1.2.2Physical quantities and kinematics
Homogeneity of Physical Equations and Powers of Ten
5 min read 10 parts Lesson 6 of 22 1 quiz
Lesson summary
1.2.2 Homogeneity of Physical Equations and Powers of Ten — AS PHYSICS 9702 notes: Test whether an equation is homogeneous using base units and handle powers of ten in estimates and unit conversions.Homogeneous equation = an equation in which every term has the same base units on…
- Homogeneity is a necessary test of correctness
- A homogeneous equation may still be wrong
- Pure numbers have no units
- Constants like 6π in Stokes' law
- Checking v2 = u2 + 2as
- Finding the unit of a constant
Section 1 of 10Homogeneous equations
1Homogeneous equation
Homogeneous equation an equation in which every term has the same base units on both sides.
2
Only quantities with the same units can be added or subtracted.
3Homogeneity is a necessary test of correctness
- Homogeneity is a necessary test of correctness a non-homogeneous equation is certainly wrong.
4A homogeneous equation may still be wrong
- A homogeneous equation may still be wrong numerical factors such as ½ in kinetic energy have no units and are invisible to the test.
Section 2 of 10Testing an equation for homogeneity
1
Replace each symbol by its SI base units.
2
Simplify the left-hand side and each term on the right-hand side separately.
3
Compare: terms that are added must match each other and the left side.
4
State the conclusion: homogeneous (possibly correct) or not homogeneous (definitely wrong).
Section 3 of 10Quantities with no units
1Pure numbers have no units
- Pure numbers have no units such as ½, 2 and π.
2
Angles in radians have no units.
3
The argument of sin, ln or an exponential must be dimensionless.
4Constants like 6π in Stokes' law
- Constants like 6π in Stokes' law have no units and do not affect the test.
Section 4 of 10Using homogeneity
Using homogeneity
Checking v2
Checking v2 u2 + 2as
- every term has m2 s-2, so it is homogeneous.
Finding the unit of a constant
- Finding the unit of a constant rearrange the equation to make the constant the subject, then substitute base units.
For drag F = kv2, k = F ÷ v2 has units kg m-1.
If a quantity must have no units
- If a quantity must have no units it is dimensionless, as for γ in the speed of sound v = √(γp ÷ ρ), where p ÷ ρ has units m2 s-2.
In an examination
- In an examination show the base units of the left side, the base units of each right-hand term and a statement that they are equal.
Section 5 of 10Dimensional analysis
1
Units alone can suggest how one quantity depends on others.
2For a pendulum period T
- For a pendulum period T depending on l (m), m (kg) and g (m s-2), mass cannot appear because nothing else contains kg to cancel it.
3To obtain seconds from m and m s−2
- T = constant × √(l ÷ g).
4The constant (2π) is found by experiment or theory
- The constant (2π) is found by experiment or theory dimensional analysis cannot give it.
Section 6 of 10Powers of ten
Powers of ten
Physical quantities span a huge range
- Physical quantities span a huge range an atom is about 10-10 m and a galaxy about 1021 m, a difference of 31 powers of ten.
To multiply add the powers
- To multiply add the powers to divide subtract them; to raise to a power multiply the exponents.
To the nearest order of magnitude
- To the nearest order of magnitude round the number to the closest power of ten.
Learn the prefixes
- Learn the prefixes 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.
Wrong prefixes are a very common examination mistake.
Section 7 of 10Typical sizes spanning many orders of magnitude
| Quantity | Size |
|---|---|
| Wavelength of a radio wave | about 102 m |
| Thickness of a human hair | about 10-4 m |
| Atomic nucleus | about 10-15 m |
| Diameter of an atom | about 10-10 m |
| Width of a galaxy | about 1021 m |
Section 8 of 10More from the specification
1Energy dimension derived in joules
- Energy dimension derived in joules the SI derived unit joule represents mechanical work done when a force of one newton acts through one metre.
2Pressure
- Force per unit area, P = F/A.What the symbols mean: p = pressure (Pa = N m−2) · F = force acting at right angles to the surface (N) · A = area (m2)
3Density
- Mass per unit volume.
4Newton
- Unit of force, kg m s-2.
5Pascal
- Unit of pressure, 1 N/m2.
6Derived unit
- A unit formed by multiplying or dividing SI base units.
Section 9 of 10Key terms
| Term | Meaning |
|---|---|
| Homogeneous equation | An equation with the same base units on both sides and in every term. |
| Dimensionless | Having no units, such as a ratio or an angle in radians. |
| Dimensional analysis | Using units to deduce how quantities depend on each other. |
| Standard form | A number written as a × 10n. |
| Order of magnitude | The power of ten closest to a number. |
| Prefix | A symbol such as k, m or μ that multiplies a unit by a power of ten. |
| Base units | The SI units from which all derived units are built. |
| Dimensional consistency | Another name for homogeneity. |
Section 10 of 10Exam technique
1
Believing a homogeneous equation must be correct — numerical factors and matching wrong powers cannot be detected.
2
Checking only the left side or one right-hand term — every added term must be tested.
3
Forgetting that radians and the arguments of logs and sines have no units.
4
Making slips with powers of ten when dividing — subtract the powers.