Questions on SI base and derived units

Question 1

Name the six SI base quantities used at A-level and the SI base unit of each.

Accepted Answer

Mass (kilogram, kg) and length (metre, m) (1); time (second, s) and electric current (ampere, A) (1); temperature (kelvin, K) and amount of substance (mole, mol) (1). One mark for each two correct pairs.

Question 2

Convert (a) 5.0 km into metres and (b) 2.5 GHz into hertz.

Accepted Answer

(a) 5.0 km = 5000 m, that is 5.0 × 10³ m (1) (b) 2.5 GHz = 2.5 × 10⁹ Hz (1)

Question 3

Express the newton (N) in terms of SI base units.

Accepted Answer

From F = ma (1), 1 N = 1 kg m s⁻² (1).

Question 4

A capacitor is labelled 85 pF and a nuclear radius is quoted as 2.4 fm. Write (a) 85 pF in farads and (b) 2.4 fm in metres, each in standard form.

Accepted Answer

(a) 85 pF = 8.5 × 10⁻¹¹ F (1). (b) 2.4 fm = 2.4 × 10⁻¹⁵ m (1).

Question 5

A photon of red light has an energy of 3.2 × 10⁻¹⁹ J. 1 eV = 1.60 × 10⁻¹⁹ J. Calculate the photon energy in electronvolts.

Accepted Answer

Energy in eV = energy in J ÷ (1.60 × 10⁻¹⁹) (1) = 3.2 × 10⁻¹⁹/(1.60 × 10⁻¹⁹) = 2.0 eV (1).

Question 6

A student records the extension of a spring, measured in millimetres, for a series of loads measured in newtons. Write the heading of each column of the results table, and state what the entries in a column headed in this way are.

Accepted Answer

Headings: load / N and extension / mm, the quantity divided by its unit (1). Dividing a quantity by its unit leaves a pure number, so every entry is a bare number and no unit is written beside it (1).

Question 7

Express (a) the joule and (b) the watt in SI base units.

Accepted Answer

(a) J = N m (1) = kg m² s⁻² (1). (b) W = J s⁻¹ = kg m² s⁻³ (1).

Question 8

Convert (a) 3.0 mA into amperes, (b) 450 nm into metres and (c) 12 MΩ into ohms.

Accepted Answer

(a) 3.0 mA = 3.0 × 10⁻³ A (1). (b) 450 nm = 4.5 × 10⁻⁷ m (1). (c) 12 MΩ = 1.2 × 10⁷ Ω (1).

Question 9

Use base units to check whether the equation v² = 2as is homogeneous (dimensionally consistent).

Accepted Answer

LHS: (m s⁻¹)² = m² s⁻² (1) RHS: a × s = (m s⁻²)(m) = m² s⁻² (1) Both sides have the same units, so the equation is homogeneous (1)

Question 10

Express the pascal (Pa) in SI base units.

Accepted Answer

Pa = N m⁻² = kg m s⁻² / m² (1) = kg m⁻¹ s⁻² (1).

Question 11

A domestic oven transfers energy at a rate of 2.4 kW. It runs for 90 minutes. Calculate the energy transferred (a) in kilowatt hours and (b) in joules. (1 kW h = 3.6 × 10⁶ J.)

Accepted Answer

(a) 90 minutes = 1.5 h (1), so energy = 2.4 × 1.5 = 3.6 kW h (1). (b) Energy = 3.6 × 3.6 × 10⁶ = 1.3 × 10⁷ J (1).

Question 12

The cross-sectional area of a copper track on a circuit board is 0.020 mm². Express this area in m², in standard form.

Accepted Answer

1 mm² = (10⁻³ m)² = 10⁻⁶ m²: the prefix must be squared with the unit (1). 0.020 mm² = 2.0 × 10⁻⁸ m² (1).

Question 13

Four lasers emit light of wavelength 580 nm, 0.65 μm, 6.2 × 10⁻⁷ m and 0.00071 mm. By converting each wavelength to metres in standard form, place the four in order of increasing wavelength.

Accepted Answer

580 nm = 5.8 × 10⁻⁷ m and 0.65 μm = 6.5 × 10⁻⁷ m (1); 0.00071 mm = 7.1 × 10⁻⁷ m (1). Order: 580 nm < 6.2 × 10⁻⁷ m < 0.65 μm < 0.00071 mm (1).

Question 14

The period of a simple pendulum is given by T = 2π√(l/g), where l is a length and g is the gravitational field strength in m s⁻². Show that the unit of √(l/g) is the second.

Accepted Answer

Unit of l/g = m/(m s⁻²) (1) = s² (1). Taking the square root gives the unit of √(l/g) as s, so T is in seconds (2π has no unit) (1).

Question 15

A trolley runs down a ramp and its speed, in m s⁻¹, is measured at intervals of time, in seconds. The speed is then plotted against the time. Write the label for each axis of the graph, and use the two labels to deduce the unit of the gradient.

Accepted Answer

Vertical axis: speed / m s⁻¹; horizontal axis: time / s (1). The gradient is the vertical label divided by the horizontal one, so its unit is (m s⁻¹)/s (1) = m s⁻², which is an acceleration (1).

Question 16

The gravitational constant G appears in F = GMm/r². Determine the SI base units of G.

Accepted Answer

Rearranging, G = Fr²/(Mm) (1) Units = (kg m s⁻² × m²)/(kg × kg) (1) = m³ kg⁻¹ s⁻² (1)

Question 17

Convert (a) a density of 2.7 g cm⁻³ into kg m⁻³ and (b) a speed of 90 km h⁻¹ into m s⁻¹.

Accepted Answer

(a) 2.7 g cm⁻³ × 1000 (1) = 2700 kg m⁻³ (1) (b) 90 km h⁻¹ = (90 × 1000)/3600 (1) = 25 m s⁻¹ (1)

Question 18

Explain why checking that an equation is homogeneous (has matching units on both sides) is a useful way to spot mistakes.

Accepted Answer

A correct physical equation must have the same units on both sides (1). If the units do not match, the equation must be wrong (1), so a units check quickly catches errors such as a wrong power or a wrongly placed quantity, before any numbers are used (1).

Question 19

A pulsed laser delivers pulses of energy 2.5 mJ, each lasting 6.0 ns. Calculate the average power of one pulse in watts, and express your answer using an appropriate SI prefix.

Accepted Answer

Convert both prefixes: 2.5 mJ = 2.5 × 10⁻³ J and 6.0 ns = 6.0 × 10⁻⁹ s (1). P = E/t = 2.5 × 10⁻³/(6.0 × 10⁻⁹) = 4.2 × 10⁵ W (1) = about 0.42 MW (or 420 kW) (1).

Question 20

A student cannot remember whether the speed of a wave on a stretched string is v = √(T/μ) or v = √(μ/T), where T is the tension in newtons and μ is the mass per unit length in kg m⁻¹. Deduce, using base units, which of the two equations could be correct.

Accepted Answer

Unit of T/μ = (kg m s⁻²)/(kg m⁻¹) = m² s⁻² (1), so √(T/μ) has unit m s⁻¹, the unit of speed (1). √(μ/T) would have unit s m⁻¹, so only v = √(T/μ) can be correct (1).

Question 21

An electricity supplier charges 28p per kilowatt hour. A 3.0 kW immersion heater runs for 2.0 hours each day. Determine the cost of running the heater for a 30-day month.

Accepted Answer

Energy each day = 3.0 × 2.0 = 6.0 kW h (1). Energy in the month = 6.0 × 30 = 180 kW h (1). Evidence of cost = 180 × 28p (1); cost = £50.40 (1).