs of forces between interacting objects.

Newton's Third Law: when two objects interact, they exert equal and opposite forces on each other.

The two forces in the pair are always the same size, act in opposite directions, act on different objects, and are of the same type (both gravitational, both contact, and so on).

Examples of these force pairs:

• A swimmer pushes backwards on the water; the water pushes the swimmer forwards with an equal force.
• A book resting on a table pushes down on the table; the table pushes up on the book with an equal force.
• A gun pushes the bullet forwards; the bullet pushes the gun backwards (recoil) with an equal force.

In an equilibrium situation, such as the book on the table, the forces on a single object balance to give zero resultant force. Take care: the book's weight and the table's upward push on the book are not a third-law pair, because they act on the same object and are different types of force. A third-law pair always acts on two different objects.

(Higher Tier only) Inertia and inertial mass are Higher Tier content.

Inertia is the tendency of an object to continue in its state of rest or of uniform motion (constant velocity). An object with large inertia resists changes to its motion.

This explains Newton's First Law: an object naturally keeps doing what it is doing unless a resultant force acts. It is why passengers lurch forwards when a car brakes suddenly; their bodies tend to keep moving at the original velocity.

Inertial mass is a measure of how difficult it is to change the velocity of an object. It is defined as the ratio of force over acceleration.

$\text{inertial mass}=\frac{F}{a}$

The larger the inertial mass, the smaller the acceleration produced by a given force. A loaded lorry has a much greater inertial mass than a bicycle, so the same force changes its velocity far more slowly.

Required practical 7 investigates how the acceleration of an object depends on the force applied and on its mass, testing Newton's Second Law.

Apparatus: a trolley on a runway, connected over a pulley by a string to a hanging mass holder; a set of masses; a light gate (or ticker tape) system to measure the trolley's acceleration.

Method (part 1 — varying force, constant mass): keep the total mass of the system fixed. To increase the accelerating force, move masses one at a time from the trolley to the hanging holder. This changes the pulling force while keeping the total mass being accelerated the same. Measure the acceleration each time.

Method (part 2 — varying mass, constant force): keep the hanging mass (the force) fixed, and add masses to the trolley to increase the mass being accelerated. Measure the acceleration each time.

Variable	Part 1	Part 2
Independent	accelerating force	mass of the trolley
Dependent	acceleration	acceleration
Control	total mass of the system	accelerating force

Expected results: acceleration is directly proportional to force (a straight line through the origin on a graph of acceleration against force), and acceleration is inversely proportional to mass (acceleration falls as mass rises). Both results confirm $F=ma$. Moving masses between the trolley and holder, rather than just adding them, is what keeps the total mass constant in part 1, so that force is the only variable being changed.

1. Using a single force instead of the resultant force

$F$ in $F=ma$ is the resultant force, the overall force after all forces are combined. Do not substitute the driving force alone if friction or air resistance also act.

2. Thinking constant velocity means no forces

At constant velocity the resultant force is zero, but individual forces (such as thrust and drag) can still act; they simply balance. Newton's First Law is about the resultant, not the absence of forces.

3. Mixing up mass and weight in F = ma

Mass in $F=ma$ is in kilograms. Weight (in newtons) is a force, not a mass. If a question gives weight, convert it to mass with $m=W/g$ before using $F=ma$.

4. Pairing the wrong forces for Newton's Third Law

A third-law pair acts on two different objects and is of the same type. The weight of a book and the table's push on the book act on the same object, so they are not a third-law pair.

5. Not keeping the total mass constant in RP7 part 1

To vary only the force, masses are moved from the trolley to the hanging holder rather than added. Simply adding masses would change both the force and the total mass, so the effect of force alone could not be isolated.