Distinguish scalars from vectors, add forces acting in a line, and state the conditions for equilibrium

What this dot point is asking

SEAB wants you to tell scalars from vectors, to add or subtract forces that act along the same straight line to find a resultant, and to state the conditions for an object to be in equilibrium. The big idea is that some quantities need a direction (vectors) and some do not (scalars), and that an object is in equilibrium when the forces on it balance.

The answer

Scalars and vectors

A scalar quantity has size (magnitude) only. Examples are mass, distance, time, speed, energy and temperature.

A vector quantity has both size and direction. Examples are force, weight, velocity, displacement and acceleration.

The test is simple: if you must state a direction for the quantity to make full sense, it is a vector. "A force of $10\ \text{N}$" is incomplete as a vector until you say which way it points.

Adding forces in a straight line

When forces act along the same line, you combine them into a single resultant force:

• Forces in the same direction add together.
• Forces in opposite directions subtract.

For example, a $30\ \text{N}$ push to the right and a $10\ \text{N}$ friction force to the left give a resultant of $30 - 10 = 20\ \text{N}$ to the right. The direction of the resultant is the direction of the larger force.

Equilibrium

An object is in equilibrium when the resultant force on it is zero. In equilibrium an object is either at rest, or moving at a constant velocity in a straight line (this is just Newton's first law again).

For forces along one line, equilibrium means the forces one way exactly balance the forces the other way. For an object resting on a surface, the upward normal force balances the downward weight.

When turning is also possible, full equilibrium needs two conditions: the resultant force is zero (no straight-line acceleration) and the resultant moment is zero (no turning), so the principle of moments also holds.

Examples in context

Example 1. A tug of war. Two teams pull a rope in opposite directions. If each pulls with $500\ \text{N}$, the resultant is zero and the rope does not move: the system is in equilibrium. As soon as one team pulls harder, a resultant force appears and the rope accelerates toward the stronger team.

Example 2. A hanging lamp. A lamp hanging still from a ceiling cord is in equilibrium. The weight pulls down and the tension in the cord pulls up with an equal force, so the resultant is zero. The cord must be strong enough that its tension equals the weight of the lamp.

Try this

• Cue. Sort these into scalars and vectors: temperature, displacement, energy, acceleration. [2 marks] Scalars: temperature, energy; vectors: displacement, acceleration.

• Cue. A $25\ \text{N}$ force acts east and a $10\ \text{N}$ force acts west on a trolley. Find the resultant. [2 marks] $25 - 10 = 15\ \text{N}$ to the east.

• Cue. A book lies still on a table. Name the two forces acting and state what their resultant must be. [3 marks] Weight downward and the normal force upward; for the book to stay still the resultant is zero, so they are equal in size.