01. Measurement

Base and Derived Quantities

• Physical quantities are classified as base (or fundamental) quantities and derived quantities.
  7 base quantities are chosen to form the base units.

• Any other physical quantities can be derived from these base quantities. These are called derived quantities.

Prefixes

• Prefixes are attached to a unit when dealing with very large or very small numbers.

Homogeneity of Units in an Equation

• A physical equation is said to be homogeneous if each of the terms, separated by plus, minus, equality or inequality signs has the same base units.

Uncertainty

• Absolute uncertainty of a measurement of $x$ can be written as $\mathrm{\Delta }x$. This means that true value of the measurement is likely to lie in the range $x-\mathrm{\Delta }x$ to $x+\mathrm{\Delta }x$.
• Fractional uncertainty = ${\displaystyle \frac{\mathrm{\Delta }x}{x}}$
• Percentage uncertainty = ${\displaystyle \frac{\mathrm{\Delta }x}{x}}\times 100$
• If the values of two or more quantities such as $a$ and $b$ are measured and then these are combined to determine another quantity $Y$, the absolute or percentage uncertainty of $Y$ can be calculated as follows:
  • If $Y=a\pm b$, then  $\mathrm{\Delta }Y=\mathrm{\Delta }a+\mathrm{\Delta }b$
  • If $Y=ab$ or $Y=\frac{a}{b}$ , then  $\frac{\mathrm{\Delta }Y}{Y}=\frac{\mathrm{\Delta }a}{a}+\frac{\mathrm{\Delta }b}{b}$
  • If $Y=a^n$ then  $\frac{\mathrm{\Delta }Y}{Y}=n\frac{\mathrm{\Delta }a}{a}$

Errors

• Systematic errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of same magnitude and sign.
• Random errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of different magnitude and sign.

Accuracy and Precision

• The accuracy of an experiment is a measure of how close a measured value is to the true value. It is a measure of the correctness of the result.
• The precision of an experiment is a measure of how exact the result is without reference to what that the result means. It is a measure of how reproducible the results are, i.e. it is a measure of how small the uncertainty is.

Vectors

• A vector quantity has magnitude and direction.
• A scalar quantity has magnitude only.
• Addition of vectors in 2D: $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$
• Subtraction of vectors in 2D: $\overrightarrow{a}-\overrightarrow{b}=\overrightarrow{d}$
  

• Methods of finding magnitudes of vectors:
  1. resolution of vectors into perpendicular components
  2. by scale drawing
  3. using:
     sine rule: $\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }$
     cosine rule: $a^2=b^2+c^2-2bc\cos \alpha$