1 Physical quantities and units#

1.1 Physical quantities#

Physical quantity := a quantity that can be measured and consists of a numerical magnitude and unit.

From Save My Exams

For AS: $g = 9.81ms^{-2}$

1.2 SI units#

Base quantities are the quantities on the basis of which other quantities are expressed.

Derived quantities are the quantities that are expressed in terms of base quantities.

A derived quantity has an equation which links to other quantities (e.g. $F=ma$).

Homogeneity of an equation

An equation is homogeneous if quantities on BOTH sides of the equation has the same unit.

A homogeneous equation may not be physically correct, but a physically correct equation will always be homogeneous.

question

The speed $v$ of a liquid leaving a tube depends on the change in pressure $\Delta P$ and the density $\rho$ of the liquid. The speed is given by the equation

$$v = k(\frac{\Delta P}{\rho})^n$$

Where k is a constant that has no units

What is the value of n?

Significant figures

Digits considered significant: non-zero digits, zeros who:

• appearing anywhere between two non-zero digits
• trailing zeros in a number containing a decimal point

Digits considered not significant: leading zeros, trailing zeros in a number without a decimal point

1.3 Errors and uncertainties#

Random errors:

• values are scattered about the true value
• can be reduced by average / take readings in different ways (e.g. different points along a wire)
• Examples: Reading scales from different angles

Systematic errors:

• the average / peak is not the true value
• the reading is larger or smaller than (or varying from) the true reading by a constant amount
• can be eliminated by careful calibration
• Examples: Zero Error, Parallax Error

Precision := the range of the values / how close the result is to each other / the size of the smallest division

• affected by random error
• improve: repeat and average

Accuracy := how close the result is to the true value

• affected by systematic error
• improve: technique, accurate instrument
• Measured by average

Uncertainty Calculation

Absolute Uncertainty (Always 1 s.f. for the final result)

$y = b \pm c \Rightarrow \Delta y = \Delta b + \Delta c$

Percentage Uncertainty (1 / 2 s.f.)

$\left. \begin{aligned} &y=b\cdot c \\ &y = \frac{b}{c} \end{aligned} \right\} \Rightarrow \frac{\Delta y}{y} = \frac{\Delta b}{b} + \frac{\Delta c}{c}$

$y = a^n \Rightarrow \frac{\Delta y}{y} = n \cdot \frac{\Delta a}{a}$

When the value times a constant, the absolute uncertainty changes but the percentage uncertainty doesn’t.

As the s.f. of the absolute uncertainty is always one, the s.f. of the value can therefore be determined.

1.4 Scalars and Vectors*#

You also need to know:

• Vector Addition & Subtraction

• Represent a vector as two perpendicular components.

Direction : N of E 30° / 30° above x-axis (math)

Remember always to include a direction when the result is a vector.

1.? Measurements#

1.?? Uncertainties#

I’ve taken these notes on class but they are neither in the syllabus nor on past papers. Anyway, I put them here.

Uncertainty

Three main types of uncertainty:

• Random Uncertainties
• Systematic Errors
• Reading Uncertainties

The Limit of Reading of a measurement is equal to the smallest graduation of the scale of an instrument.

The Degree of Uncertainty of a reading (end reading) is equal to half the smallest graduation of the scale of an instrument.

Absolute - fractional errors - percentage errors

1 mm - 1/208 - 0.48%