16.1 Force and Newton's first law — Learn

Prerequisites: adding and subtracting vectors and finding a resultant (Section 10.3); Pythagoras' theorem; the tangent ratio.

By the end of this section you can: list the forces that act in common situations; draw a clear force diagram for a body; state Newton's first law; use a zero resultant to find an unknown force; and explain what a resultant force does to a moving body.

1. What a force is

A force is a push or a pull. It is a vector, so it has a size and a direction. The unit of force is the newton (N). One newton is the force that gives a mass of one kilogram an acceleration of one metre per second squared.

Forces come in two broad groups.

• Contact forces act where two surfaces or bodies touch. Examples are the normal reaction from a floor, tension in a rope, thrust from an engine, friction and air resistance.
• Non-contact forces act at a distance. Weight is the most important one here, and it acts on a body from the Earth without touching it.

Some forces have standard names. The normal reaction is the push from a surface at right angles to that surface. Tension is the pull in a string or rope. Weight is the force of gravity on a body. Friction opposes sliding along a surface. Air resistance, or drag, opposes motion through air.

2. Force diagrams

A force diagram, also called a free-body diagram, shows one chosen body on its own with every force that acts on it.

• Draw one arrow for each force.
• Start each arrow on the body.
• Point the arrow the way the force acts.
• Label each arrow with the force and its size.
• Do not draw a force that acts on a different body.

With one body on a table, the weight acts downwards and the normal reaction acts upwards at right angles to the table.

3. Newton's first law

Newton's first law says that a body stays at rest, or keeps moving with a constant velocity, unless a resultant force acts on it.

Two useful ways to read the law:

• If the resultant force on a body is zero, its velocity does not change. It stays still, or it keeps the same speed in the same straight line.
• If the resultant force is not zero, the velocity changes. The body speeds up, slows down, or changes direction.

The tendency of a body to keep its velocity is called its inertia. A body with more mass has more inertia, so a larger force is needed to change its velocity by the same amount.

Example 1. A book rests on a table. Draw a force diagram and explain why the book stays at rest.

Two forces act on the book:

• its weight $W$, pulling downwards;
• the normal reaction $R$ from the table, pushing upwards.

The book stays at rest, so by Newton's first law the resultant force is zero. The two forces have equal sizes and opposite directions, so $R = W$.

Example 2. Forces of $5$ N and $12$ N act at right angles at a point. Find the resultant force.

Add the two perpendicular forces as vectors. The resultant has size

The direction is $\tan^{-1}\left(\frac{5}{12}\right) = 22.6^\circ$ from the $12$ N force, towards the $5$ N force.

Example 3. A particle is in equilibrium under three forces. Two of them are $7$ N to the right and $4$ N upwards. Find the third force.

The word equilibrium means the resultant force is zero. The three forces are coplanar, meaning they act in one plane. For zero resultant, the third force must cancel the other two. Write the two known forces as a column vector:

The third force is

It has size $\sqrt{7^2 + 4^2} = \sqrt{65} \approx 8.06$ N and points to the left and downwards.

Example 4. A cyclist moves at a constant speed of $6$ m/s in a straight line. The total driving force is $120$ N. Find the total resistance.

A constant velocity means the resultant force is zero. The resistance must balance the driving force, so the resistance is $120$ N backwards.

4. Try it: the resultant force explorer

Predict first. Two forces act at a point. Before you move anything, predict what the resultant is when force B is the exact opposite of force A. Write down the size you expect for the resultant.

Test. Change the size and the direction of each force. Watch the resultant arrow. Then make force B opposite to force A and watch the resultant shrink. Change the size of force B only and watch what the resultant does.

Explain. Write two sentences. The first says what has to be true about two forces for the point to be in equilibrium. The second says what a non-zero resultant does to the velocity of the point.

Watch out. Three mistakes are common on this page.

Thinking that motion needs a force. A body moving at a constant velocity has a zero resultant force. The engine's thrust balances the resistance exactly. A resultant force changes the velocity, and it does not keep the velocity steady.

Adding sizes that are not in the same direction. A $6$ N force and an $8$ N force at right angles do not give $14$ N. Add them as vectors, using Pythagoras, to get $10$ N. Re-check: if the two forces point along the same line, the sizes add or subtract; if not, they do not.

Leaving a force off a diagram. A body at rest on a table has two forces on it, not one. If you show only the weight, the diagram says the book should fall. Every force that acts on the chosen body belongs on the diagram.

Progress check

Return to the chapter problem. The $45$ kg crate rests on the warehouse floor before anyone touches it. The floor is horizontal.

1. Draw the crate on its own. Add an arrow for every force that acts on the crate.
2. Say why the crate stays at rest, using Newton's first law.
3. The weight of the crate turns out to be $441$ N. What is the normal reaction from the floor, and why?

You should now be able to

• say what a force is, and give its unit;
• name the common contact and non-contact forces;
• draw a labelled force diagram for one chosen body;
• state Newton's first law and use it in words;
• decide what a zero resultant means for a body at rest and for a body moving;
• add perpendicular forces to find a resultant, and its direction;
• use equilibrium to find a missing third force.