Kinematics describes motion; dynamics studies its cause, force. Newton’s three laws (1687, “Mathematical Principles of Natural Philosophy”) are the key to almost every problem in mechanics: draw the forces on a body correctly, write the second law in components, and half the problem is solved. These laws work for everything from cars and lifts to the planets, and they underlie the topics of energy, momentum and oscillations.
§1. The first law: the law of inertia
Since Aristotle, people had thought that a force is needed to keep a body moving. Galileo’s experiments disproved this: a ball rolling on a smooth surface hardly slows down; it is friction that stops it.
If no forces act on a body, or the forces on it balance, it stays at rest or moves in a straight line at constant speed.
The property of keeping its velocity is inertia. Passengers lurching forward when a bus brakes and dust leaving a carpet when it is beaten both show inertia.
Frames in which the first law holds are inertial frames. A frame attached to the Earth is treated as inertial in most problems; the frame of an accelerating bus or a spinning roundabout is non-inertial.
If the resultant force is zero, the acceleration is zero and the velocity is constant. So in a car moving steadily, the driving force balances the resistive forces.
§2. Mass and force
Mass measures a body’s inertia: it is harder to change the velocity of a more massive body. The unit is the kilogram (kg). Mass is a scalar and is additive: the mass of a body is the sum of the masses of its parts. Force measures the action of other bodies; it is a vector, measured in newtons (N). The same force gives a small mass a large acceleration and a large mass a small one. When two bodies interact, their accelerations are inversely proportional to their masses: a₁/a₂ = m₂/m₁.
§3. The second law
a⃗ = F⃗_net/m, i.e. F⃗_net = ma⃗
F_net is the vector sum of all the forces on the body.
The acceleration always points along the resultant force (not necessarily along the velocity).
1 N gives a 1 kg mass an acceleration of 1 m/s².
The general form in terms of momentum: FΔt = Δ(mv).
Method:
For each body draw the forces on it: weight, tension, normal reaction, friction, applied force.
Choose axes, one along the acceleration.
Write ΣF = ma for each axis.
Solve the equations and check the answer for units and sense.
Worked example
Horizontal forces of 20 N and 55 N act on a 5 kg body on a smooth surface. What is the smallest possible acceleration?
The resultant is smallest when the forces are opposite: 55 − 20 = 35 N.
a = F/m = 35/5 = 7 m/s² (the largest is 75/5 = 15 m/s²).
Worked example
A 4 kg body starting from rest reaches 6 m/s in 3 s. What resultant force acted on it?
a = (v − v₀)/t = 6/3 = 2 m/s².
F = ma = 4·2 = 8 N.
§4. The third law
Two bodies act on each other with forces equal in size and opposite in direction: F⃗₁₂ = −F⃗₂₁
These forces act on different bodies, so they do not balance each other.
They are of the same nature: both gravitational, or both elastic, and so on.
A horse pulls a cart and the cart pulls the horse with the same force; the horse moves forward because its hooves get a forward friction force from the ground. A rocket pushes gas backwards, and the gas pushes the rocket forwards.
A book on a table: the partner of its weight (the Earth pulls the book) is the book pulling the Earth; the partner of the normal reaction is the book pressing on the table.
§5. Connected bodies
Bodies joined by a light, inextensible string move with the same acceleration, and the tension is the same all along the string. Either write an equation for each body or treat the whole system as one body (the internal forces, the string tension, then cancel).
Connected bodies on a horizontal surface: pulled by a force F, they accelerate at a = F/(m₁ + m₂), and the string tension accelerates the rear body: T = m₁a.
Two masses over a pulley (Atwood machine):
a = (m₂ − m₁)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂)
The axle of a fixed pulley feels a force 2T.
Worked example
Bodies of 2 kg and 3 kg joined by a string lie on a smooth table; the 3 kg body is pulled with a horizontal force of 10 N. Find the acceleration and the tension.
The whole system: a = F/(m₁ + m₂) = 10/5 = 2 m/s².
For the rear 2 kg body: T = m₁a = 2·2 = 4 N.
Worked example
Loads of weight P and 20 N hang over a pulley. In 2.5 s the 20 N load falls 6.25 m. Find the acceleration and the force on the pulley’s axle.
s = at²/2 → a = 2s/t² = 12.5/6.25 = 2 m/s².
For the 20 N load (m = 2 kg): mg − T = ma → T = 2·(10 − 2) = 16 N.
On the axle: 2T = 32 N.
§6. Limits of Newton’s laws
Newtonian (classical) mechanics gives accurate results over a huge range: designing machines and structures, the motion of cars and aircraft, the orbits of satellites and planets. But it has limits:
at speeds close to the speed of light, special relativity is needed;
processes inside atoms and molecules are described by quantum mechanics;
the laws hold in their simple form only in inertial frames; in a non-inertial frame, inertial forces must be added.
At everyday speeds and sizes these corrections are so small that they can be ignored.
Worked example
A horizontal force of 20 N is applied to a 5 kg box on a horizontal table; the coefficient of friction is 0.2. Find the acceleration.
Friction: F_fr = μmg = 0.2·5·10 = 10 N.
Resultant: 20 − 10 = 10 N.
a = 10/5 = 2 m/s².
Check yourself
A resultant force of 10 N acts on a 2 kg body. Acceleration?
A 60 kg person rises in a lift accelerating upwards at 2 m/s². Weight?
Masses of 1 kg and 3 kg hang over a fixed pulley. Acceleration?
For a book lying on a table, which force is the third-law partner of its weight?
Resultant force on a car moving at constant velocity?
Masses of 1 kg and 3 kg hang over a pulley. String tension?
The same force gives a 2 kg body an acceleration of 6 m/s². What acceleration does it give a 3 kg body?
Which way does acceleration point: along the velocity or along the resultant force?
Common mistakes
Putting both third-law forces on one body and saying they “cancel”.
Setting a single force equal to ma instead of the resultant.
Confusing the force of gravity (mg, Earth on body) with weight (P, body on support).
Using the direction of motion in a lift instead of the direction of acceleration.
Thinking motion always needs a force: constant velocity needs no resultant force.