Laws of Motion

Aristotle believed that a force is needed to keep a body moving, and that a moving object comes to rest naturally because that is its 'natural state'. This looks true in daily life — push a book and it stops when you stop pushing — only because friction is quietly acting on the book and slowing it down. Aristotle never separated the effect of friction from the effect of the applied force.

Galileo corrected this with thought experiments on inclined planes. He argued that if a ball rolls down one incline and up another with no friction, it will rise to the same height it started from, however gentle the second slope is made. Push the second slope's angle to zero and the ball would keep moving forever at constant speed, needing no force at all to sustain that motion. Friction and other resistive forces, not motion itself, are what bring bodies to rest — motion needs no cause, only a change in motion does.

This is the seed of the law of inertia: an external force is needed to change a body's state, not to maintain it.

Newton's first law: every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by an external force.

This law does two jobs. First, it defines force qualitatively — force is whatever changes a body's state of rest or uniform velocity. Second, it defines inertia: the natural tendency of a body to resist any change in its state of motion. Inertia is not a force; it is a property, and its measure is mass — the more massive a body, the harder it is to change its velocity.

There are three kinds of inertia, and exam papers like naming them with an everyday example: inertia of rest (a coin on a card flicked off a glass does not move with the card and falls into the glass; passengers jerk backward when a bus starts suddenly), inertia of motion (a passenger standing in a moving bus falls forward when the bus brakes suddenly — the body keeps moving while the feet are stopped by friction), and inertia of direction (mud flies off a spinning cycle tyre tangentially; a stone tied to a string flies off tangentially if the string snaps).

The first law holds only in an inertial frame of reference — one that is at rest or moving with constant velocity. In a frame that itself accelerates (a braking bus, a merry-go-round) objects appear to move without any real force acting on them; that apparent effect is explained using a 'pseudo force', which is not a genuine force and has no reaction pair.

Momentum p = mv combines the two quantities — mass and velocity — that together decide how hard it is to stop or deflect a moving body. Newton's second law states: 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.

Derivation of F = ma. Let a body of mass m move with velocity v under a force F. Its momentum is p = mv. By the second law,

F ∝ dp/dt, so F = k·dp/dt = k·d(mv)/dt.

For a body of constant mass, F = k·m(dv/dt) = k·ma. The unit of force (the newton) is chosen so that a force of 1 N gives a mass of 1 kg an acceleration of 1 m s⁻², which fixes k = 1. So F = ma, or more generally F = dp/dt when mass is not constant.

The first law is contained inside the second: if F = 0 then dp/dt = 0, so p (and hence v, for constant mass) stays unchanged — that is exactly the statement of the first law. The second law is a vector law: it applies separately along each of the x, y and z directions, which is why component-wise resolution (F_x = ma_x, etc.) is the standard method for solving inclined-plane and projectile-type force problems.

Impulse. When a large force acts for a very short time (a bat hitting a ball, a collision), it is easier to measure the total effect than the force at each instant. Impulse J = F·Δt = Δp, the change in momentum. This is why cricketers draw their hands back while catching a fast ball, and why vehicles have crumple zones and airbags — increasing Δt for the same Δp reduces the peak force.

Newton's third law: to every action there is an equal and opposite reaction; action and reaction act on two different bodies, never on the same body, and they act simultaneously — neither is 'before' the other.

Because action and reaction act on different bodies, they never cancel each other even though they are equal and opposite; a pair of forces cancels only when both act on the same body. This is the most tested idea from this section: a book resting on a table experiences its weight (from the Earth) and a normal reaction (from the table); these two forces on the book are equal and opposite but they are not an action–reaction pair — they act on the same body and happen to balance by the first law's equilibrium condition, not by the third law. The true third-law partner of the table's normal force on the book is the book pressing down on the table; the true partner of the book's weight is the book pulling the Earth upward with an equal force.

Everyday illustrations examiners like: a swimmer pushes water backward and water pushes the swimmer forward; a gun recoils when a bullet is fired forward; walking is possible because your foot pushes the ground backward and the ground pushes you forward.

Derivation. Consider two bodies A and B, of masses m₁ and m₂, colliding for a short time Δt with no external force on the system. Let F_AB be the force on A due to B, and F_BA the force on B due to A. By Newton's second law,

F_AB = (p_1f − p_1i)/Δt and F_BA = (p_2f − p_2i)/Δt.

By Newton's third law, F_AB = −F_BA, so (p_1f − p_1i) = −(p_2f − p_2i). Rearranging, p_1i + p_2i = p_1f + p_2f — the total momentum of an isolated system (no external force) stays constant. This is the law of conservation of linear momentum, and it holds for any number of bodies, colliding, exploding or separating, as long as no external force acts on the system.

The classic application is recoil: a gun and bullet start at rest, so their total initial momentum is zero. When the bullet is fired forward, the gun must recoil backward with exactly enough momentum to keep the total at zero — m_bulletv_bullet = −m_gunv_gun. The same idea explains why a rocket accelerates forward by ejecting exhaust gas backward, and why two ice skaters pushing off from each other move apart with momenta equal in magnitude but opposite in direction.

A particle is in equilibrium when the vector sum of all forces acting on it is zero: ΣF = 0, which by the second law means its acceleration is zero — it is either at rest or moving with constant velocity.

For two forces, equilibrium needs them equal in magnitude and opposite in direction. For three concurrent forces, equilibrium needs each force to be equal in magnitude to the resultant of the other two but opposite in direction — equivalently, the three force vectors, placed head to tail, must form a closed triangle (this is the triangle law used for equilibrium problems, and it generalises to a closed polygon for more forces). In component form, equilibrium simply requires ΣF_x = 0 and ΣF_y = 0 separately, which is the practical method used for problems with strings, weights and inclined surfaces.

A weight hanging from two strings attached to a wall or ceiling is the standard equilibrium problem: resolve each string's tension into horizontal and vertical components, set the sum of horizontal components to zero and the sum of vertical components equal to the weight, and solve the two equations together.

Four forces recur through mechanics: weight (mg, always vertically downward, due to gravity), normal reaction N (perpendicular to the contact surface, a manifestation of the electromagnetic forces between surface atoms resisting inter-penetration), tension T (along a string, pulling inward on whatever it is attached to, same magnitude throughout an ideal massless, inextensible string over a frictionless pulley), and friction f (along the contact surface, opposing relative sliding or its tendency).

Friction is tested the most. Static friction f_s is a self-adjusting force that opposes the tendency of relative sliding between surfaces still at rest with respect to each other; it rises to match the applied force up to a maximum value f_s,max = μ_sN, beyond which sliding begins. Kinetic friction f_k = μ_kN acts once sliding has started, and μ_k is slightly less than μ_s — this is why it takes a bigger push to start moving a heavy box than to keep it moving. Both laws of friction state that f_max is proportional to the normal reaction N and (to a good approximation) independent of the apparent area of contact.

The angle of friction λ is the angle the resultant of N and f_max makes with N; tan λ = μ_s. The angle of repose θ is the angle of an incline at which a body just begins to slide under gravity alone; at that angle mg sin θ = f_s,max = μ_smg cos θ, so tan θ = μ_s — numerically the angle of repose equals the angle of friction.

Friction is not always unwanted: walking, writing and the grip of tyres on a road all depend on it. Where it is unwanted (machine bearings), it is reduced by lubrication or by replacing sliding friction with rolling friction, which is much smaller because rolling contact deforms the surfaces far less than sliding contact.

A car going around a circular path needs a centripetal force mv²/r directed toward the centre. On a level (unbanked) road, only friction between the tyres and the road can supply this force, so mv²/r ≤ μ_smg, giving a maximum safe speed v_max = √(μ_srg). Beyond this speed the car skids outward.

Banked road (no friction) — derivation. Bank the road at angle θ to the horizontal. The two forces on the car are its weight mg (down) and the normal reaction N (perpendicular to the road surface, so tilted at θ to the vertical). Resolving N into components: N cos θ balances the weight, and N sin θ supplies the centripetal force.

N cos θ = mg …(i)
N sin θ = mv²/r …(ii)

Dividing (ii) by (i): tan θ = v²/(rg), so the optimum (safe, no-friction-needed) speed is v = √(rg tan θ). At this exact speed, friction is not needed at all; below it the car tends to slide down the slope (friction must act up the slope), above it the car tends to skid outward up the slope (friction must act down the slope).

Banked road with friction. Including the maximum static friction μ_sN acting down the slope (to prevent outward skidding at high speed) gives the maximum speed:

v_max = √[rg(μ_s + tan θ)/(1 − μ_stan θ)].

Banking is used on real roads and railway tracks so that a safe range of speeds is possible with less dependence on friction, which reduces tyre wear and the risk of skidding, especially on a wet road where μ_s drops sharply.

A fixed method solves almost every problem in this chapter: draw a free-body diagram for each body separately, mark every real force (never a fictitious one, unless you have deliberately chosen a non-inertial frame), choose a positive direction along the expected motion, and apply F_net = ma along that direction for each body. Then combine the equations.

Two blocks connected by a string over a pulley (Atwood machine), hanging on either side. For masses m₁ and m₂ (m₂ > m₁) connected by a light, inextensible string over a frictionless, massless pulley: taking the direction of actual motion as positive for each mass,

m₂g − T = m₂a and T − m₁g = m₁a.

Adding the two equations eliminates T: a = (m₂ − m₁)g/(m₁ + m₂), and substituting back gives T = m₁m₂(1 + g/... simplified as T = m₁(g + a) = 2m₁m₂g/(m₁ + m₂).

Apparent weight in a lift. A person of mass m stands on a weighing machine inside a lift; the machine reads the normal reaction N it exerts on the person, which by the third law equals the force the person exerts on it. Applying F = ma to the person (taking upward as positive when the lift accelerates upward with acceleration a): N − mg = ma, so N = m(g + a) — the person feels heavier. If the lift accelerates downward, N = m(g − a) — the person feels lighter. If the lift falls freely (a = g, cable snapped), N = 0 — apparent weightlessness, even though gravity is still fully acting.

The same free-body method extends to blocks on inclined planes (resolve mg into mg sin θ along the slope and mg cos θ perpendicular to it) and to blocks connected on a table with a hanging mass over a pulley at the edge.