Cambridge IGCSE0625

Physical quantities and measurement techniques

Physics 0625 Chapter Notes

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Physical quantities and measurement techniques
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In this chapter

  1. 1Physical Quantities, Units and Prefixes
  2. 2Measuring Length and Time Accurately
  3. 3Determining the Volume of Objects
  4. 4Understanding and Calculating Density
  5. 5Handling Data and Drawing Graphs

1. Physical Quantities, Units and Prefixes

In physics, every measurement is a 'physical quantity', which has two parts: a number (its magnitude) and a unit. For example, a length of 10 metres. To avoid confusion, scientists worldwide use a standard set of units called the SI system (Système International d'Unités). The base SI units for the most common quantities are the metre(m) for length, the kilogram (kg) for mass, and the second(s) for time. For very large or very small quantities, we use prefixes. A prefix is a multiplier. For example, 'kilo-'(k) means 1000, so 1 kilometre (km) is 1000 metres. 'Centi-'(c) means 1/100, so 1 centimetre (cm) is 0.01 metres. 'Milli-'(m) means 1/1000, so 1 millimetre (mm) is 0.001 metres.

Key term

Physical Quantity: A property of a material or system that can be quantified by measurement, consisting of a magnitude and a unit.

Examiner insight

Examiners frequently award marks for correctly converting units (e.g., cm to m or g to kg) as the first step in a calculation question.

Common pitfall

Forgetting that the base SI unit for mass is the kilogram (kg), not the gram (g), which is crucial for equations like F=ma or E=mc².

Fun fact

The metre was originally defined in the 1790s as one ten-millionth of the distance from the Earth's equator to the North Pole.

Worked example 12 marks

A student measures the mass of a textbook to be 2350 g and its width to be 21 cm. Convert these measurements into the standard SI base units.

  1. 1

    Step 1: Convert the mass from grams(g) to kilograms (kg). The SI base unit for mass is the kilogram.

  2. 2

    There are 1000 g in 1 kg. To convert from g to kg, we divide by 1000.

  3. 3

    Mass in kg = 2350 g / 1000 = 2.35 kg.

  4. 4

    Step 2: Convert the width from centimetres (cm) to metres (m). The SI base unit for length is the metre.

  5. 5

    There are 100 cm in 1 m. To convert from cm to m, we divide by 100.

  6. 6

    Width in m = 21 cm / 100 = 0.21 m.

  7. 7

    Answer: The mass is 2.35 kg and the width is 0.21 m.

Recap

  • A physical quantity always includes both a number and a unit.
  • The SI system provides standard units for scientific measurements.
  • The SI base unit for length is the metre (m), for mass is the kilogram (kg), and for time is the second (s).
  • Prefixes are used to scale units for very large or small measurements, e.g., kilo (x1000) and milli (x0.001).
  • Always check and convert units to be consistent before starting a calculation.

Quick check

  1. How many millilitres (mL) are in a 2.5 litre (L) bottle of water?1 mark
  2. State the SI base unit for mass.1 mark

2. Measuring Length and Time Accurately

Accurate measurement is key in physics. For length, a standard metre rule can measure to the nearest millimetre (mm). To avoid 'parallax error', you must position your eye directly above the marking on the scale. Parallax error is the apparent shift in position of an object when viewed from different angles. For time, a digital stopwatch is used. However, human reaction time introduces error when measuring very short intervals. To measure the period of an oscillation (like a swinging pendulum) accurately, you should time a large number of complete swings (e.g., 20 or 50), and then divide the total time by the number of swings. This method minimizes the effect of reaction time error on the final result. A complete swing or oscillation is from one side, to the other, and back to the start.

Period (T) = Total time for N oscillations / Number of oscillations (N)

Key term

Parallax Error: An apparent shift in the position of an object when viewed from different angles, which can lead to inaccurate readings from a scale.

Examiner insight

For pendulum experiments, marks are consistently awarded for the method of timing multiple oscillations and then dividing to find the period, as this demonstrates an understanding of how to reduce random error.

Common pitfall

Measuring the time for just one swing of a pendulum. This is highly inaccurate because the human reaction time error in starting and stopping the watch is a large fraction of the total time measured.

Worked example 12 marks

A student measures the time for 25 complete oscillations of a simple pendulum. The stopwatch reading is 40.0 s. Calculate the period (T) of the pendulum.

  1. 1

    Step 1: Identify the given values. Number of oscillations (N) = 25. Total time(t) = 40.0 s.

  2. 2

    Step 2: State the formula for the period, T.

  3. 3

    T = Total time / Number of oscillations.

  4. 4

    Step 3: Substitute the values into the formula and calculate.

  5. 5

    T = 40.0 s / 25 = 1.6 s.

  6. 6

    Step 4: State the final answer with the correct unit.

  7. 7

    The period of the pendulum is 1.6 s.

Worked example 23 marks

Describe how to measure the length of a pendulum accurately, mentioning one potential source of error and how to avoid it.

  1. 1

    Step 1: Place a metre rule alongside the pendulum string, ensuring the zero mark is at the point of suspension (the clamp).

  2. 2

    Step 2: Measure the length from the point of suspension to the centre of the pendulum bob.

  3. 3

    Step 3: A potential source of error is parallax error. To avoid this, position your eye level with the measurement on the rule to ensure you are looking at the scale perpendicularly.

Recap

  • Always view a measurement scale from directly in front (perpendicularly) to avoid parallax error.
  • A metre rule is used to measure length to the nearest millimetre.
  • To measure the period of a pendulum, time a large number of oscillations (e.g., 20) and divide the total time by the count.
  • This averaging technique reduces the impact of random errors like human reaction time.

Quick check

  1. What is the name of the error caused by viewing a ruler's scale from an angle?1 mark
  2. A pendulum swings 50 times in 75 seconds. What is its period?1 mark

3. Determining the Volume of Objects

Volume is the amount of three-dimensional space an object occupies. The method to measure it depends on the object's state and shape. For a liquid, use a measuring cylinder. Place it on a flat surface and read the volume from the bottom of the curved surface, called the 'meniscus', with your eye at the same level. For a regular solid, like a cuboid, measure its length (l), width (w), and height(h) with a ruler and calculate the volume using V = l × w × h. For an irregular solid, like a stone, use the displacement method. Partially fill a measuring cylinder with water and record the initial volume. Carefully submerge the stone completely and record the new, final volume. The volume of the stone is the difference between the final and initial water levels.

Volume of a cuboid (V) = length × width × height

Volume of an irregular object = Final water level - Initial water level

Key term

Displacement: The technique of finding the volume of an object by measuring the volume of fluid it displaces when fully submerged.

Examiner insight

When describing the displacement method, students must state that the object should be fully submerged in the water to get full marks.

Common pitfall

When using a measuring cylinder for displacement, dropping the object in can cause water to splash out, leading to an inaccurate and smaller reading for the object's volume.

Fun fact

The story of Archimedes shouting 'Eureka!' comes from him discovering the principle of displacement while getting into a bath, which he then used to determine if a king's crown was made of pure gold.

Worked example 12 marks

A student wants to find the volume of a small, irregularly shaped rock. They pour 40 cm³ of water into a measuring cylinder. After carefully placing the rock in the cylinder, the water level rises to 58 cm³. What is the volume of the rock?

  1. 1

    Step 1: Identify the initial and final volumes of the water.

  2. 2

    Initial volume = 40 cm³.

  3. 3

    Final volume = 58 cm³.

  4. 4

    Step 2: The volume of the rock is the change in the water level (the volume of water displaced).

  5. 5

    Volume of rock = Final volume - Initial volume.

  6. 6

    Step 3: Calculate the difference.

  7. 7

    Volume of rock = 58 cm³ - 40 cm³ = 18 cm³.

Worked example 22 marks

A rectangular block of metal measures 5.0 cm by 2.0 cm by 3.0 cm. Calculate its volume.

  1. 1

    Step 1: Identify the dimensions of the block.

  2. 2

    Length = 5.0 cm, Width = 2.0 cm, Height = 3.0 cm.

  3. 3

    Step 2: State the formula for the volume of a rectangular block (cuboid).

  4. 4

    Volume = length × width × height.

  5. 5

    Step 3: Substitute the values and calculate.

  6. 6

    Volume = 5.0 cm × 2.0 cm × 3.0 cm = 30.0 cm³.

Recap

  • Measure the volume of a liquid with a measuring cylinder, reading the bottom of the meniscus at eye level.
  • Calculate the volume of a regular solid like a cuboid using V = l × w × h.
  • Find the volume of an irregular solid by submerging it in water and measuring the volume of water it displaces.
  • The unit for volume is often cubic centimetres (cm³) or cubic metres (m³).

Quick check

  1. What is the name for the curved surface of a liquid in a measuring cylinder?1 mark
  2. A box is 10 cm long, 5 cm wide and 4 cm high. What is its volume?1 mark

4. Understanding and Calculating Density

Density is a measure of how much 'stuff' (mass) is packed into a given space (volume). A small, heavy object is very dense, while a large, light object is not. It is defined by the equation: Density = Mass / Volume. The symbol for density is the Greek letter rho (ρ). The standard SI unit for density is kilograms per cubic metre (kg/m³), but grams per cubic centimetre (g/cm³) is also commonly used. To find the density of any object, you need to measure its mass (using a top-pan balance) and its volume (using one of the methods from the previous topic). An object will float in a fluid if its density is less than the fluid's density. For example, wood (density ≈ 0.7 g/cm³) floats on water (density ≈ 1.0 g/cm³), but a rock (density ≈ 3 g/cm³) sinks.

Density (ρ) = Mass (m) / Volume (V)

Key term

Density: The mass per unit volume of a substance, which is a measure of how tightly matter is packed together.

Examiner insight

Candidates often lose marks by using inconsistent units, for example, mixing a mass in kilograms with a volume in cubic centimetres. Always convert units to be consistent before calculating density.

Common pitfall

Incorrectly rearranging the density formula. Remember the triangle: cover ρ to get m/V, cover m to get ρ×V, cover V to get m/ρ.

Worked example 13 marks

A piece of aluminium has a mass of 135 g and a volume of 50 cm³.(a) Calculate the density of aluminium in g/cm³.(b) If the density of water is 1.0 g/cm³, will the aluminium sink or float? Explain your answer.

  1. 1

    Part (a):

  2. 2

    Step 1: State the formula for density.

  3. 3

    Density (ρ) = Mass(m) / Volume (V).

  4. 4

    Step 2: Substitute the given values.

  5. 5

    ρ = 135 g / 50 cm³.

  6. 6

    Step 3: Calculate the density.

  7. 7

    ρ = 2.7 g/cm³.

  8. 8

    Part (b):

  9. 9

    Step 4: Compare the density of aluminium to the density of water.

  10. 10

    The density of aluminium (2.7 g/cm³) is greater than the density of water (1.0 g/cm³).

  11. 11

    Step 5: State the conclusion.

  12. 12

    Therefore, the aluminium will sink.

Worked example 24 marks

Describe the experimental procedure to determine the density of a small, irregularly shaped stone.

  1. 1

    Step 1: Measure the mass of the stone using a top-pan balance and record it in grams (g).

  2. 2

    Step 2: Measure the volume of the stone using the displacement method. Partially fill a measuring cylinder with water and record the initial volume (V₁).

  3. 3

    Step 3: Carefully slide the stone into the cylinder, ensuring it is fully submerged and no water splashes out. Record the new, final volume (V₂).

  4. 4

    Step 4: Calculate the volume of the stone by finding the difference: Volume = V₂ - V₁.

  5. 5

    Step 5: Calculate the density using the formula ρ = mass / volume.

Recap

  • Density is defined as mass per unit volume (ρ = m/V).
  • To find density, you must measure both the mass and volume of the object.
  • Mass is measured with a balance; volume is measured using a ruler and formula (regular) or displacement (irregular).
  • An object sinks if its density is greater than the density of the fluid it is in.
  • Common units for density are g/cm³ and kg/m³.

Quick check

  1. An object has a mass of 200 g and a volume of 25 cm³. What is its density?1 mark
  2. The density of oil is 0.8 g/cm³. Will it float on water (density 1.0 g/cm³)?1 mark

5. Handling Data and Drawing Graphs

Good experimental technique involves handling data correctly. To improve reliability, you should repeat readings and calculate an average. This reduces the effect of random errors. Some instruments have a 'zero error', meaning they don't read zero when they should. For example, a balance might show 0.2 g with nothing on it. This value should be subtracted from all subsequent readings. When recording results, use a sensible number of significant figures, usually matching the precision of your measuring instrument. In many experiments, you will plot a graph. Always label your axes with the quantity and its unit (e.g., 'Load / N'). Plot points as small, neat crosses. Draw a single, thin 'line of best fit' that shows the trend of the data, with roughly an equal number of points on either side of the line. Do not just connect the dots. The gradient (steepness) of a straight-line graph is often important. It's calculated by 'rise over run' (change in y / change in x). Use a large triangle on your best-fit line to calculate it accurately.

Average = Sum of readings / Number of readings

Gradient = Change in y / Change in x = (y₂ - y₁) / (x₂ - x₁)

Key term

Line of Best Fit: A straight line or smooth curve drawn through the centre of a group of data points on a scatter graph to show the underlying relationship.

Examiner insight

When calculating a gradient, examiners expect to see a large triangle drawn on the graph and the coordinates used for the calculation read directly from the best-fit line.

Common pitfall

Calculating the gradient of a graph using two of the original plotted data points, instead of using two points that lie on the line of best fit itself.

Worked example 14 marks

A student investigates how a spring stretches. They record the following data for the extension(e) at different loads (L). Plot a graph of Load (y-axis) against Extension (x-axis) and draw a line of best fit. Data: (e=10mm, L=2.0N), (e=20mm, L=4.0N), (e=30mm, L=5.8N), (e=40mm, L=8.2N), (e=50mm, L=10.0N).

  1. 1

    Step 1: Draw the axes. Label the y-axis 'Load / N' and the x-axis 'Extension / mm'. Choose a sensible scale for each axis that uses at least half of the graph paper.

  2. 2

    Step 2: Plot each data point carefully using a small 'x' or a circled dot.

  3. 3

    Step 3: Observe the trend of the points. They appear to form a straight line.

  4. 4

    Step 4: Use a ruler to draw a single straight line of best fit. The line should pass as close as possible to all points, with an approximately equal number of points above and below the line. Note that the point (30mm, 5.8N) might be slightly off the line.

  5. 5

    Step 5: The line should represent the general trend and does not have to pass through every single point or the origin unless the data suggests it.

Worked example 23 marks

Using the graph from the previous example, calculate the gradient of the line of best fit. State its units.

  1. 1

    Step 1: Choose two points that are far apart ON THE LINE OF BEST FIT, not from the original data points. Let's pick (x₁, y₁) = (10, 2.0) and (x₂, y₂) = (50, 10.0), assuming they lie on our line.

  2. 2

    Step 2: Draw a large triangle on the graph connecting these two points to visualize the 'rise' and 'run'.

  3. 3

    Step 3: State the formula for the gradient.

  4. 4

    Gradient = (y₂ - y₁) / (x₂ - x₁).

  5. 5

    Step 4: Substitute the coordinates from the line.

  6. 6

    Gradient = (10.0 N - 2.0 N) / (50 mm - 10 mm) = 8.0 N / 40 mm.

  7. 7

    Step 5: Calculate the value and determine the units.

  8. 8

    Gradient = 0.2 N/mm.

Recap

  • Repeat readings and calculate an average to improve the reliability of your results.
  • Identify and correct for zero errors on measuring instruments.
  • On a graph, always label axes with the quantity and unit.
  • Draw a single line of best fit to show the data trend; do not connect the dots.
  • Calculate the gradient using a large triangle drawn on your line of best fit, not using plotted points.

Quick check

  1. When plotting a graph, what two things must every axis label include?2 marks
  2. What is the main reason for taking repeat readings and calculating an average?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A student measures the time for 50 swings of a long pendulum to be 1 minute and 15 seconds. Calculate the period of the pendulum in seconds.3 marks
  2. A rectangular block of perspex has dimensions 2.0 cm x 4.0 cm x 5.0 cm. Its mass is 48.0 g. Calculate the density of perspex.3 marks
  3. Describe, with the aid of diagrams if you wish, how you would use a measuring cylinder and a top-pan balance to find the density of a liquid like olive oil.4 marks
  4. A micrometer screw gauge has a zero error of +0.02 mm. It is used to measure the diameter of a wire, and the reading on the scale is 0.76 mm. What is the actual diameter of the wire?1 mark
  5. A rock has a volume of 60 cm³ and a density of 2.5 g/cm³. Calculate the mass of the rock in grams.2 marks
  6. A student is finding the density of an irregularly-shaped piece of metal. List the two physical quantities they must measure and the instrument used for each measurement.4 marks
  7. Explain why it is better to measure the thickness of 100 sheets of paper and divide by 100, rather than measuring the thickness of a single sheet.2 marks
  8. A block of ice has a mass of 460 g. The density of ice is 0.92 g/cm³. Calculate the volume of the ice block.2 marks
  9. A student plots a graph of voltage (V) on the y-axis against current (I) on the x-axis. They draw a line of best fit and calculate the gradient. The points used from the line are (0.2 A, 3.0 V) and (0.8 A, 9.0 V). Calculate the gradient of the graph and state its units.3 marks
  10. A cube of a certain metal has sides of length 5.0 cm. The mass of the cube is 962.5 g. Calculate the density of the metal in g/cm³. Would this cube float or sink in mercury, which has a density of 13.6 g/cm³? Justify your answer.4 marks

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