Processing results: units, prefixes and powers of ten

Converting before you calculate

Convert every value to SI base units first:

• mm to m: x 10^-3; mm² to m²: x 10^-6; mm³ to m³: x 10^-9.
• cm to m: x 10^-2; cm² to m²: x 10^-4; cm³ to m³: x 10^-6.
• g to kg: x 10^-3. nm to m: x 10^-9. μF to F: x 10^-6. GPa to Pa: x 10⁹.

A squared or cubed unit needs the factor squared or cubed.

Classic traps

• Using the radius formula with a diameter: A = πd² / 4 = πr².
• Dividing again by the number of items when the total mass and total size are already used (the 500 sheets mistake).
• Using power for energy: input energy = VIt, output energy = mgh. Mix them and the efficiency is meaningless.
• An efficiency above 100% means a mistake.

Showing units

To show that a quantity has a given unit, substitute the units of every quantity in the equation, use powers correctly, and simplify step by step. For η = 2r^2g(ρ_s - ρ_l) / 9v: m² x m s^-2 x kg m^-3 / (m s^-1) = kg m^-1 s^-1, which is Pa s.

Worked example

A motor connected to 3.0 V draws 0.40 A for 2.4 s while lifting a 0.150 kg mass through 0.80 m. Find its efficiency.

1. Input energy = VIt = 3.0 x 0.40 x 2.4 = 2.88 J
2. Useful output = mgh = 0.150 x 9.81 x 0.80 = 1.177 J
3. Efficiency = 1.177 / 2.88 x 100 = 40.9%

Answer: Efficiency = 40.9% (2 or 3 s.f.)

Check yourself

1. A wire has diameter 0.38 mm. Find its cross-sectional area, in units of 10^-7 m², to 3 s.f.
   Show answer

   1.13 x 10^-7 m². A = π(0.38 x 10^-3)² / 4 = 1.13 x 10^-7 m².

2. A block measures 2.50 cm x 4.00 cm x 1.20 cm and has a mass of 32.4 g. Find its density in kg m^-3.
   Show answer

   2700 kg m^-3. V = 0.0250 x 0.0400 x 0.0120 = 1.20 x 10^-5 m³; ρ = 0.0324 / 1.20 x 10^-5 = 2700 kg m^-3.

3. A motor on 6.0 V draws 0.50 A for 4.0 s and lifts a 0.40 kg mass through 1.5 m. Find its efficiency as a percentage, to 3 s.f. (g = 9.81 N/kg)
   Show answer

   49.1 %. Input = 6.0 x 0.50 x 4.0 = 12 J; output = 0.40 x 9.81 x 1.5 = 5.886 J; efficiency = 49.1%.

4. Convert 450 nm into metres, in units of 10^-7 m.
   Show answer

   4.5 x 10^-7 m. 450 x 10^-9 m = 4.5 x 10^-7 m.

5. Convert 25.0 cm³ into m³, in units of 10^-5 m³.
   Show answer

   2.50 x 10^-5 m³. 25.0 x 10^-6 m³ = 2.50 x 10^-5 m³.

6. Which is the pascal second (Pa s) written in SI base units?
   • kg m s^-1
   • kg m^-2 s^-1
   • kg m^-1 s^-2
   • kg m^-1 s^-1
   Show answer

   kg m^-1 s^-1. Pa = N m^-2 = kg m^-1 s^-2; multiplying by s gives kg m^-1 s^-1.

7. A student calculates the efficiency of a motor as 140%. What is the most likely cause?
   • The motor is very well designed
   • A mistake, such as mixing power and energy or a wrong power of ten
   • Friction added energy
   • The mass was lifted too slowly
   Show answer

   A mistake, such as mixing power and energy or a wrong power of ten. No device can be more than 100% efficient, so a calculation error is certain.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson examiners' report: IAL Physics WPH13/01, January 2024
• Pearson examiners' report: IAL Physics WPH16/01, June 2024

Last updated 8 November 2026. Spotted a mistake? Tell us.