Newton's laws

A trolley stops quickly on a rough track but rolls much longer on a smooth one. Must a force keep motion going, or is force needed for something else: changing velocity?

Reducing friction lets the trolley keep moving longer after a push. Imagine removing resistance entirely: nothing remains to change its velocity. The persistence of rest or uniform straight-line motion without external influence is called inertia. Frames in which a free body behaves this way are inertial frames. An Earth-based laboratory is approximately inertial in many school experiments; a sharply braking carriage is not a suitable example of this model.

When a hand pushes a trolley, they interact. The effect on the chosen body is described by force \vec F, in newtons. First isolate the trolley mentally: which objects act on it—the hand, Earth, the floor? Add these forces as vectors to obtain the net force \sum\vec F. Opposing effects can balance partly or fully. The trolley's force on the hand is not on this list: it acts on a different recipient.

The same trolley can be accelerated more or less strongly. Experiments show that doubling the net force doubles acceleration. Doubling mass at the same force makes velocity change half as fast. Newton's second law combines these relations: \sum\vec F=m\vec a. Mass m, in kg, measures inertia; acceleration \vec a, in m/s², points along the net force, not necessarily along the existing velocity.

Why not simply take one prominent force and write F=ma? Acceleration comes from all the forces acting together. Here F is the magnitude of their vector sum. We use \sum\vec F=m\vec a for a constant-mass body in an inertial frame, at speeds much smaller than light speed. When fuel leaves a rocket, the chosen body's composition changes; this equation cannot be transferred to it without accounting for mass flow.

The force unit is consistent with mass and acceleration units: 1\,\text{Н}=1\,\text{кг}\cdot\text{м/с}^2. A net force of 1 N gives a 1 kg body acceleration 1 m/s²: its velocity changes by 1 m/s each second. The same force gives a 2 kg body acceleration 0.5 m/s². If the net force falls to zero, acceleration vanishes for finite positive mass, but the velocity already gained remains.

A book rests on a table: gravity acts, but the support balances it. A trolley can also move at constant velocity with zero net force. In both cases the law gives \vec a=0, so the velocity vector stays constant; rest is the special case of zero initial velocity. Motion alone therefore does not show that the net force points forwards. We must observe the change in velocity.

Push a wall with your palm: your palm feels a force too. For a hand and trolley, call the hand's force on the trolley \vec F_1 and the trolley's force on the hand \vec F_2. Newton's third law relates them: \vec F_1=-\vec F_2. Their magnitudes are equal and their directions opposite at the same instant. They are two sides of one interaction, not a sequence of action followed by response. Different masses mean equal forces need not give equal accelerations.

Why does the trolley accelerate anyway? Only the hand's force from the interaction pair acts on it; the other force acts on the hand. Forces can balance on the same chosen body, so this pair cannot cancel in the trolley's equation. For a system containing person and trolley, the pair becomes internal and cancels in the total; external forces, such as those from the floor, remain. The chosen system changes the force list, not the motion itself.

Horizontal forces of 6 N right and 2 N left act on a 2 kg trolley. Find its acceleration.

Choose the trolley and take right as positive.

Add force components: ΣFx=6−2=4 Н.

By Newton's second law, ax=ΣFx/m=4/2=2 м/с².

Check: the net force points right, so acceleration also points right.

Acceleration is 2 m/s² to the right: each second the rightward velocity component increases by 2 m/s. If the trolley initially moved left, it would first slow down.

Force changes velocity. With zero net force, both rest and existing uniform straight-line motion persist.

Acceleration increases with net force and decreases with mass; write the equation for a specific body in a chosen reference frame.

Interaction forces are equal but act on different bodies. They cannot cancel in the equation for one body.

Next we study models for particular forces and conditions for equilibrium.