List the six SI base quantities required at this level and give the unit of each.
Explain the difference between a base unit and a derived unit.
State what it means for a physical equation to be homogeneous.
Express each of the following in SI base units.
Show that the equation v2 = u2 + 2 a s is homogeneous.
A force F is related to an extension x by F = k x. Find the SI base units of the constant k.
(Challenge) The gravitational force between two masses is F = G m1 m2 / r2. Find the SI base units of the gravitational constant G.
A learner writes the kinetic energy of a body as E = m v2. Show that this equation is homogeneous, then explain why it is still not correct. What does this tell you about the limits of a homogeneity check?
Total: 32 marks. Original work by the TheLucidSTEM team. Written in the style of the papers; no past paper question is reproduced.
Answer key · full worked solutionsclick to reveal
mass (kg), length (m), time (s), electric current (A), thermodynamic temperature (K), amount of substance (mol).
a base unit is one of the fixed SI units from which others are built; a derived unit is a combination of base units (for example the newton, kg m s−2).
an equation is homogeneous when every term has the same SI base units, so the two sides are dimensionally consistent.
N = kg m s−2; J = kg m2 s−2; W = kg m2 s−3; Pa = kg m−1 s−2; C = A s; V = kg m2 s−3 A−1.
C1. 2.5 GW = 2.5 × 109 W.
C2. 470 nm = 470 × 10−9 m = 4.7 × 10−7 m.
C3. 0.05 ms = 0.05 × 10−3 s = 5 × 10−5 s.
C4. 6 500 000 Hz = 6.5 × 106 Hz = 6.5 MHz.
v2: (m s−1)2 = m2 s−2.
u2: (m s−1)2 = m2 s−2.
2 a s: (m s−2)(m) = m2 s−2 (the 2 is a pure number).
all three terms are m2 s−2, so the equation is homogeneous.
k = F / x = (kg m s−2) / m = kg s−2.
rearrange: G = F r2 / (m1 m2).
base units: (kg m s−2)(m2) / (kg2) = m3 kg−1 s−2.
m v2 has base units kg × (m s−1)2 = kg m2 s−2, which equals the base units of energy, so the equation is homogeneous. It is still wrong because the correct kinetic energy is ½ m v2; the equation is missing the factor of one half. A homogeneity check confirms the units match but cannot detect a wrong pure number, so it is necessary but not sufficient for an equation to be correct.