Physical quantities and measurement techniques

Key points

Measuring length and volume

• A ruler or metre rule measures length. Most rulers are marked in millimetres (mm).
• Line up the zero mark of the ruler with one end of the object. If the zero mark is worn or is not at the end of the ruler, read the scale at both ends of the object. Then subtract the two readings.
• Put your eye directly in line with the mark you are reading, at right angles to the scale. If you look from an angle, the reading is wrong. This is called a parallax error.
• Hold the ruler flat against the object. This keeps the scale close to the object and reduces parallax error.
• Write a reading from a millimetre scale to the nearest millimetre, for example 12.3 cm.
• A measuring cylinder measures the volume of a liquid. Its scale is usually in cm³. 1 cm³ is the same as 1 ml.
• Stand the measuring cylinder on a flat, level surface.
• The surface of water in a cylinder is curved. This curved surface is called the meniscus. Read the scale at the bottom of the meniscus, with your eye level with the liquid surface.
• Choose the smallest measuring cylinder that will hold all the liquid. A narrow cylinder has smaller divisions on its scale, so you can read the volume more precisely.
• To find the volume of a regular solid, measure its sides with a ruler and calculate the volume. To find the volume of an irregular solid, use displacement in a measuring cylinder (see 1.4).

Measuring time

• A clock is used for long time intervals, such as minutes, hours or days.
• A stopwatch is a digital timer that you start and stop by hand. It is used for intervals from a few seconds to a few minutes.
• A stopwatch may show 0.01 s, but you cannot start and stop it that precisely. There is a short delay between seeing an event and pressing the button. This delay is your reaction time.
• For a very short interval, the reaction-time error is large compared with the interval itself.
• For very short intervals, use a digital timer connected to light gates. A light gate has a beam of light that shines onto a sensor. The moving object breaks the beam, and this starts or stops the timer automatically. There is no reaction time, so the reading is more accurate.
• Check that the timer reads zero before you start it.

Measuring multiples

• Some quantities are too small, or too short in time, to measure accurately on their own. Measure many of them together, then divide by the number.
• A small distance: to find the thickness of one sheet of paper, measure the thickness of a stack of 100 sheets with a ruler. Then divide by 100.
• A short time: an oscillation is one complete swing of a pendulum: from one side, to the other side, and back again.
• The period of a pendulum is the time for one complete oscillation.
• Time 20 oscillations with a stopwatch. Then divide the time by 20 to get the period.
• This works because your reaction time affects only the start and the stop. The uncertainty of a reading is how far it may be from the true value. The typical uncertainty from starting and stopping the stopwatch is about the same whether you time 1 oscillation or 20.
• Dividing the total time by 20 therefore makes this uncertainty in one period about 20 times smaller.
• Repeat the whole measurement several times. Then calculate the mean (average). This reduces the effect of single readings that are a little too high or too low.
• Good practice when timing a pendulum:
  • Put a fixed marker (a fiducial marker) at the centre of the swing. Start and stop the timer when the bob passes the marker. The bob moves fastest at the centre, so the moment it passes is easiest to judge.
  • Say “zero” as you start the timer. Count “one” when the bob next passes the marker going the same way.
  • Release the bob from a small angle, and use the same angle each time.
• Extra detail: for small angles, the period hardly depends on the angle. It depends mainly on the length of the pendulum.

Scalars and vectors Extended

• A scalar quantity has a size, called its magnitude, but no direction. Extended
• A vector quantity has both a magnitude and a direction. Extended
• Scalars: distance, speed, time, mass, energy, temperature. Extended
• Vectors: force, weight, velocity, acceleration, momentum, electric field strength, gravitational field strength. Extended
• Learn the pairs: speed is a scalar but velocity is a vector; mass is a scalar but weight is a vector. Extended
• A vector is drawn as an arrow. The length of the arrow shows the magnitude. The arrow points in the direction of the vector. Extended
• The resultant of two vectors is the single vector that has the same effect as the two vectors together. Extended
• In this syllabus, you find the resultant of two vectors at right angles only for forces or velocities. Extended
• By calculation: use Pythagoras’ theorem for the size of the resultant. Use tan θ to find its direction. Extended
• Graphically (by scale drawing): Extended
  1. Choose a scale, for example 1 cm represents 2 N. Write the scale down.
  2. Draw the first vector as an arrow of the right length and direction.
  3. From the tip of the first arrow, draw the second arrow at right angles.
  4. Draw the resultant from the start of the first arrow to the tip of the second arrow.
  5. Measure the length of the resultant with a ruler. Use the scale to change it into newtons or m/s.
  6. Measure the angle with a protractor to give the direction.
• Check your answer: the resultant of two vectors at right angles is always larger than either vector, but smaller than their sum. Extended

Common mistakes

• Students read the measuring cylinder at the top of the curved surface, or look down from above. / The mark scheme wants: eye level with the liquid surface, reading the bottom of the meniscus.
• Students time a single oscillation and use it as the period. / The mark scheme wants: time many oscillations (for example 20), divide by the number, then repeat and take the mean.
• Students say “one” as they start the stopwatch, so they time one oscillation fewer than they count. / The mark scheme wants: count from zero at the start.
• Students write that a more precise stopwatch removes reaction time. / The mark scheme wants: reaction time comes from the person, not the stopwatch. Measuring multiples or using light gates reduces its effect.
• Students list weight or velocity as scalars. / The mark scheme wants: weight and velocity are vectors, because they have a direction. Extended
• Students add two forces at right angles, for example 3 N + 4 N = 7 N. / The mark scheme wants Pythagoras: √(3² + 4²) = 5 N, and a direction. Extended

Exam tips

• Describe how to measure something: name the instrument, say what you read, and say how you avoid an error (eye level, zero mark, bottom of the meniscus).
• Explain why you measure multiples: the uncertainty from starting and stopping is about the same for 1 or 20 oscillations. Dividing by 20 shares it among all the oscillations, so the uncertainty in one period is much smaller.
• When you calculate a period, show the division by the number of oscillations. Give the answer with the unit, s.
• State whether a quantity is a scalar or a vector, and give the reason: “it has direction” or “it has magnitude only”. Extended
• Determine a resultant: give its size with a unit and its direction as an angle from a named direction. Extended
• For a scale drawing, write your scale, make the diagram large, and measure carefully with a ruler and a protractor. Extended
• A typical 1-mark answer: “The period is the time for one complete oscillation.”
• A typical 2-mark answer to “How do you find the period of a pendulum accurately?”: “Time 20 oscillations with a stopwatch (1). Divide the total time by 20 (1).”
• A typical 2-mark answer to “Describe how to read the volume of water in a measuring cylinder”: “Put your eye level with the water surface (1). Read the scale at the bottom of the meniscus (1).”
• A typical 1-mark answer: “A vector quantity has both magnitude and direction.” Extended