Force and inertia
A force is a push or a pull. It can start motion, stop it, change speed, change direction or change shape. Its SI unit is the newton (N). Force is a vector: it has size and direction.
Inertia means a body resists any change in its state. It does not like to start, stop or turn by itself.
- Inertia of rest: a coin on a card falls into the glass when you flick the card away.
- Inertia of motion: passengers lean forward when a bus stops suddenly.
- Inertia of direction: mud flies off a spinning wheel along the tangent.
Mass is the measure of inertia. More mass means more force is needed for the same change.
Aristotle thought a force is needed to keep a body moving. Galileo rolled balls on smooth slopes and showed that motion stops only because of friction. With no friction, a ball would roll on for ever.
Newton's first law of motion
First law: every body stays at rest, or keeps moving with the same speed in a straight line, unless a net external force makes it change.
So if the net force is zero, the acceleration is zero. The first law gives the meaning of force (the thing that causes acceleration) and it is also called the law of inertia.
Note: a body can have many forces on it and still move uniformly, as long as they cancel. A car at steady speed on a highway has engine force and friction balancing each other.
The first law holds in an inertial frame (a frame that is not accelerating). Inside a braking bus, things seem to move without any force because the bus itself is accelerating.
Momentum and Newton's second law
Linear momentum p = m v. It is a vector along the velocity. SI unit: kg m/s. A slow truck and a fast bullet can both be hard to stop because both have large momentum.
Second law: the rate of change of momentum of a body is proportional to the net force on it and happens in the direction of the force. Choosing the unit of force suitably, the constant is 1:
F = dp/dt
Derivation of F = ma
If the mass does not change, dp/dt = d(mv)/dt = m (dv/dt) = m a. So F = m a.
1 newton is the force that gives a 1 kg mass an acceleration of 1 m/s².
Points to remember
- F = ma is a vector equation. You can apply it along x, y and z separately: Fx = m ax, Fy = m ay.
- F is the net external force. Forces inside a body cancel.
- The law is local: the acceleration now depends on the force now, not earlier.
- If F = 0 then a = 0, which is the first law again.
Free-body diagram
To solve problems, draw the body alone and mark every force on it (weight mg, normal force N, tension T, friction f, applied force). Then write F = ma along each axis.
Impulse
When a large force acts for a very short time (a bat hitting a ball, a hammer on a nail), we measure its effect by impulse:
J = F × Δt = Δp = m v − m u
Unit: N s (same as kg m/s). This is the impulse–momentum theorem. For a changing force, impulse is the area under the force–time graph.
For the same change in momentum, a longer time means a smaller force. That is why we bend our knees when we jump down, why cricket players pull the hands back, why cars have crumple zones and air bags, and why glassware is packed in straw or foam.
Try it at home
Coin and card: put a playing card on a glass and a coin on the card. Flick the card sideways quickly. The coin drops into the glass (inertia of rest).
Egg catch: throw a raw egg gently onto a stretched bedsheet held by two friends. It does not break, because the sheet stops it slowly (impulse). Do this outdoors!
In the 3D: on the last step, keep F = 10 N and change m from 1 kg to 5 kg. Predict a each time, then check.
Key formulas and definitions
- p = m v
- F = dp/dt = m a (constant mass)
- 1 N = 1 kg × 1 m/s²
- Impulse J = F × Δt = Δp = m v − m u
- Weight W = m g
- Impulse = area under F–t graph
Worked examples
1. A 5 kg box is pushed with a net force of 20 N. Find its acceleration.
a = F/m = 20/5 = 4 m/s² in the direction of the force.
2. Find the momentum of a 0.16 kg cricket ball moving at 25 m/s.
p = m v = 0.16 × 25 = 4 kg m/s.
3. A 1000 kg car speeds up from 10 m/s to 20 m/s in 5 s. Find the net force.
a = (20 − 10)/5 = 2 m/s². F = m a = 1000 × 2 = 2000 N. Check with momentum: Δp = 1000 × 10 = 10 000 kg m/s; F = Δp/Δt = 10 000/5 = 2000 N.
4. A 0.15 kg ball comes at 20 m/s and is hit straight back at 30 m/s. Find the impulse given by the bat.
Take the return direction as +. u = −20 m/s, v = +30 m/s. J = m(v − u) = 0.15 × (30 − (−20)) = 0.15 × 50 = 7.5 N s, towards the bowler's side.
5. In the example above, the bat touches the ball for 0.005 s. Find the average force.
F = J/Δt = 7.5/0.005 = 1500 N.
6. A fielder stops a 0.15 kg ball moving at 20 m/s. (i) With stiff hands in 0.02 s, (ii) pulling hands back in 0.12 s. Compare the forces.
Δp = 0.15 × 20 = 3 kg m/s. (i) F = 3/0.02 = 150 N. (ii) F = 3/0.12 = 25 N. Pulling back makes the force 6 times smaller.
7. A 2 kg body is at rest. A force F = 6t newton (t in s) acts on it. Find its speed at t = 2 s.
Impulse = area under F–t graph from 0 to 2 s = ½ × 2 × 12 = 12 N s. Δp = 12 kg m/s, so v = 12/2 = 6 m/s.
8. A 60 kg person stands in a lift going up with acceleration 2 m/s². Find the normal force from the floor (g = 10 m/s²).
Free-body diagram: N upward, mg downward. N − mg = m a → N = m(g + a) = 60 × 12 = 720 N. The person feels heavier.
Common mistakes
- Thinking a moving body needs a force to keep moving. It needs a force only to change its velocity.
- Using a force that is not the net force in F = ma. Add all forces (with directions) first.
- Forgetting the sign of velocity when it reverses. A ball bouncing back has Δv = v − (−u) = v + u.
- Thinking impulse and force are the same. Impulse = force × time; a small force for long time can give the same impulse as a big force for short time.