Chapter 3: Force and Motion

• The concept of force and how it is measured.
• Newton's first law and the phenomenon of inertia.
• Momentum.
• Newton's second law (F = ma; force as the rate of change of momentum); the newton.
• Explaining everyday phenomena (collisions, airbags, falling on different surfaces).
• Newton's third law (action and reaction) and its applications.

• In the previous chapter we described motion by speed, velocity and acceleration, without asking why things move.
• In this chapter we ask what produces acceleration.
• The answer is force.
• Newton's three laws of motion, which we study here, are among the most important laws in physics.

• Before the seventeenth century no organised theory of motion existed, although many ancient philosophers had tried to explain it.
• The greatest credit for founding it goes to two scientists: Galileo and Newton.

"Force" is a word we use every day: your muscles pull a bag, a car engine starts the car moving, a brake stops it.

Force is an external influence acting on an object that changes, or tries to change, its state of motion (its velocity in magnitude or direction).

• Force is a vector quantity (it has magnitude and direction).
• It is measured by a spring balance (spring scale).
• Its SI unit is the newton (N).

"Tries to change": if you push a heavy wall and it does not move, your push is still a force, but another force (the reaction of the floor/friction) balances it.

• Often several forces act on one body.
• Their vector sum is the resultant (net) force, written ΣF (read "sigma F").
• Forces in the same line are added with signs (+ in one direction, − in the opposite one).

Example 1. A box on the floor is pulled to the right by 30 N and to the left by 10 N (friction). ΣF = 30 − 10 = 20 N to the right.

Example 2. The same box is pulled by 30 N to the right and 30 N to the left. ΣF = 0: the forces are balanced and the box keeps its state of motion.

• Imagine returning home after a long trip: everything is as you left it.
• This simple fact, "things stay as they are unless something changes them", is the basis of Newton's first law.

• Everyday experience seems to contradict it: if you push a box on the floor, it slides some distance, slows down and stops.
• Ancient thinkers concluded that rest is the natural state of matter and that every motion dies away.
• Experiments showed that this is wrong: the box stops because of friction, a force opposing its sliding.
• If friction did not exist (as on very smooth ice, or in space) the box would go on moving for ever without stopping.

Newton's first law of motion: An object remains at rest, or moves with constant velocity in a straight line, unless a resultant force acts on it and changes this state.

Mathematical form: ΣF = 0 (the object keeps its state)

where ΣF is the resultant of all forces acting on it. Several forces may act, but if they cancel each other the resultant is zero.

• If ΣF = 0, then a = 0: the velocity does not change, whether the object is at rest or moving.
• A force is needed to start a stationary object moving or to stop a moving one (that is, to change its velocity).
• No force is needed to keep an object moving at constant velocity (only to cancel friction/air resistance). A car moving steadily on a straight road has an engine force that exactly balances the resistances: ΣF = 0.

Example 1. A car moves at a steady 25 m/s along a straight road. The engine force is 900 N forward. What is the resistance to motion? Since the velocity is constant, ΣF = 0, so the resistance is 900 N backward.

Example 2. A book rests on a table: its weight (downward) is balanced by the table's upward force; ΣF = 0, so it remains at rest.

Newton's first law is also called the law of inertia.

Inertia is the tendency of a stationary object to remain at rest, and of a moving object to continue moving with the same velocity in a straight line; that is, objects resist any change in their state of rest or motion.

Everyday examples of inertia

• A passenger standing in a bus is thrown forward when the bus brakes suddenly: his body tends to keep moving forward.
• When a train suddenly moves forward, a small bag lying under a seat appears to slide backward relative to the train, because it tends to stay at rest while the train moves forward under it.
• A card under a coin on a glass: flick the card away quickly and the coin drops into the glass, since it tends to remain at rest.

Measure of inertia. Observation shows that the difficulty of stopping a moving body depends on its mass and its velocity:

• A large truck is hard to stop, a small bicycle is easy, if they move at the same velocity.
• A car moving fast is hard to stop; the same car moving slowly is easy to stop. The two factors together define a new quantity, the momentum.

Momentum (p) is the product of the mass of a body and its velocity:

p = m v

• It is a vector (same direction as the velocity).
• Unit: kg·m/s.
• Dimensional formula: MLT⁻¹.

A body with larger momentum needs a larger force or a longer time to be stopped.

Example 1. A truck of mass 8000 kg moves at 15 m/s: p = 8000 × 15 = 1.2 × 10⁵ kg·m/s. A cyclist and cycle of total mass 90 kg at 6 m/s: p = 540 kg·m/s. The truck's momentum is more than 200 times larger.

Example 2. A 0.15 kg ball moves at 20 m/s to the right: p = +3 kg·m/s. After it rebounds at 20 m/s to the left, p = −3 kg·m/s; the change of momentum is −3 − (+3) = −6 kg·m/s.

Example 3. What speed gives a 1200 kg car the same momentum as the truck in Example 1? v = p/m = 1.2 × 10⁵/1200 = 100 m/s (360 km/h).

• By the first law, an object with no resultant force has no acceleration.
• So if a resultant force acts (ΣF ≠ 0) the velocity changes and the object accelerates (a ≠ 0).
• Newton's second law gives the exact relation.

Newton's second law: The resultant force acting on an object is equal to the time rate of change of its momentum.

ΣF = Δp/Δt = (m v_f − m v_i)/Δt

If the mass is constant:

ΣF = m (v_f − v_i)/Δt = m Δv/Δt

and since Δv/Δt is the acceleration,

ΣF = m a or a = ΣF / m

When a resultant force acts on an object it gives the object an acceleration that is directly proportional to the force and inversely proportional to its mass, and in the direction of the force.

• Same mass, greater force → greater acceleration.
• Same force, greater mass → smaller acceleration.

Graph of acceleration against force.

• For a given mass the graph a–F is a straight line through the origin whose slope is 1/m.
• For a smaller mass the slope (and so the acceleration for the same force) is larger.
• For example a 100 kg body has twice the acceleration of a 200 kg body under the same force.

From F = ma: 1 newton is the force that gives a mass of 1 kg an acceleration of 1 m/s². 1 N = 1 kg·m/s²

The dimensional formula of force is MLT⁻².

A force F on a wooden cube gives it a certain acceleration. The same force on another cube gives it three times the acceleration. Since a = F/m with F constant, a ∝ 1/m, so the second cube has one-third of the mass of the first.

Example 1 — F = ma. A car of mass 1200 kg accelerates uniformly at 2.5 m/s². The resultant force = 1200 × 2.5 = 3000 N.

Example 2 — mass. A resultant force of 45 N gives a body an acceleration of 1.5 m/s². m = F/a = 45/1.5 = 30 kg.

Example 3 — net force with friction. A 50 kg crate is pushed by a 200 N force while friction opposes the motion with 120 N. ΣF = 200 − 120 = 80 N; a = 80/50 = 1.6 m/s².

Example 4 — from the change of momentum. A force acts on a 2 kg body, changing its velocity from 3 m/s to 15 m/s in 4 s. Δp = 2 × (15 − 3) = 24 kg·m/s; F = 24/4 = 6 N. (Check: a = 12/4 = 3 m/s²; F = 2 × 3 = 6 N.)

Example 5 — velocity after a time. A 5 kg trolley at rest is pulled by a net force of 10 N for 6 s. a = 10/5 = 2 m/s²; v = at = 2 × 6 = 12 m/s.

From F = Δ(mv)/Δt, i.e. F·Δt = Δ(mv) (the product Ft is called the impulse), the force needed to bring a given moving body to rest depends on the time it takes:

• If the change of momentum happens over a longer time, the force is smaller (the effect of the impact is less).
• If it happens in a shorter time, the force is larger.

This explains:

1. A car hitting a wall is more destructive than a car hitting a heap of straw: the wall stops it in a very short time (large force); straw takes longer (smaller force).
2. A person falling from a height into water is not hurt but falling onto ground may hurt: water brings him to rest more slowly.
3. The severity of the injury increases with the height of the fall, because the velocity (and so the momentum) at impact increases.
4. An egg does not break when it falls on a cushion, but it breaks on the floor.
5. A collision of a large truck with a wall is more destructive than that of a small truck: larger mass → larger momentum → larger force for the same stopping time.
6. Airbags in cars increase the time over which the driver's body comes to rest in a collision, reducing the force on him. Similarly, a cricket or football goalkeeper pulls the hands back while catching a fast ball to lengthen the stopping time.

If ΣF = 0 then, from ΣF = ma, a = 0 and the velocity is constant. So the first law follows from the second law as a special case (zero resultant force).

• A balloon blown up and released flies off as the air rushes out.
• A person on a rolling chair who pushes a wall with his feet moves away from the wall.
• A rifle "kicks back" when a bullet is fired. All these cases are explained by Newton's third law.

Newton's third law: When one object exerts a force on a second object, the second exerts a force on the first that is equal in magnitude and opposite in direction. In short: for every action there is an equal and opposite reaction.

Mathematical form: F₁ = − F₂

• Forces occur in pairs.
• Example: two spring balances hooked together and pulled give identical readings: the force of the first on the second equals that of the second on the first.

1. There is no single force in the universe; action and reaction appear together and disappear together.
2. They are equal in magnitude, opposite in direction, and have the same line of action.
3. They have the same nature: if the action is a gravitational force, so is the reaction.
4. They do not cancel each other, because they act on two different bodies. (Forces cancel only if they act on the same body.)

Thinking corner: when a large truck collides with a small car, which receives the larger force? Both receive forces of equal magnitude. The car is damaged more, and gains a larger acceleration (a = F/m), because its mass is smaller.

• Rocket propulsion: burning gas of large mass is expelled backwards (downwards) at high speed; the reaction on the rocket is a thrust forward (upwards). A rocket needs no air to push against, so it works in space.
• Recoil of a gun: the gun pushes the bullet forward; the bullet pushes the gun backward with an equal force, but the gun's much larger mass gives it a very small acceleration.
• Swimming/walking: a swimmer pushes the water back, the water pushes him forward; when walking your foot pushes the ground backward and the ground pushes you forward.
• Space: an astronaut who throws an object in one direction moves in the opposite direction. This suggests how a spacecraft can change its direction outside the atmosphere: by ejecting gas in the direction opposite to the desired change.

• A train suddenly moves forward; where does a small bag under a seat go? Relative to the train it moves backward, because of its inertia (it tends to stay at rest).
• Why is the first law a special case of the second? When ΣF = 0, ma = 0 so a = 0 — constant velocity.
• What happens when an astronaut throws an object? He moves in the opposite direction (third law); a spacecraft can change direction by expelling gas.

• Force: an external influence that changes or tries to change the state of motion; unit newton (N), measured by a spring balance.
• First law: ΣF = 0 → rest or constant velocity; inertia is the resistance to change of the state of motion.
• Momentum p = mv (kg·m/s).
• Second law: ΣF = Δp/Δt = ma; a ∝ F, a ∝ 1/m; 1 N = 1 kg·m/s². A longer time of impact gives a smaller force.
• Third law: F₁ = −F₂; equal, opposite, same nature, on different bodies, appear and disappear together.