Newton's first law
Every body continues in its state of rest or of uniform motion in a straight line unless compelled by an external force to change that state.
The first law defines force (something that changes a body's motion) and identifies the property of inertia. It also defines inertial frames: frames of reference in which the law holds. A frame that is accelerating is not inertial.
Newton's second law
F_net = dp/dt, and for constant mass, F_net = m a
Here p = m v is the linear momentum. Force is a vector, so the law holds separately for each component:
F_x = m a_x, F_y = m a_y, F_z = m a_z
Only the net external force on a body determines its acceleration. Internal forces between parts of a system cancel in pairs.
Impulse
When a large force acts for a short time, as in a bat hitting a ball, it is easier to work with the impulse:
J = F_avg Δt = Δp
Impulse equals the change in momentum. Its unit is N s, which is the same as kg m/s.
Example. A 0.15 kg ball arrives at 20 m/s and is hit straight back at 30 m/s. Impulse = m(v − u) = 0.15 × (−30 − 20) = −7.5 N s, directed back towards the bowler.
Newton's third law
To every action there is always an equal and opposite reaction.
Action and reaction act on different bodies, so they never cancel each other. They also act at the same instant: there is no delay between "cause" and "effect".
Conservation of momentum
For an isolated system (no net external force), the total momentum is constant:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This follows from the second and third laws. It explains gun recoil, rocket propulsion, and how momentum is shared in collisions.
Free-body diagrams: a method
Isolate the body you are interested in.
Draw all the external forces acting on it: weight mg, the normal reaction N, tension T, friction f, and any applied forces.
Choose axes, usually along and perpendicular to the direction of motion.
Apply F_net = m a along each axis.
Example: connected bodies. Two blocks of mass 4 kg and 6 kg lie side by side on a frictionless floor. A 20 N force pushes the 4 kg block, which pushes the 6 kg block.
For the whole system: a = 20 / (4 + 6) = 2 m/s².
For the 6 kg block alone, the only horizontal force is the contact force from the 4 kg block: F = 6 × 2 = 12 N.
Example: a lift. A 60 kg person stands on a scale in a lift that accelerates upward at 2 m/s² (g = 10 m/s²). N − mg = ma, so N = 60 × (10 + 2) = 720 N, and the scale reads more than the person's weight. In free fall (a = g downward), N = 0 and the person feels weightless.
Friction
Friction opposes relative motion (or the tendency of it) between surfaces in contact.
Static friction adjusts itself to prevent sliding, up to a maximum value: f_s ≤ μ_s N.
Kinetic friction acts during sliding: f_k = μ_k N, and is roughly independent of the speed.
Usually μ_k < μ_s, so it takes more force to start sliding than to keep sliding.
Friction is, to a good approximation, independent of the area of contact.
Example. A block rests on a slope that is slowly tilted. It just begins to slide when the angle reaches θ. Along the slope, mg sin θ = μ_s mg cos θ, so μ_s = tan θ. This angle is called the angle of repose.
Circular motion and banking
An object moving in a circle needs a net centripetal force m v²/r towards the centre. For a car on a flat curve, that force is supplied by friction, so the maximum safe speed is v_max = √(μ_s r g).
Roads are banked (tilted inwards) on curves so that part of the normal reaction supplies the centripetal force. Without friction, the ideal speed on a road banked at angle θ satisfies tan θ = v²/(r g).