Laws of Motion

It seems obvious from everyday experience that keeping something moving takes a continuous push, a toy car needs its string constantly pulled, or it stops. This was essentially Aristotle’s view: an external force is required to keep a body in motion. It is also wrong, and finding the flaw took nearly two thousand years.

The flaw is friction. The toy car stops because the floor’s friction opposes it, and the child’s pull exists only to cancel that friction, not to sustain motion by itself. Galileo saw this by imagining away the friction entirely: rolling a ball down one incline and up another, he noted it always climbed back to nearly its starting height, and reasoned that in the ideal, frictionless limit it would climb to exactly that height, however gentle the second slope, meaning on a perfectly flat frictionless surface it would simply keep rolling forever.

This is the law of inertia: a state of rest and a state of uniform straight-line motion are physically equivalent, and in both, the net external force is zero. Inertia is a body’s built-in resistance to having that state changed.

Newton took Galileo’s insight and stated it as a law:

Every body continues to be in its state of rest or of uniform motion in a straight line, unless compelled by some external force to act otherwise.

Equivalently: if the net external force on a body is zero, its acceleration is zero, and acceleration can be nonzero only if a net force acts. This runs both ways in practice. Given the forces, the first law predicts the motion (a spaceship with engines off and nothing nearby must move at constant velocity). Given the motion, it lets you infer the forces: a book lying still on a table tells you the table’s normal force $R$ must exactly cancel its weight $W$, not because of any separate law linking $R$ and $W$, but purely because the book is observed to be unaccelerated.

The first law only covers the special case of zero net force. The second law covers everything else, by relating force to momentum, $\mathbf{p}=m\mathbf{v}$:

The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.

choosing the constant of proportionality to be 1, which is what defines the SI unit of force: one newton is the force that gives a 1 kg mass an acceleration of 1 m/s². A few things worth holding onto about this law: it is a vector equation, so it applies separately to each of the x-, y-, and z-components; it is a local and instantaneous relation, meaning the force here and now determines the acceleration here and now, with no memory of the body’s past motion, a ball at the top of its throw is momentarily at rest but is not force-free, gravity and the acceleration $g$ act on it exactly as they did a moment earlier; and it applies to a system of particles too, with $\mathbf{F}$ as the total external force (internal forces are excluded entirely).

To every action, there is always an equal and opposite reaction.

Stated more carefully: the force on body A due to body B is equal and opposite to the force on body B due to A, $\mathbf{F}_{AB} = -\mathbf{F}_{BA}$. Three points that the words “action” and “reaction” tend to obscure:

• The two forces are simultaneous, not cause-and-effect; either one may be called the action and the other the reaction.
• They act on two different bodies, never on the same body, so they can never cancel each other out for either body individually. If you push a wall, the wall pushes back on you, not on itself.
• Two forces on the same body that happen to be equal and opposite (like a book’s weight and the table’s normal force on it) are not a third-law pair, they are just a coincidence of that body being in equilibrium. A genuine third-law pair is always “force on A by B” and “force on B by A”, on two different objects.

Combine the second and third laws and something powerful falls out. For two colliding bodies A and B, Newton’s third law gives $\mathbf{F}_{AB}=-\mathbf{F}_{BA}$, and the second law turns each side into a rate of change of momentum over the same contact time $\Delta t$:

The total momentum after the interaction equals the total momentum before it, for any pair of mutual forces, elastic or inelastic. Extended to any number of particles, this is the law of conservation of momentum: in an isolated system, with no external force, mutual internal forces can shuffle momentum between individual particles, but the total never changes, because every internal push has an equal and opposite partner that cancels it out overall. Firing a bullet is the classic case: bullet and gun start with zero total momentum, and they must end with zero total momentum too, so the gun recoils backward with exactly the momentum the bullet carries forward.

A particle is in equilibrium when the net external force on it is zero, which by the first law means it is either at rest or moving with constant velocity. For two forces, equilibrium simply requires $\mathbf{F}_1 = -\mathbf{F}_2$. For three or more concurrent forces, it requires the vector sum to vanish:

which geometrically means the forces, drawn head to tail, close up into a polygon with no gap left over. In components, this is three independent scalar conditions, one per axis, all of which must hold simultaneously.

Only two forces are truly fundamental at the scale of ordinary mechanics: gravity, and the electrical forces between charged particles inside matter. Every other everyday force, normal reaction, tension, the spring force $F=-kx$, buoyancy, air resistance, and friction, is really electrical in origin, arising from how atoms and molecules resist being pushed into or pulled apart from each other, just expressed at a much simpler, macroscopic level because tracking the underlying electrical interactions directly would be hopeless.

A body moving in a circle of radius $R$ at speed $v$ needs a centripetal acceleration $v^2/R$ toward the centre, which by the second law needs a matching centripetal force:

“Centripetal force” is not a new, separate kind of force, it is simply the name for whichever real force happens to be providing that inward pull: tension for a whirled stone, gravity for an orbiting planet, and for a car turning on a flat road, friction.

On a level road, the only horizontal force available is friction, so the car cannot turn faster than the speed at which the required centripetal force would exceed the maximum static friction $\mu_s N = \mu_s mg$:

which, notice, does not depend on the car’s mass at all. A banked road tilts the normal force itself inward, so it can supply part or all of the centripetal force without relying on friction. Balancing forces along the vertical and horizontal directions and eliminating the normal force $N$ gives the maximum safe speed on a bank of angle $\theta$:

which is always greater than the flat-road value for the same $\mu_s$. Setting $\mu_s = 0$ in the same derivation gives the single speed at which the bank alone, with zero friction, supplies exactly the needed centripetal force, the optimum speed:

Driving at exactly $v_0$ causes the least wear on the tyres, since no sideways friction is needed at all.

A few of the chapter’s own exercise questions, with original worked solutions. (g = 10 m/s² throughout, as the chapter’s exercises specify.)