Forces and motion, Year 11: scalars, vectors, weight and work done

A vector quantity

HintForce, velocity and displacement all belong to this family.

The whyTo state one fully you need a number, a unit and a direction: 20 N downwards, or 5 m/s due east. Leave the direction out and the description is incomplete. Two of them can cancel if they point opposite ways, which never happens with quantities that have size alone.

A scalar quantity

HintMass, time, energy and temperature all belong to this family.

The whyA number and a unit say everything there is to say: 3 kg, 20 s, 500 J. Such quantities simply add up like ordinary numbers. Speed and distance are in this family too, which is why GCSE introduces velocity and displacement as their partners with direction.

length

HintThink of what you would measure with a ruler on the drawing.

The whyDrawn to a scale, an arrow twice as long stands for a quantity twice as big. This is the idea behind every force diagram: the arrow carries both pieces of information that a vector needs.

30

HintFind how many newtons each centimetre stands for.

The whyThe scale is 10 N ÷ 2 cm = 5 N for every centimetre. A 6 cm arrow therefore stands for 6 × 5 N. Because both arrows share one scale, the second being three times as long means the force is three times as big.

Weight — it is a force, so it has a direction (downwards)

HintOne of them is measured in newtons.

The whyMass is just an amount of matter, in kilograms, with no direction. Weight is the pull of gravity on that mass, and a pull always acts in some direction — towards the centre of the Earth. Every force is a vector.

50

HintRearrange the weight equation so that mass stands alone, then divide.

The whyWeight = mass × gravitational field strength, or W = m g, with W in newtons, m in kilograms and g in newtons per kilogram. Rearranged, m = W ÷ g = 490 ÷ 9.8. AQA expects this equation to be recalled and says the value of g will always be given.

88.2 N

HintCompare the two masses before you reach for any equation.

The whyThe symbol ∝ means 'is directly proportional to': multiply the mass by some number and the weight is multiplied by the same number. Here the mass is three times bigger, so the weight is 3 × 29.4 N. The constant linking them is g, which works out as 29.4 ÷ 3 = 9.8 N/kg.

A free body diagram

HintIts name says the object has been cut loose from everything around it.

The whyEverything else — the table, the rope, the Earth — is left out and replaced by the force it exerts on the object. Each arrow starts on the object and points the way that force acts, with its length showing the size. If the arrows balance, the resultant force is zero.

Weight downwards and the normal contact force upwards — arrows of equal length

HintThe book is not speeding up in any direction, so think about what the resultant must be.

The whyThe Earth pulls the book down and the table pushes it up. The book stays at rest, so the resultant force is zero: the two arrows are the same length and point in opposite directions. A free body diagram with a zero resultant shows balanced forces.

Resolving a force

HintThe two new forces are called components.

The whyA force pulling up and to the right does two jobs at once: part of it pulls sideways and part pulls upwards. Splitting it into a horizontal and a vertical part lets you deal with each direction separately. AQA asks for this by scale drawing only, with no trigonometry.

50 N at 53° east of north

HintTurn the measured line back into newtons using the scale, and say which way it points.

The whyDrawing the two arrows tip to tail and joining start to finish gives the single force with the same effect. The line is 5.0 cm, and each centimetre stands for 10 N, so the resultant is 50 N. A resultant is a vector, so the answer needs its direction as well as its size.

87 N horizontally and 50 N vertically

HintEach side of the triangle is a force drawn to the same scale as the arrow.

The whyThe two shorter sides of the triangle are the components: one along the ground and one straight up. Convert each with the scale: 8.7 cm is 87 N and 5.0 cm is 50 N. Together these two forces have exactly the same effect as the single 100 N pull.

15 m

HintRearrange the work equation so that distance stands alone.

The whyWork done = force × distance, or W = F s, with W in joules, F in newtons and s in metres measured in the direction in which the force acts. Rearranged, s = W ÷ F = 600 ÷ 40. AQA expects this equation to be recalled.

250

HintThink about what one newton moving through one metre is called.

The whyOne joule is the work done when a force of one newton moves something one metre, so 1 J = 1 N m exactly. Multiplying newtons by metres in W = F s therefore gives joules directly, with no conversion factor.