Scalars, vectors, weight and resultant forces

Physics · Forces. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A scalar has size only, such as mass, speed, distance, energy and time. A vector has size and direction, such as force, velocity, displacement, acceleration and weight. A vector is drawn as an arrow: its length shows the size and it points in the direction.

A force is a push or a pull caused by an interaction between two objects. Contact forces need the objects to touch (friction, air resistance, tension, normal contact force). Non-contact forces act at a distance (gravitational, magnetic, electrostatic).

Weight is the force of gravity on an object. It acts at the centre of mass. The resultant force is the single force that has the same effect as all the forces acting together.

Rules to remember

weight = mass × gravitational field strength (W = mg); W in N, m in kg, g in N/kg.
On Earth, g = 9.8 N/kg. Mass stays the same everywhere; weight changes with g.
Weight is measured with a newtonmeter (a calibrated spring balance).
Forces in the same direction: add. Forces in opposite directions: subtract.
Two forces at right angles: resultant² = F₁² + F₂², or use a scale drawing.
A force can be resolved into two components at right angles that together have the same effect as the force.

Common mistake

Students give mass in newtons or weight in kilograms. Mass is in kg and does not change on the Moon; weight is a force in N and does change.

Worked example

A bag of mass 12 kg hangs from a rope. The rope pulls upwards with a force of 150 N. What is the resultant force on the bag? (g = 9.8 N/kg)

Weight = m × g = 12 × 9.8 = 117.6 N, downwards.
The forces act in opposite directions, so subtract: 150 − 117.6 = 32.4 N.
The upward force is the larger one, so the resultant is upwards.

Answer: 32.4 N upwards (about 32 N)

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