Newton's Laws of Motion & Contact Forces

Statement of the First Law

"An object at rest remains at rest, and an object in motion continues to move in a straight line at constant speed, unless acted upon by an external unbalanced force."

This is the law of inertia. The word inertia describes the natural tendency of an object to resist any change to its state of motion.

The need for the First Law

Newton's First Law can be simplified to: objects do not need a force to keep moving. Left entirely alone, an object travelling through space will continue at exactly the same speed in exactly the same straight line, forever. A force is required to change the motion of an object, whether that means speeding up, slowing down, or changing direction. This was a revolutionary idea. For over a thousand years the prevailing view, inherited from Aristotle, held that motion required a continuous cause. Newton showed this was wrong: force is not the cause of motion; it is the cause of change in motion.

Friction and why the ancient view seemed correct

On Earth, Aristotle's model feels intuitively right. Push a book across a table, remove your hand, and it stops almost immediately. Maintaining motion seems to require a continuous push. So why is this view wrong?

The answer is friction. Friction is itself a force, and it acts on the book from the moment it begins to move. When you stop pushing, the only remaining horizontal force is friction, acting opposite to the direction of motion. The book decelerates and stops, not because motion naturally ends, but because friction causes it to decelerate.

Once the push is removed, only friction acts horizontally between the block and the surface in the opposite direction to the motion of the block. The block stops because of friction, not because motion naturally ceases. This is exactly what Newton's model predicts.

Statement of the Second Law

"The rate of change of momentum of an object is directly proportional to the resultant external force acting on it, and takes place in the direction of that force."

For an object of constant mass \(m\), this gives:

where \(\mathbf{F}\) is the resultant force (N), \(m\) is the mass (kg), and \(\mathbf{a}\) is the acceleration (m s\(^{-2}\)). Both \(\mathbf{F}\) and \(\mathbf{a}\) are vectors: the acceleration is always in the same direction as the resultant force.

A simple interpretation

The First Law tells us that a force is needed to change motion. The Second Law tells us how much the motion changes. The key relationship is one of direct proportionality: the larger the resultant force, the larger the acceleration. Double the force, double the acceleration. Triple the force, triple the acceleration.

At the same time, for a fixed force, a more massive object accelerates less. Mass measures the object's inertia, its resistance to being accelerated.

Both lines pass through the origin. For the same force, the less massive object accelerates more.

Only the resultant force matters

The \(F\) in \(F = ma\) is always the resultant (net) force, the vector sum of all forces acting on the object. If forces act in opposite directions, their difference determines the acceleration. Rearranging:

If the resultant is zero then \(a = 0\): the object is in equilibrium, at rest or moving at constant velocity. This is entirely consistent with the First Law.

Before Newton: a long tradition of observation

When Newton published his Philosophiæ Naturalis Principia Mathematica in 1687, he was not working in isolation. Over the preceding century and a half, a remarkable series of experiments and observations had steadily accumulated, each contributing a piece of the puzzle that Newton finally assembled into a single, precise law. Newton himself acknowledged this debt, writing: "If I have seen further, it is by standing on the shoulders of giants."

The key question each generation of investigators wrestled with was this: exactly how does motion change when a force acts? Is the change in speed proportional to the force? To the time? To the distance? Answering these questions required moving beyond Aristotle's purely philosophical approach and developing systematic, quantitative experiment.

Galileo Galilei (1564–1642), the inclined plane experiments

The most significant experimental groundwork was laid by Galileo. Aristotle had taught that heavier objects fall faster than lighter ones, and that the speed of a falling body is proportional to its weight. Galileo set out to test this systematically.

Direct free-fall experiments were impractical with sixteenth-century timing methods, so Galileo ingeniously diluted gravity by rolling brass balls down smooth inclined planes. By tilting the plane at a shallow angle, the effective accelerating force was reduced to a manageable fraction of gravity, making the motion slow enough to time with a water clock.

Galileo's key findings were:

Galileo, free fall and the equivalence of falling bodies

Galileo also investigated free fall directly, both theoretically and, according to the famous (though possibly apocryphal) account, by dropping objects from the Tower of Pisa. Whether or not that particular experiment occurred, Galileo established clearly through careful argument and experiment that:

All objects in free fall accelerate at the same rate, regardless of their mass. A cannonball and a musket ball, released simultaneously from the same height, land simultaneously. This was a direct contradiction of Aristotle's claim that the speed of fall is proportional to weight, and it implied that the relationship between force and acceleration must involve mass as a separate factor, an insight Newton later formalised as \(a = F/m\).

Galileo also extended his analysis to projectile motion, showing that a projectile follows a parabolic path because the horizontal motion is uniform (no horizontal force) and the vertical motion is uniformly accelerated (constant downward force of gravity). This decomposition of motion into independent components was essential to Newton's later vector treatment of force and acceleration.

Simon Stevin (1548–1620), statics and the parallelogram of forces

The Flemish mathematician Simon Stevin made important contributions to understanding how forces combine. His work on the inclined plane, predating Galileo's, used a clever thought experiment involving a chain draped over a triangular prism. From the condition that the chain must be in equilibrium (otherwise it would spin forever), Stevin correctly derived the force components along inclined surfaces.

More importantly, Stevin established the parallelogram law for forces: when two forces act at a point, their combined effect is represented by the diagonal of the parallelogram formed by the two force vectors. This was the foundation of the vector treatment of force that Newton would employ in the Principia, and it implied that force has both magnitude and direction, it is not a simple scalar quantity.

René Descartes (1596–1650), momentum and the conservation principle

Descartes, in his Principia Philosophiae (1644), he proposed that the total quantity of motion in the universe is conserved, the first formulation of what would become the conservation of momentum. He defined "quantity of motion" as mass multiplied by speed, \(mv\).

Descartes also formulated a version of the law of inertia: that a body in motion continues in a straight line at constant speed unless acted upon by an external cause. This was a crucial conceptual step, though Descartes did not have a clear quantitative account of how forces change that motion.

Christiaan Huygens (1629–1695), centripetal force and collision theory

Huygens made two contributions that directly shaped Newton's Second Law. First, studying the pendulum clock (which he invented in 1656), he derived the formula for the period of a pendulum and showed that the restoring force is proportional to the displacement for small angles, an early quantitative relationship between force and the resulting motion.

Second, and more significantly, Huygens derived the formula for centripetal acceleration, the acceleration of a body moving in a circle. He showed that the inward acceleration required to maintain circular motion at speed \(v\) and radius \(r\) is:

This result, published in Horologium Oscillatorium (1673), was essential to Newton's derivation of the inverse-square law of gravity. By combining Huygens' centripetal formula with Kepler's Third Law, Newton could calculate the force required to keep the Moon in its orbit and show it matched the force of gravity at the Earth's surface, a decisive test of \(F = ma\).

Huygens also worked extensively on elastic collisions, correctly formulating conservation of momentum as a vector quantity and showing that kinetic energy is also conserved in perfectly elastic collisions, laying groundwork for Newton's treatment of impulse and the change in momentum.

Johannes Kepler (1571–1630), the laws of planetary motion

Working from Tycho Brahe's precise astronomical observations, Kepler published his three laws of planetary motion between 1609 and 1619. The third law states:

where \(T\) is the orbital period and \(r\) is the mean orbital radius. Kepler derived this empirically from data, with no underlying physical explanation.

The importance for Newton's Second Law was indirect but decisive. Newton showed that Kepler's Third Law, combined with the assumption of circular orbits and Huygens' centripetal acceleration formula, implied that the gravitational force must vary as the inverse square of distance. He then used \(F = ma\) to connect the gravitational force to the resulting planetary accelerations, demonstrating that a single force law could explain all of Kepler's empirical results.

Robert Hooke and the concept of central force (1660s–1680s)

Robert Hooke, a contemporary and rival of Newton, contributed the crucial physical insight that planetary motion results from two components: an inertial tendency to continue in a straight line (Descartes' principle) and a continuous attractive force directed toward the Sun. In a letter to Newton in 1679, Hooke suggested that this attractive force might vary as the inverse square of distance, though he could not prove it mathematically.

Hooke's own law of elasticity (1678), \(F \propto x\), force proportional to extension , was another direct quantitative relationship between force and the resulting deformation, reinforcing the emerging view that nature's force laws take simple proportional forms.

What Newton synthesised in 1687

By the time Newton wrote the Principia, the following pieces were in place from his predecessors:

Newton's achievement was to unify these disparate results into a single, precise, mathematical statement: the rate of change of momentum equals the resultant force, \(\mathbf{F} = m\mathbf{a}\). He then demonstrated that this one law, applied consistently, reproduced every known result, from Galileo's inclined planes to Kepler's planetary orbits and predicted new ones, including the precession of equinoxes and the shape of the Earth.

Statement of the Third Law

"For every action there is an equal and opposite reaction: if object A exerts a force on object B, then object B exerts an equal and opposite force on object A."

Forces never exist alone. Every force is one half of an interaction pair between two objects. A Newton 3 pair must satisfy all of the following:

Charged particles in space

Two charged particles floating in empty space, far from any other influence, provide the clearest possible example. The electric force between them is the only force present. In both cases, same charge or opposite charge, the force on each particle from the other is exactly equal in magnitude and opposite in direction, regardless of the size or mass of the particles.

An object released above the Earth

When an object is released above the Earth, gravity pulls the object downward toward the Earth's centre. By Newton's Third Law, the object simultaneously pulls the Earth upward toward itself, with exactly the same force. Both forces are gravitational, they act on different objects, and they are equal and opposite: a genuine Newton 3 pair.

The Earth pulls the object downward (solid red) and the object pulls the Earth upward (dashed blue) with exactly the same force. This is a true Newton 3 pair: same type of force (gravitational), equal magnitude, opposite direction, acting on different objects.

What is a contact force?

At the atomic level, every solid object is made of atoms surrounded by electron clouds. When two objects are brought together until they "touch," their outer electron clouds begin to overlap. The electromagnetic repulsion between these like-charged electron clouds produces a force that pushes the objects apart.

This is the contact force, also called the normal reaction force or normal contact force. It acts perpendicular to the surface of contact and only while the objects remain in contact or very close proximity.

The contact force is a Newton 3 pair

When two objects A and B are in contact, A pushes B away and B pushes A away. The two contact forces, one on A, one on B, are equal in magnitude and opposite in direction, both are electromagnetic in origin, and they act on different objects. This pair is a genuine Newton 3 pair.

The contact forces on A and B are equal in magnitude, opposite in direction, a Newton 3 pair.

Independence from gravity

The contact force is entirely independent of the gravitational attraction between the objects. Two objects resting on a surface experience a contact force determined only by how firmly they are pressed together, not by the local gravitational field strength. This independence is fundamental to understanding the next section.

Forces acting on an object resting on the ground

Consider an object of mass \(m\) resting on a horizontal surface. We focus only on the forces acting on the object. Other forces exist, the object pulls the Earth upward, the object pushes the ground downward, but these act on other bodies. When analysing the object, we consider only the forces acting on it.

There are exactly two forces acting on the object:

The object is in vertical equilibrium

Since the object is stationary it has zero acceleration. By Newton's Second Law, \(\sum F = ma = 0\), so the resultant force must be zero. The upward contact force and the downward weight must therefore be equal:

The two forces are equal in magnitude and opposite in direction. Vertically, the object is in equilibrium.

Why \(N\) and \(W\) are NOT a Newton 3 pair

This is one of the most important distinctions in A-level mechanics. Although \(N\) and \(W\) are equal and opposite, they are not a Newton 3 pair. They fail on two separate counts:

The actual Newton 3 pairs in this situation

There are two genuine Newton 3 pairs in this scenario, each involving forces on different objects:

The equilibrium of the object arises because two unrelated forces, gravity pulling it down and contact pushing it up, happen to be equal and opposite when the object rests at the surface. They come from completely different physical origins. This is why they cannot form a Newton 3 pair.