A crate rests on a table. List every force on it and what exerts each one.

The Earth pulls the crate down — its weight. The table pushes the crate up — the normal force. There is no motion and no push, so there is nothing else. Two forces only, and at rest they are equal and opposite

A body has 30 N acting east and 40 N acting north. What is the net force?

The forces are perpendicular. The magnitude is the square root of 30 squared plus 40 squared. The root of 900 plus 1600 is the root of 2500. 50 N, north-east, at the angle whose tangent is 40 over 30

A 10 kg body sits on a slope of 30 degrees. What is the component of weight along the slope?

The weight is 10 times 10, so 100 N straight down. The component along the slope is the magnitude times the sine of the angle. The sine of 30 degrees is one half. 50 N down the slope

Newton's Laws · Grade 11

Why does every mechanics problem start with a free-body diagram?

Because Newton's laws are about the net force, and you cannot work that out before you know which forces are acting. The diagram forces you to answer the most important question in any problem: which bodies are touching or pulling this one, and what exactly is each of them doing to it.

Learning objectives

Draw a complete free-body diagram for a body resting on a table, on a slope or hanging from a rope
Add two forces acting at an angle to one another and find the net force in size and direction
Resolve a force into components along a chosen direction and perpendicular to it
State for any named force which body exerts it and on which body it acts

A force is always between two bodies

Every force has something exerting it and something receiving it. 'Weight' means the force the Earth exerts on the body; 'normal force' means the force the surface exerts on the body. Who exerts it, on what.

That rule filters out most invented forces. There is no 'force of motion': no body exerts it, nothing pushes a ball that is already flying through the air forward, and so no arrow is drawn in the direction of travel.

In the diagram the body is a dot or a box, and one arrow leaves it for every force acting on it. Forces the body exerts on other things are not in its diagram — they belong to the other body's.

Three situations that recur in every problem

A body resting on a table: weight down, normal force from the table up. If it is at rest the two are equal in size — not because that is a law, but because otherwise it would be accelerating.

A body hanging from a rope: weight down, rope tension up. A body on a slope: weight still straight down, towards the centre of the Earth, and the normal force perpendicular to the slope rather than vertical.

That last point is what makes slopes hard. Weight does not change direction when the surface tilts; what changes is the angle between it and the surface, which is why it is convenient to resolve it along the slope and perpendicular to it.

Adding and resolving

Two forces in the same direction add as numbers. Two in opposite directions subtract. Two at an angle add as vectors — by the parallelogram, or by resolving both into components and adding the components separately.

Resolving is the tool that always works. Choose two perpendicular directions, usually along the motion and across it, and split every force into two components. The component along a chosen direction is the magnitude times the cosine of the angle from it, and the perpendicular component is the magnitude times the sine.

After resolving, the problem becomes two simple problems in perpendicular directions, and Newton's laws can be applied to each one separately.

Worked examples

A crate rests on a table. List every force on it and what exerts each one.
1. The Earth pulls the crate down — its weight
2. The table pushes the crate up — the normal force
3. There is no motion and no push, so there is nothing else
Answer: Two forces only, and at rest they are equal and opposite
A body has 30 N acting east and 40 N acting north. What is the net force?
1. The forces are perpendicular
2. The magnitude is the square root of 30 squared plus 40 squared
3. The root of 900 plus 1600 is the root of 2500
Answer: 50 N, north-east, at the angle whose tangent is 40 over 30
A 10 kg body sits on a slope of 30 degrees. What is the component of weight along the slope?
1. The weight is 10 times 10, so 100 N straight down
2. The component along the slope is the magnitude times the sine of the angle
3. The sine of 30 degrees is one half
Answer: 50 N down the slope

Common mistakes

Drawing an arrow in the direction of motion: Motion is not a force and no body exerts it. A ball flying through the air is pushed forward by nothing, so its diagram has no forward arrow.
Putting a force the body exerts into its own diagram: A body's diagram holds only the forces acting on it. The force it exerts belongs to the other body's diagram, and mixing the two is the source of most wrong cancellations.
Drawing the normal force straight up on a slope: The normal force is perpendicular to the surface, not vertical. On a tilted surface the two are not on the same line, and that is exactly what leaves a component to accelerate the body downhill.

What to remember

Every force is between two bodies: who exerts it, on what.
A body's diagram holds only forces acting on it.
The normal force is perpendicular to the surface, not always vertical.
Resolving turns one problem into two simple ones.