Newton's laws

Why this matters

Kinematics describes motion; Newton's laws explain it. Every structural load calculation, every machine part sizing and every orbit calculation is built on three short statements Isaac Newton published in 1687. The tool that makes them usable is the free-body diagramA sketch of one object showing every force acting on it, drawn before writing any equations.: a sketch of one object with every force acting on it. Engineers draw these before they write a single equation, because a missing or invented force makes every number after it wrong.

The three laws

1. First law (inertiaThe tendency of an object to keep its velocity unless a net force acts on it (Newton's first law).). An object keeps moving at a constant velocity (which includes staying at rest) unless a net force acts on it. A hockey puck slides a long way on ice because there is very little friction to change its velocity.
2. Second law. The net force on an object equals its mass times its acceleration:

ΣF = m·a

Force is measured in newtonsThe unit of force: 1 N gives 1 kg an acceleration of 1 m/s², so 1 N = 1 kg·m/s². (N). One newton is the force that gives 1 kg an acceleration of 1 m/s², so 1 N = 1 kg·m/s². The Σ means "add up every force, as vectors".

3. Third law. When object A pushes on object B, B pushes back on A with a force of equal size in the opposite direction. The two forces act on different objects, so they never cancel each other in a single free-body diagram.

Mass and weight

Mass (kg) is how much matter an object has and how strongly it resists acceleration. WeightThe gravitational force on an object, W = m·g, measured in newtons. is the gravitational force on it: W = m·g, with g ≈ 9.81 m/s² near Earth's surface. A 2.0 kg box weighs 2.0 × 9.81 = 19.6 N. On the Moon the same box has the same mass but weighs only about a sixth as much.

Drawing a free-body diagram

1. Pick one object and draw it as a dot or a box.
2. Draw every force acting on that object, starting at the object and pointing in the direction of the force. Common ones:
   • Weight W, straight down.
   • Normal forceThe push of a surface on an object, perpendicular to the surface and away from it. N, perpendicular to any surface it rests on, pushing away from the surface.
   • TensionThe pulling force along a rope or cable, directed away from the object. T, along a rope or cable, pulling away from the object.
   • Friction f, along the surface, opposing sliding (next lesson).
   • Applied forces, such as a push.
3. Choose axes (usually x along the motion, y perpendicular to it).
4. Write ΣF = m·a separately for each axis.

Only forces from things touching the object (plus gravity, which acts at a distance) belong on the diagram. "The force of motion" is not a force; velocity is not a force.

Worked example 1: pushing a box

You push a 2.0 kg box across a floor with a horizontal force of 10 N. Friction opposes the motion with 4.0 N. What is the box's acceleration?

Free-body diagram: push 10 N to the right, friction 4.0 N to the left, weight 19.6 N down, normal force up.

y-axis:  N − 19.6 = m·0      →  N = 19.6 N   (the box does not leave the floor)
x-axis:  10 − 4.0 = 2.0·a    →  a = 3.0 m/s² to the right

Worked example 2: an elevator cable

A 500 kg elevator car hangs from a cable. What is the cable tension when the car (a) sits still, (b) accelerates upward at 2.0 m/s², and (c) accelerates downward at 2.0 m/s²?

Take up as positive. Two forces act on the car: tension T up and weight m·g down.

ΣF = T − m·g = m·a     →     T = m·(g + a)

(a) a = 0:      T = 500 × 9.81        = 4905 N
(b) a = +2.0:   T = 500 × (9.81 + 2.0) = 5905 N
(c) a = −2.0:   T = 500 × (9.81 − 2.0) = 3905 N

This is why you feel heavier when an elevator starts upward: the floor has to push harder on you to accelerate you. If the car were in free fall (a = −9.81 m/s²), the tension would be zero.

Real elevator cables, cranes and lifting gear are designed with large safety factors and to published standards, and are signed off by qualified engineers and inspectors. This example shows the physics, not a design method.

Writing it in code

Treat each force as an (x, y) pair. Adding forces is then adding pairs:

forces = [(10, 0), (-4, 0), (0, -19.62), (0, 19.62)]
fx = sum(f[0] for f in forces)
fy = sum(f[1] for f in forces)
print(fx / 2.0, fy / 2.0)   # 3.0 0.0  (acceleration in m/s²)

Common mistakes

• Putting both third-law forces on one diagram. The floor pushes on the box (on the box's diagram); the box pushes on the floor (on the floor's diagram).
• Assuming the normal force always equals the weight. It does on a flat floor with no vertical acceleration. On a ramp, in an elevator, or with someone pressing down, it does not.
• Adding a "force of motion". A moving object with no net force keeps moving; it does not need a force to sustain its velocity.
• Mixing up mass and weight. Mass in kg, weight in N.

Quick recap

• ΣF = m·a, applied separately along each axis.
• Weight is m·g; 1 N = 1 kg·m/s².
• Third-law pairs act on different objects.
• Draw the free-body diagram first, every time.

Try it

In the Do tab, finish the elevator-tension function and test it for an elevator that is still, speeding up and in free fall.