Measuring length with a ruler

A ruler measures length using a graduated scale: a series of marks showing distances in units such as centimetres and millimetres. Before measuring, check what each interval on the scale represents.

Place the ruler alongside the object, with the scale parallel to the length being measured. Align one end of the object with the zero mark, which may not be at the physical end of the ruler. Read the scale at the other end of the object; this reading is its length.

Look directly above the mark when taking the reading. Viewing it from an angle can make the end of the object appear to line up with a different mark. This is called parallax error.

Measuring when the zero mark cannot be used

If one end of the object cannot be aligned with zero, read the scale at both ends. Subtract the smaller reading from the larger reading to find the length. For example, if the ends line up with $$2,\text{cm}$$ and $$7,\text{cm}$$, the length is $$7 - 2 = 5,\text{cm}$$.

Record the length with its unit, such as $$\text{cm}$$ or $$\text{mm}$$.

Measuring volume with a measuring cylinder

A measuring cylinder measures the volume of a liquid using a graduated scale: a series of marked divisions along its side. The liquid’s surface shows where to read the volume on the scale.

Place the cylinder upright on a level surface and allow the liquid to settle. Bring your eyes level with the liquid’s surface before reading the scale. Looking from above or below can make the reading appear higher or lower than it is.

The surface of a liquid often curves where it meets the sides of the cylinder. This curved surface is called the meniscus. Read the scale at the bottom of the meniscus, not at its raised edges.

Record the reading with its unit, such as cubic centimetres ($$\text{cm}^3$$) or millilitres ($$\text{mL}$$). One cubic centimetre is equal to one millilitre: $$1,\text{cm}^3 = 1,\text{mL}$$.

Measuring time intervals

A time interval is the time that elapses between two events. For example, the events could be the start and end of a run.

To measure an interval with a clock, record the clock reading at each event. Then subtract the start reading from the end reading:

For example, if a run starts at 2:10:05 and ends at 2:10:17, the time interval is 12 seconds.

A digital timer can measure the interval directly. Start it at the first event and stop it at the second; its display shows the elapsed time. Record the result with an appropriate unit, such as seconds ($$\text{s}$$).

Measurement by multiples

A single small length or short time interval can be difficult to measure accurately. Instead, measure several equal repetitions together, then divide the total by the number of repetitions. The result is the average measurement for one repetition.

Measuring length by multiples

Place identical items end to end in a straight line, without gaps or overlaps. Measure their combined length and divide by the number of items to find the average length of one item.

For example, if 10 identical items have a combined length of $$45\ \text{mm}$$, their average length is $$45\ \text{mm} \div 10 = 4.5\ \text{mm}$$.

Measuring time by multiples

For a repeated event, count a number of complete repetitions and measure their total time. Divide the total time by the count to find the average time per event.

A pendulum’s period is the time for one complete oscillation. For example, an oscillation can be counted from one extreme of its swing to the other extreme and back. If 20 complete oscillations take $$32\ \text{s}$$, the average period is $$32\ \text{s} \div 20 = 1.6\ \text{s}$$.

Why measuring multiples helps

A ruler reading or stopwatch timing has some uncertainty. When a larger total is measured, the same uncertainty is a smaller fraction of that total. Dividing the total by the number of repetitions therefore reduces the effect of that reading or timing uncertainty on the average. It does not necessarily remove errors that affect every repetition in the same way.

Scalars and vectors

A physical quantity can be described by its magnitude: its size, expressed as a number with a unit. Some quantities also need a direction to specify them fully.

A scalar has magnitude only. For example, a time of 5 seconds is specified without a direction.

A vector has both magnitude and direction. For example, a velocity of 5 metres per second east is different from a velocity of 5 metres per second west, even though their magnitudes are the same.

Scalar quantities

Distance, speed, time, mass, energy and temperature are scalars. For example, speed tells us how fast an object moves, but not the direction in which it moves.

Vector quantities

Force, weight, velocity, acceleration, momentum, electric field strength and gravitational field strength are vectors. For example, velocity specifies both how fast an object moves and its direction of motion.

Weight is a force and has a direction; mass is a scalar and does not.

Resultant of perpendicular vectors

The resultant of two vectors is a single vector that has the same combined effect as the two original vectors. For two forces or two velocities at right angles, the resultant has both a magnitude and a direction.

Finding the resultant by drawing

Draw the first vector to scale. From its head, draw the second vector to the same scale, keeping it at right angles to the first. The resultant is the arrow from the tail of the first vector to the head of the second.

The two vectors and the resultant form a right-angled triangle. Measure the resultant arrow’s length and use the scale to find its magnitude. Measure its angle from a stated direction to find its direction.

Calculating the resultant

If the perpendicular vectors have magnitudes $$A$$ and $$B$$, the resultant is the hypotenuse of the right-angled triangle. By Pythagoras, its magnitude $$R$$ is:

The direction can be expressed as an angle $$\theta$$ measured from the direction of $$A$$ towards the direction of $$B$$. For this choice of angle:

Use the inverse tangent to find $$\theta$$, and state the direction from which the angle is measured.

Resultant of two perpendicular forces

Suppose a force of $$3,\mathrm{N}$$ acts east and a force of $$4,\mathrm{N}$$ acts north. Their resultant magnitude is:

Taking east as the direction of $$A$$, the angle towards north is $$\theta = \tan^{-1}(4/3) \approx 53^\circ$$. The resultant is therefore $$5,\mathrm{N}$$ at $$53^\circ$$ north of east.