Cambridge AS & A Level9702

Physical quantities and units

Physics 9702 Chapter Notes

What this chapter covers

Physical quantities and units - Physical quantitiesPhysical quantities and units - SI unitsPhysical quantities and units - Errors and uncertaintiesPhysical quantities and units - Scalars and vectors
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In this chapter

  1. 1Physical Quantities and SI Units
  2. 2SI Base and Derived Units
  3. 3Checking Equations with Homogeneity
  4. 4Scientific Notation and Prefixes
  5. 5Estimation and Orders of Magnitude

1. Physical Quantities and SI Units

In physics, a physical quantity is any property that can be measured. It's not enough to just give a number; every measurement must have two parts: a numerical value (its magnitude) and a unit. For example, stating a mass is '10' is meaningless. Is it 10 grams, 10 kilograms, or 10 tonnes? Writing 'mass = 10 kg' is clear and complete. To ensure scientists and engineers worldwide can compare their results without confusion, a global standard system of units is used. This is called the Système Internationale d’Unités, or SI system. Using correct SI units is a fundamental skill in physics.

Key term

Physical Quantity: A property of an object or phenomenon that can be measured and expressed as a numerical magnitude and a unit.

Examiner insight

Examiners will penalise answers that omit or use incorrect units for a final calculated value. Getting the number right is only half the job.

Common pitfall

A very common mistake is to perform a calculation and write down only the numerical answer, forgetting to include the correct unit. This will always lose marks in an exam.

Fun fact

The original standard for the metre was defined in 1793 as one ten-millionth of the distance from the Earth's equator to the North Pole, a definition that required a huge surveying effort across France.

Worked example 12 marks

A student measures the time taken for a ball to drop from a height. They record the height as 2.50 m and the time as 0.71 s. Identify the physical quantities measured and state their magnitudes and units.

  1. 1

    Step 1: Identify the first measurement. The student measured the height.

  2. 2

    Step 2: State the physical quantity, its magnitude, and its unit. The quantity is length (or height), the magnitude is 2.50, and the unit is the metre (m).

  3. 3

    Step 3: Identify the second measurement. The student measured the time.

  4. 4

    Step 4: State the physical quantity, its magnitude, and its unit. The quantity is time, the magnitude is 0.71, and the unit is the second (s).

Recap

  • A physical quantity consists of a numerical magnitude and a unit.
  • Omitting the unit from a measurement makes it incomplete and scientifically invalid.
  • The SI system (Système Internationale d’Unités) is the standard system of units used globally in science.
  • Always write the unit after the numerical value.

Quick check

  1. What are the two essential components of any physical quantity measurement?1 mark

2. SI Base and Derived Units

The entire SI system is built upon seven fundamental quantities called base quantities. Each has a corresponding base unit. The three you will use most in mechanics are the metre(m) for length, the kilogram (kg) for mass, and the second(s) for time. The other four are the ampere (A) for electric current, the kelvin (K) for temperature, the mole (mol) for the amount of substance, and the candela (cd) for luminous intensity (which is not required for A-Level Physics). All other physical quantities have units that are combinations of these seven base units. These are called derived units. For example, speed is distance/time, so its unit is m/s (or m s⁻¹). Some derived units are used so often they are given a special name, like the newton (N) for force or the joule (J) for energy. A key skill is to be able to break down these special names into their fundamental base units.

speed = distance / time → unit: m s⁻¹

acceleration = change in velocity / time → unit: m s⁻²

force = mass × acceleration → unit: kg m s⁻² (which is the newton, N)

energy = force × distance → unit: kg m² s⁻² (which is the joule, J)

Key term

Base Unit: One of the seven fundamental units of measurement in the SI system from which all other units are derived.

Examiner insight

Questions often require you to express a derived unit, like the volt or the pascal, in terms of its fundamental SI base units. Practice this skill until it is second nature.

Common pitfall

Confusing base quantities (e.g., mass) with base units (e.g., kilogram). Mass is the concept, kilogram is the unit.

Worked example 13 marks

The unit of pressure is the pascal (Pa). One pascal is defined as a force of one newton per square metre (N m⁻²). Express the pascal in terms of SI base units.

  1. 1

    Step 1: Write down the definition of the pascal in terms of other units. Pa = N / m².

  2. 2

    Step 2: Express the newton (N) in terms of SI base units. We know Force = mass × acceleration (F=ma).

  3. 3

    Step 3: Substitute the base units for mass (kg), and acceleration (m s⁻²). So, N = kg m s⁻².

  4. 4

    Step 4: Substitute this expression for the newton back into the equation for the pascal. Pa = (kg m s⁻²) / m².

  5. 5

    Step 5: Simplify the expression by cancelling units. Pa = kg m⁻¹ s⁻².

Worked example 24 marks

The unit of electrical potential difference is the volt (V). A volt is defined as a joule per coulomb (J C⁻¹). Given that a coulomb is an ampere-second (A s), express the volt in SI base units.

  1. 1

    Step 1: Write the definition of the volt in terms of other units. V = J / C.

  2. 2

    Step 2: Express the joule (J) in base units. Energy (joule) = Force × distance = (kg m s⁻²) × m = kg m² s⁻².

  3. 3

    Step 3: Express the coulomb (C) in base units. C = A s.

  4. 4

    Step 4: Substitute the base unit expressions for J and C into the equation for the volt. V = (kg m² s⁻²) / (A s).

  5. 5

    Step 5: Simplify the expression. V = kg m² s⁻³ A⁻¹.

Recap

  • There are seven SI base units: m, kg, s, A, K, mol, cd.
  • All other units are called derived units and are combinations of base units.
  • Common derived units like the newton (N) and joule (J) can be expressed in terms of base units.
  • To find the base units of a quantity, use its defining equation.

Quick check

  1. List the three SI base units used in mechanics.1 mark
  2. Is the joule (J) a base unit or a derived unit?1 mark

3. Checking Equations with Homogeneity

The principle of homogeneity is a powerful tool for checking your work in physics. It states that for any valid physical equation, the base units on both sides of the equals sign must be identical. Furthermore, in equations involving addition or subtraction (e.g., A = B + C), every term (A, B, and C) must have the same units. You cannot add a length to a time! This principle can be used to: 1) Check if an equation you have written or remembered is correct. 2) Determine the units of an unknown constant within an equation.

Principle of Homogeneity: For an equation to be valid, [LHS units] = [RHS units].

Key term

Homogeneity: The principle stating that for a physical equation to be dimensionally consistent, all terms being added or equated must have the same base units.

Examiner insight

In homogeneity questions, examiners look for clear working where you analyse each term in the equation separately before comparing them. Lay out your work logically.

Common pitfall

Forgetting that in an equation like A = B + C, you must check that the units of A, B, and C are all identical, not just that A's units equal the combined units of B+C.

Worked example 13 marks

One of the equations of motion is s = ut + ½at². Show that this equation is homogeneous. (s = displacement, u = initial velocity, t = time, a = acceleration).

  1. 1

    Step 1: Determine the base units of the term on the left-hand side (LHS). The term is displacement, s. Its unit is the metre (m). So, [LHS] = m.

  2. 2

    Step 2: Determine the base units of the first term on the right-hand side (RHS), which is 'ut'. Velocity 'u' is in m s⁻¹ and time 't' is in s. So, [ut] = (m s⁻¹) × s = m.

  3. 3

    Step 3: Determine the base units of the second term on the RHS, which is '½at²'. The constant '½' has no units. Acceleration 'a' is in m s⁻² and time squared 't²' is in s². So, [½at²] = (m s⁻²) × s² = m.

  4. 4

    Step 4: Compare the units of all terms. The LHS has units of m. Both terms on the RHS have units of m. Since all terms have the same units, the equation is homogeneous.

Worked example 24 marks

The period T of a simple pendulum is thought to depend on its length l and the acceleration of free fall g. The proposed equation is T = k * l^x * g^y, where k is a dimensionless constant. By considering the homogeneity of the equation, find the values of x and y.

  1. 1

    Step 1: Write down the base units for each quantity. [T] = s. [l] = m. [g] = m s⁻². k is dimensionless.

  2. 2

    Step 2: Substitute the base units into the equation. s = (m)^x * (m s⁻²)^y.

  3. 3

    Step 3: Group the units on the RHS by base unit type (m and s). s = m^x * m^y * (s⁻²)^y = m^(x+y) * s^(-2y).

  4. 4

    Step 4: Equate the powers of each base unit on the LHS and RHS. For seconds (s): The power on the LHS is 1. The power on the RHS is -2y. So, 1 = -2y, which gives y = -1/2.

  5. 5

    Step 5: For metres (m): The power on the LHS is 0 (as m is not present). The power on the RHS is x+y. So, 0 = x + y.

  6. 6

    Step 6: Substitute the value of y into the equation for m. 0 = x + (-1/2). This gives x = 1/2. So the formula is T = k * l^(1/2) * g^(-1/2) or T = k * sqrt(l/g).

Recap

  • An equation is homogeneous if the base units are the same on both sides.
  • In sums or differences, every individual term must have the same units.
  • Use homogeneity to check if an equation you've recalled is dimensionally correct.
  • You can use homogeneity to find the units of a constant or the powers in a formula.

Quick check

  1. If an equation is homogeneous, does that guarantee it is physically correct?1 mark

4. Scientific Notation and Prefixes

Physics deals with quantities that can be astronomically large (like the distance to a star) or incredibly small (like the diameter of an atom). Writing these numbers with lots of zeros is clumsy and prone to error. We use scientific notation (or standard form) to express them as a number between 1 and 10 multiplied by a power of 10. For example, the speed of light is 300,000,000 m s⁻¹ which is written as 3.0 × 10⁸ m s⁻¹. To make writing units easier, we also use standard prefixes which represent specific powers of 10. For example, 'kilo'(k) means 10³, so 5 km is 5 × 10³ m. 'Micro' (µ) means 10⁻⁶, so 12 µs is 12 × 10⁻⁶ s. It is crucial to convert all prefixed units into their standard SI base unit equivalent before starting a calculation.

tera (T): 10¹²

giga (G): 10⁹

mega (M): 10⁶

kilo (k): 10³

centi (c): 10⁻²

milli (m): 10⁻³

micro (µ): 10⁻⁶

nano (n): 10⁻⁹

pico (p): 10⁻¹²

Key term

Prefix: A multiplier placed before a unit to indicate a multiple or submultiple of that unit, usually representing a specific power of ten.

Examiner insight

Examiners often set questions that require careful conversion of prefixes. Double-check that you have converted every quantity to its base SI unit (metres, kilograms, seconds, etc.) before substituting into a formula.

Common pitfall

Errors when dealing with areas and volumes. For example, converting 1 cm² to m²: it's not 10⁻² m², it's (10⁻² m)² = 10⁻⁴ m². Similarly, 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³.

Fun fact

The memory in your phone is measured in gigabytes (GB) and the processor speed in gigahertz (GHz). One gigabyte is 10⁹ bytes, enough to store about 500,000 pages of text!

Worked example 14 marks

The resistance R of a wire is given by R = ρL/A, where ρ is the resistivity of the material, L is its length and A is its cross-sectional area. A copper wire has a resistivity of 1.7 × 10⁻⁸ Ω m, a length of 2.5 km and a diameter of 0.50 mm. Calculate its resistance.

  1. 1

    Step 1: Convert all quantities to standard SI units. Length L = 2.5 km = 2.5 × 10³ m.

  2. 2

    Step 2: Convert the diameter to radius in metres. Diameter = 0.50 mm = 0.50 × 10⁻³ m. Radius r = diameter / 2 = 0.25 × 10⁻³ m.

  3. 3

    Step 3: Calculate the cross-sectional area A using A = πr². A = π × (0.25 × 10⁻³ m)² = π × (6.25 × 10⁻⁸ m²) ≈ 1.963 × 10⁻⁷ m².

  4. 4

    Step 4: Substitute all the values in SI units into the resistance formula. R = (1.7 × 10⁻⁸ Ωm) × (2.5 × 10³m) / (1.963 × 10⁻⁷ m²).

  5. 5

    Step 5: Calculate the final result. R = (4.25 × 10⁻⁵) / (1.963 × 10⁻⁷) Ω ≈ 216.5 Ω. Giving the answer to 2 significant figures as per the data: R = 220 Ω.

Recap

  • Scientific notation (e.g., 3.0 × 10⁸) is used for very large or small numbers.
  • Prefixes are shortcuts for powers of 10 (e.g., k for 10³, µ for 10⁻⁶).
  • Always convert all quantities to their base SI units before performing calculations.
  • Be extra careful when squaring or cubing prefixed units, e.g., (cm)³ to (m)³.

Quick check

  1. Express 4700 nF (nanofarads) in farads (F) using scientific notation.1 mark
  2. Convert an area of 5.0 cm² into m².1 mark

5. Estimation and Orders of Magnitude

In physics, it's vital to have a 'feel' for numbers. Before you do a precise calculation, it's good practice to estimate the answer. This helps you spot mistakes if your final calculated answer is wildly different. An 'order of magnitude' estimate is a rough approximation of a value to the nearest power of 10. For example, the height of a person is closer to 10⁰ m (1m) than 10¹ m (10 m), so its order of magnitude is 10⁰ m. To make good estimates, you need to know some benchmark values. For example, knowing a car travels at about 30 m s⁻¹ on a motorway helps you realise a calculated speed of 3000 m s⁻¹ for a car is impossible. When asked to make an estimate, you must always state the assumptions and values you use.

Key term

Order of Magnitude: An estimate of a quantity given as the nearest power of ten.

Examiner insight

In estimation questions, examiners award more marks for the logical process and reasonable assumptions than for the final answer itself. Clearly lay out the steps of your thinking.

Common pitfall

Giving an estimate with too many significant figures. An estimate is by nature approximate, so an answer like '362.87 kg' is inappropriate; 'about 350 kg' or '400 kg' is better.

Worked example 14 marks

Estimate the number of breaths you will take in a lifetime. State your assumptions clearly.

  1. 1

    Step 1: State assumptions for calculation. Assume an average breathing rate and an average lifespan. Let's assume: Average breathing rate = 15 breaths per minute. Average lifespan = 80 years.

  2. 2

    Step 2: Calculate the number of minutes in a lifetime. Minutes in lifetime = (80 years) × (365 days/year) × (24 hours/day) × (60 minutes/hour).

  3. 3

    Step 3: Perform the calculation for total minutes. Minutes ≈ 80 × 365 × 24 × 60 ≈ 4.2 × 10⁷ minutes.

  4. 4

    Step 4: Calculate the total number of breaths. Total breaths = (Breathing rate) × (Total minutes) = 15 × (4.2 × 10⁷) ≈ 6.3 × 10⁸ breaths.

  5. 5

    Step 5: State the final answer as an estimate. The number of breaths in a lifetime is approximately 600 million. (An order of magnitude estimate would be 10⁹ breaths).

Worked example 23 marks

Estimate the mass of water in a fully filled domestic bathtub. State your assumptions.

  1. 1

    Step 1: State assumptions for the dimensions of a bathtub. A bathtub is a rectangular prism. Let's assume its dimensions are: Length ≈ 1.5 m, Width ≈ 0.6 m, Depth of water ≈ 0.4 m.

  2. 2

    Step 2: Calculate the volume of water. Volume = Length × Width × Depth = 1.5 m × 0.6 m × 0.4 m = 0.36 m³.

  3. 3

    Step 3: State the assumption for the density of water. The density of water is approximately 1000 kg m⁻³.

  4. 4

    Step 4: Calculate the mass of the water using the formula mass = density × volume. Mass = 1000 kg m⁻³ × 0.36 m³ = 360 kg.

  5. 5

    Step 5: State the final answer as a reasonable estimate. The mass of water in the bathtub is approximately 300-400 kg.

Recap

  • Estimation is used to check if a calculated answer is sensible.
  • An order of magnitude is a power-of-10 approximation of a value.
  • Memorise some benchmark values (e.g., density of water, speed of a car).
  • When making an estimate, always state the assumptions and values you are using.
  • An estimate should be given to one or two significant figures only.

Quick check

  1. What is a reasonable estimate for the mass of an adult person in kilograms?1 mark
  2. What is the order of magnitude for the height of a standard door in metres?1 mark

End-of-chapter exercise

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

  1. The unit of charge is the coulomb (C). Given that electric current (in amperes, A) is the rate of flow of charge, express the coulomb in terms of SI base units.2 marks
  2. A student measures a current as 40.0 mA and a time as 2.50 µs. Calculate the total charge that has flowed in this time. Give your answer in coulombs (C).3 marks
  3. The universal gravitational constant, G, has a value of 6.67 × 10⁻¹¹ N m² kg⁻². Express the units of G in terms of SI base units only.3 marks
  4. Einstein's famous equation is E = mc². Use the principle of homogeneity to show that the units are consistent. (The unit of energy, E, is the joule, J).4 marks
  5. A spherical water droplet has a radius of 0.50 mm. The density of water is 1000 kg m⁻³. Calculate the mass of the water droplet. (The volume of a sphere is given by V = (4/3)πr³).4 marks
  6. Estimate the number of cans of drink that would be required to fill a small car. State all the assumptions you make in your calculation.4 marks
  7. The pressure P exerted by an ideal gas is given by the equation P = (1/3)ρ<c²>, where ρ is the density of the gas and <c²> is the mean square speed of the gas molecules. Show that this equation is homogeneous.4 marks
  8. The intensity I of a wave is defined as the power P transmitted per unit area A. If the intensity of a laser beam is 2.0 GW cm⁻², convert this value into the standard SI unit W m⁻².3 marks
  9. The energy stored in a capacitor is given by E = ½QV, where Q is the charge stored and V is the potential difference across it. Use this equation and your knowledge of base units to show that the unit of capacitance, the farad (F), defined by C = Q/V, has base units of kg⁻¹ m⁻² s⁴ A².5 marks
  10. A car is travelling at a speed of 72 km h⁻¹. The driver sees an obstacle and applies the brakes, decelerating at 5.0 m s⁻². Calculate the distance the car travels before stopping, using the equation v² = u² + 2as. Ensure you use consistent units.3 marks

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