Forces

Task 1 - What is a force?

Hook

A hockey puck slides across ice and keeps going for a very long way. The same puck flicked across a carpet stops almost at once. Nothing is pushing the puck forward in either case, so why do they behave so differently?

Intuition

A force is a push or a pull. Contact forces need two objects to touch, such as friction, a push from a hand, the tension in a rope, or the support from a floor. Non-contact forces act across a gap, such as gravity, magnetism and electric forces.

The puck does not stop because its motion "runs out". It stops because friction pushes backward on it. Ice gives very little friction, so the puck keeps going. Carpet gives a lot, so it stops quickly. Take friction away completely and the puck would never stop.

Formalise

Force is measured in newtons (N). One newton is the force that gives a 1 kg mass an acceleration of $1\text{ m/s}^2$. An object's weight is the force of gravity on it, its mass $m$ multiplied by the gravitational field strength $g$, which is $9.81\text{ N/kg}$ on Earth.

Worked example

What is the weight of a 70 kg person on Earth and on the Moon, where $g = 1.62\text{ N/kg}$?

1. On Earth. $W = 70 \times 9.81 = 686.7\text{ N}$.
2. On the Moon. $W = 70 \times 1.62 = 113.4\text{ N}$.

The person's mass, 70 kg, is the same in both places because it measures how much matter they contain. Their weight changes because it is a force that depends on local gravity.

Your turn

Task 2 - Free body diagrams and net force

Hook

A book rests on a table. Gravity is pulling it downward the whole time. Why does it not fall through?

Intuition

The table pushes up on the book with a normal force. The book squashes the table's surface very slightly, and the surface pushes back just enough to support it. The upward normal force and the downward weight are equal, so they cancel. The net force on the book is zero.

A free body diagram shows this clearly. Experiment with the diagram below. Try the presets first, then change the sliders to make the forces unbalanced and watch the gold net force arrow appear.

Formalise

To find the net force along one line, choose a positive direction. Forces pointing that way are positive and forces pointing the other way are negative. Then add them up. The symbol $\Sigma$ means "the sum of".

Worked example

A 20 kg crate is pushed to the right with a force of 100 N while friction acts on it with 60 N. Take right and up as positive.

1. Vertical forces. Weight is $20 \times 9.81 = 196.2\text{ N}$ down. The floor supplies a normal force of 196.2 N up. Net vertical force is $196.2 - 196.2 = 0$.
2. Horizontal forces. Push is +100 N, friction is -60 N. Net horizontal force is $100 - 60 = 40\text{ N}$ to the right.

The crate does not move up or down, but it does speed up to the right. This is the "Crate being pushed" preset above.

Your turn

Task 3 - Newton's first law

Hook

An astronaut on a spacewalk lets go of a spanner and gives it a gentle push. Far from any planet, with no air, what happens to the spanner?

Intuition

It keeps going in a straight line at the same speed, forever, until something hits it. Objects resist changes to their motion. This property is called inertia, and the more mass an object has, the more inertia it has.

The surprising part is that motion does not need a force to keep it going. On Earth this is hard to see, because friction and air resistance are almost always acting. A car on a motorway needs its engine running to keep a steady speed, but only because the engine's driving force has to cancel the backward forces. If the net force is zero, the velocity stays constant, and that includes staying at rest.

Formalise

Newton's first law. An object stays at rest, or keeps moving at a constant velocity, unless a non-zero net force acts on it.

Worked example

A car cruises at a steady 30 m/s on a straight, level motorway. The engine produces a driving force of 900 N. How big are the resistive forces?

1. Constant velocity means the net force is zero.
2. So the backward forces, air resistance plus rolling resistance, must total exactly 900 N.

If the driver presses harder and the driving force rises to 1200 N, the forces become unbalanced and the car accelerates. As it speeds up, air resistance grows, until the forces balance again at a new, higher steady speed.

Your turn

Task 4 - Newton's second law

Hook

An empty shopping trolley is easy to get moving. A full one, pushed just as hard, picks up speed much more slowly. Why?

Intuition

The acceleration an object gets depends on two things. A larger net force gives a larger acceleration. A larger mass gives a smaller acceleration for the same force. The full trolley has more mass, so the same push accelerates it less.

Friction often decides whether anything moves at all. Below a certain push, friction matches the push exactly and nothing happens. Once the push is large enough, the object starts to slide and friction settles at a fixed value of $\mu$ multiplied by the normal force. The Greek letter $\mu$, pronounced "mew", is the coefficient of friction. It is around 0.05 for ice and around 0.6 for rubber on dry tarmac.

Use the diagram to explore. Try finding the smallest push that just gets a 10 kg cart moving when $\mu = 0.2$.

Formalise

Newton's second law, and the simple friction model used for sliding objects on a level surface, where $N$ is the normal force.

Worked example 1

A 1200 kg car has a net forward force of 3000 N. What is its acceleration?

$a = F_{net} / m = 3000 / 1200 = 2.5\text{ m/s}^2$.

Worked example 2

A 20 kg box is pushed across a level floor with 100 N. The coefficient of friction is 0.3. What is the acceleration?

1. Normal force. On a level floor it balances the weight. $N = 20 \times 9.81 = 196.2\text{ N}$.
2. Friction. $f = 0.3 \times 196.2 = 58.9\text{ N}$.
3. Net force. $100 - 58.9 = 41.1\text{ N}$.
4. Acceleration. $a = 41.1 / 20 = 2.06\text{ m/s}^2$.

Your turn

Task 5 - Newton's third law

Hook

A horse pulls a cart forward. By Newton's third law, the cart pulls back on the horse with an equal force. If the two forces are equal and opposite, how does anything ever move?

Intuition

The two forces in a third law pair act on different objects. One acts on the cart and the other acts on the horse. Forces only cancel when they act on the same object, so a third law pair never cancels.

To decide whether the cart accelerates, look only at forces on the cart. The horse pulls it forward and friction pulls it back. If the horse's pull is bigger, the cart speeds up. The horse moves forward because the ground pushes it forward, in response to the horse's hooves pushing backward on the ground.

Formalise

Newton's third law. If object A exerts a force on object B, then B exerts a force of the same size, in the opposite direction, on A. The two forces are always the same type, so a gravitational pull is paired with a gravitational pull.

Worked example

A rocket in space burns fuel. How does it accelerate with nothing to push against?

1. The rocket engine pushes exhaust gas backward, out of the nozzle.
2. By the third law, the gas pushes the rocket forward with an equal force.
3. That forward force is the only force on the rocket, so the rocket accelerates forward.

Your turn

Task 6 - Practice set and game connection

Game connection

A physics engine handles many forces at once with a simple pattern. Each object has a force accumulator, a single variable that starts every frame at zero. Gravity, thrusters, wind and friction each add their force into it. At the end of the frame the engine divides the total by the object's mass to get the acceleration, updates the velocity and position, and clears the accumulator ready for the next frame.

Consider a 2 kg crate pushed with 10 N against 4 N of friction, with $\Delta t = 0.1\text{ s}$. Complete the first frames.

Practice set