The weight of an object and the mass of an object are directly proportional.

The idea

Weight is directly proportional to mass, written W ∝ m, because W = m g and g is constant at a given place. Directly proportional has a precise meaning: double the mass and the weight doubles exactly, treble it and the weight trebles. On a graph of weight against mass, this shows as a straight line that passes through the origin, and the gradient of that line is the gravitational field strength. The relationship is not the same as weight being approximately equal to mass; they are different quantities in different units, and the constant linking them changes from place to place, so the same masses plotted on the Moon give a straight line through the origin with a much smaller gradient.

What earns the marks

• Reading W ∝ m as 'weight is directly proportional to mass'.
• Recognising a graph of weight against mass as a straight line through the origin.
• Identifying the gradient of that line as the gravitational field strength.
• Using the proportionality to scale a value, such as doubling the mass to double the weight.

Common mistakes and how to put them right

• Reading directly proportional as approximately equal.

• Accepting a straight line that does not pass through the origin as showing proportionality.

• Assuming the constant of proportionality is the same on every planet.

• Treating weight and mass as the same quantity in different units.

• Reading a single weight value off the graph as the gradient without dividing by the corresponding mass.

Exam tips

• In a multiple choice question, the phrase to look for is 'directly proportional to', not 'approximately equal to'.
• Check the line passes through the origin before claiming proportionality from a graph.
• Use a large triangle when finding the gradient, so the reading error is a smaller fraction of the result.