Motion, Force and Laws of Motion

A train standing on a platform is at rest for the passengers sitting in it, but it is moving for the person standing on the moon, because the Earth itself is moving. So rest and motion are relative: an object is in motion if its position changes with time with respect to a chosen reference point. To describe motion we need four ideas, and each has a scalar version and a vector version: distance and displacement, speed and velocity.

Here is the classic picture. Radhika runs one full lap of a 400 m circular track and returns to the starting line. Her distance is 400 m, but her displacement is zero, because the final position coincides with the initial position. So her average velocity for the lap is also zero, while her average speed is certainly not zero. Remember: displacement depends only on the start and the end; distance depends on the whole path.

Acceleration is the rate of change of velocity: a = (v − u) / t, where u is the initial velocity, v the final velocity and t the time. Its SI unit is m/s². If velocity decreases, acceleration is negative and is called retardation or deceleration. Because velocity has direction, acceleration happens even when only the direction changes, which we will use in circular motion. Motion is uniform when equal distances are covered in equal intervals of time (however small); otherwise it is non-uniform.

For motion in a straight line with constant acceleration, three equations tie together u, v, a, t and s (the displacement). Learn when to use which: pick the equation that has exactly the quantities given and the one asked.

Tie a stone to a string and whirl it above your head. The stone does not fly off in a straight line; it keeps going round. If you let the string go, it shoots off along the tangent to the circle, in a straight line. This tells us something deep: to move in a circle, something must keep pulling the object towards the centre. Motion along a circle is called circular motion; when the speed stays constant, it is uniform circular motion.

In uniform circular motion the speed is constant but the velocity keeps changing, because the direction changes at every instant. A changing velocity means acceleration. This acceleration always points towards the centre and is called centripetal acceleration: a = v² / r = ω²r. The inward force that produces it is the centripetal force: F = mv² / r = mω²r. It acts perpendicular to the velocity, so it changes only the direction and never the speed; hence it does zero work.

Centripetal force is not a new kind of force. It is just the name for whichever real force points to the centre. For a stone on a string it is the tension; for the Moon it is Earth's gravity; for a car on a flat turn it is friction between tyres and road; for an electron round a nucleus it is the electric attraction.

To measure how fast something goes round we use angular velocity ω = θ / t, in radians per second. One full turn is 2π radians (360°). If T is the time period (time for one turn) and f is the frequency (turns per second), then ω = 2π / T = 2πf and f = 1 / T. The linear speed is v = rω. For a rigid body, every particle has the same angular velocity, but particles farther from the axis have a larger linear speed. Think of a spinning merry-go-round: horses on the outer ring cover a bigger circle in the same time, so they move faster.

Before Galileo, people believed (as Aristotle said) that a body needs a continuous push to keep moving, because everyday objects do stop. Galileo rolled balls down one inclined plane and up another and saw that the smoother the surfaces, the closer the ball rose to its original height. In the limit of no friction it would roll for ever. The ball stops in real life not because 'motion dies out' but because friction acts on it. Newton put this insight into his First Law of Motion.

A force is a push or a pull. It can start a motion, stop it, change its speed, change its direction, or change the shape of a body. The SI unit of force is the newton (N); 1 N is the force that gives a mass of 1 kg an acceleration of 1 m/s². When several forces act together, what matters is the net force. If the net force is zero the forces are balanced and the state of the body does not change; if it is not zero the forces are unbalanced and the body accelerates. A force needs at least two objects to interact.

Nature has just four fundamental forces: gravitational (weakest, but acts over any distance and holds planets together), electromagnetic (covers electricity, magnetism, and also friction and tension at the atomic level), the strong nuclear force (strongest, holds the nucleus together, very short range) and the weak nuclear force (causes beta decay). Every push or pull we see is one of these in disguise.

Newton's First Law: A body remains at rest, or in uniform motion along a straight line, unless an external unbalanced force acts on it. It is also called the law of inertia. Mass is the measure of inertia: a heavier body has more inertia and is harder to start or stop.

A spring balance measures force. Inside it is a coiled spring that stretches when a force is applied; the stretch is proportional to the force (Hooke's law), and the pointer shows the value on a scale. Remember also the difference between mass and inertia: mass is the quantity of matter and measures inertia; it does not change from place to place.

Which hurts more: a cricket ball thrown gently at you, or the same ball hit hard by a batsman? And a slow-moving truck or a fast-moving bullet? Both mass and speed matter. Newton combined them into one quantity called momentum: p = m × v. It is a vector (it points along the velocity), its SI unit is kg m/s, and its dimension is [M L T⁻¹]. A heavy body or a fast body, or both, has a large momentum, and is harder to stop.

Newton's Second Law: the rate of change of momentum of a body is directly proportional to the applied unbalanced force and takes place in the direction of the force. In symbols, F = Δp / Δt. For a body of constant mass, Δp = m(v − u), so F = m(v − u)/t = ma. Hence: for the same force, a lighter body gets a bigger acceleration, and a = F/m.

Now the second law gives a neat definition of the newton: 1 N = 1 kg × 1 m/s². The product of force and time is impulse: J = F × Δt = Δp. Impulse equals the change in momentum, its unit is N s (the same as kg m/s), and it is a vector. A large force acting for a very short time, such as a bat hitting a ball, gives a large impulse.

From F Δt = Δp, if the change in momentum is fixed, then a longer time means a smaller force. This single idea explains many safety devices and techniques: a cricketer pulls his hands back while catching a fast ball to lengthen the stopping time; a high-jumper lands on a thick cushion or sand; airbags, helmets, crumple zones and seat belts stretch the collision time; we bend our knees while landing from a jump. The opposite is also true: a nail is driven by a hammer in a very short time to produce a very large force.

Push a wall hard while standing on roller skates: you roll backwards, though the wall did not move. Why? Because when you push the wall, the wall pushes you back equally hard. Forces always come in pairs. This is Newton's Third Law: to every action there is an equal and opposite reaction.

Now combine the second and third laws and you get the Law of Conservation of Momentum: in an isolated system (no external force) the total momentum before an interaction equals the total momentum after it. Take two bodies colliding. During contact, body A pushes B with force F for a time t, and B pushes A with −F for the same time. The impulses are equal and opposite, so one body gains exactly the momentum the other loses: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.

Recoil of a gun: before firing, gun and bullet are at rest, so total momentum is zero. After firing, bullet (mass m, velocity v) moves forward and the gun (mass M, velocity V) moves backward so that mv + MV = 0, giving V = − mv / M. The gun is much heavier, so its recoil velocity is small. Rockets work the same way: hot gases thrown backward push the rocket forward, and a rocket needs no air to push against, so it works in vacuum.

Try pushing a heavy almirah across the floor. It refuses to move at first, then slides with some effort, and stops soon after you stop pushing. Something is always opposing you. That something is friction: the force that opposes the relative motion, or the tendency of relative motion, between two surfaces in contact. Friction is a contact force and it arises because even surfaces that look smooth are rough at the microscopic level, so their tiny bumps interlock and stick to each other.

Friction is self-adjusting. If you push a box gently with 5 N and it does not move, the friction is also exactly 5 N. Push with 8 N, still no motion, and friction becomes 8 N. This goes on until a maximum value is reached, called limiting friction. Push even slightly harder and the box starts sliding. So friction is not a fixed number; it matches your push until the limit is crossed.

Laws of (solid) friction. (1) Friction always acts parallel to the surfaces and opposes the relative motion. (2) It depends on the nature of the two surfaces (rough or smooth, dry or lubricated). (3) It is directly proportional to the normal reaction N that presses the surfaces together: f = μN, where μ is the coefficient of friction. μ is a pure ratio, so it has no unit. (4) To a good approximation it does not depend on the area of contact and, for sliding, not on the speed. A wide tyre therefore does not give more friction just because it is wide; it gives better grip mainly because of its tread and rubber.

Fluid friction (drag). Friction is not only between solids. When an object moves through a liquid or gas, the fluid opposes its motion; this force is called fluid friction or drag. It increases with the speed of the object and depends on its shape. That is why boats, fish, aeroplanes and racing cars have streamlined shapes, and why a parachute, which is designed to catch a lot of air, slows a falling person. Energy lost to drag is the reason a moving object in air or water gradually slows down.