This physics grades 11-12 lesson on uncertainty and vectors notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on uncertainty and vectors and check yourself with the quiz.
Quick answer: Homogeneity means every term of an equation has the same base units, which lets you test formulae. Uncertainty states how far a reading may be wrong: add absolute uncertainties for sums and percentage uncertainties for products. A vector is resolved into components with F cos θ and F sin θ, then added by components.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: A Level 1, IB A
Uncertainty and vectors notes: the key ideas
Why do units have to match in every equation?
Physics lets you add or compare only quantities that have the same units. Every unit in the SI system is built from seven base units; in mechanics you meet the metre (m), kilogram (kg) and second (s), together with the ampere (A) and kelvin (K) later on. All other units, the derived units, are combinations of these.
An equation is homogeneous when every term has the same base units on both sides. A homogeneous equation can still be wrong, but an equation that is not homogeneous must be wrong.
| Quantity | Derived unit | In base units |
|---|---|---|
| Force | newton (N) | kg m s⁻² |
| Pressure | pascal (Pa) | kg m⁻¹ s⁻² |
| Energy | joule (J) | kg m² s⁻² |
| Power | watt (W) | kg m² s⁻³ |
| Frequency | hertz (Hz) | s⁻¹ |
Test s = ut + ½at². Base units: u t gives (m s⁻¹)(s) = m, and a t² gives (m s⁻²)(s²) = m. Both terms are lengths, so the equation is homogeneous. Now test v = u + at²: the first two terms are in m s⁻¹, but at² is in m, so the equation cannot be correct.
How do you work out an uncertainty?
Every measurement has an absolute uncertainty, written ±Δx, in the same unit as the reading. For one reading it is usually half the smallest division, or the full division when you must judge the start and the end. For repeated readings, take the mean and use half the range (largest minus smallest) as the uncertainty. The percentage uncertainty compares the uncertainty with the size of the reading.
Percentage uncertainty = (Δx ÷ x) × 100 %. Here Δx is the absolute uncertainty and x is the measured value, both in the same unit.
Sum or difference: add the absolute uncertainties. Product or quotient: add the percentage uncertainties. Power xn: multiply the percentage uncertainty by n.
A plate measures 12.0 ± 0.1 cm by 8.0 ± 0.1 cm. Percentage uncertainties: 0.1 ÷ 12.0 = 0.83 % and 0.1 ÷ 8.0 = 1.25 %. The area is 12.0 × 8.0 = 96.0 cm² with percentage uncertainty 0.83 + 1.25 = 2.08 %. Absolute uncertainty = 0.0208 × 96.0 = 2.0 cm². Area = 96 ± 2 cm².
Students add percentage uncertainties for a sum, or add absolute uncertainties for a product. Remember: adding or subtracting quantities means absolute uncertainties; multiplying or dividing means percentages. Subtracting two close readings makes the percentage uncertainty much larger.
What is the difference between a scalar and a vector?
A scalar has magnitude only, while a vector has magnitude and direction. Mass, time, speed, distance, energy and temperature are scalars. Displacement, velocity, acceleration, force and momentum are vectors, so they must be added with their directions.
How do you resolve a vector into components?
Any vector can be replaced by two perpendicular components that together have the same effect. For a vector F at an angle θ above the horizontal, the horizontal component is F cos θ and the vertical component is F sin θ. Always sketch the right-angled triangle first, because the angle you are given decides which function goes with which component.
A rope pulls a cart with 40 N at 35° above the horizontal. Fx = 40 cos 35° = 40 × 0.819 = 32.8 N. Fy = 40 sin 35° = 40 × 0.574 = 22.9 N. Check: √(32.8² + 22.9²) = 40.0 N. The horizontal component is 32.8 N and the vertical component is 22.9 N.
How do you add vectors?
Place the vectors head to tail; the resultant runs from the first tail to the last head. For two perpendicular vectors, use Pythagoras for the size and the tangent for the direction. For any number of vectors, add all the x-components, add all the y-components, then combine the two totals.
Before you calculate, write the unit of the answer you expect. If your final line gives joules when you wanted newtons, you know straight away that a term has been left out or inverted.
- Base units: m, kg, s, A, K; newton = kg m s⁻²
- Percentage uncertainty = (Δx ÷ x) × 100 %
- Sum or difference: add absolute uncertainties; product or quotient: add percentages
- Component next to the angle: F cos θ; opposite the angle: F sin θ
- Resultant of perpendicular vectors: R = √(x² + y²), tan θ = y ÷ x
Checking units, uncertainties and vector components: frequently asked questions
What does homogeneous mean in physics?
An equation is homogeneous when all its terms reduce to the same base units. It is a quick test: if the units on the two sides differ, the equation is wrong. Matching units do not prove it is right, because numerical factors such as ½ cannot be checked this way.
How do you find a percentage uncertainty?
Divide the absolute uncertainty by the measured value and multiply by 100. For a reading of 24.0 cm ± 0.2 cm the result is 0.83 %. Keep one or two significant figures only.
What is the uncertainty of a power such as r³?
Multiply the percentage uncertainty of r by the power. If r has 2 % uncertainty, r³ has 6 %. This works because r³ is r × r × r and the percentage uncertainties add.
How do you resolve a force at an angle?
Draw a right-angled triangle with the force as the hypotenuse. The component next to the angle is F cos θ and the component opposite is F sin θ. Check by confirming that the components give back the original force.
How do you find the uncertainty in a gradient from a graph?
Draw the best-fit line, then the steepest and shallowest lines that still pass through all the error bars. The uncertainty in the gradient is half the difference between the steepest and shallowest gradients.
