1.1 – Measurements in physics

1.1.1 Fundamental and derived SI units

1.1.1.1 Fundamental units:

The SI system (short for Système International d’Unités) has seven fundamental units. These are:

1 The metre (m). This is the unit of distance.

2 The kilogram (kg). This is the unit of mass.

3 The second (s). This is the unit of time.

4 The ampere (A). This is the unit of electric current.

5 The kelvin (K). This is the unit of temperature.

6 The mole (mol). One mole of a substance contains as many particles as there are atoms in 12 g of carbon-12. This special number of particles is called Avogadro’s number

7 The candela (cd). This is a unit of luminous intensity, which not required in IB syllabus.

1.1.1.2 Derived units:

Physical quantities other than those above have units that are combinations of the seven fundamental units. They have derived units. For example,

1. speed has units of distance over time, metres per second m/s

2. the unit of force is the newton (N). It equals the combination kgms⁻²

1.1.2 Scientific notation and metric multipliers

1.1.2.1 Scientific notation

Scientific notation means writing a number in the form a × 10^b , where a is decimal such that 1 ≤ a < 10 and b is a positive or negative integer. The number of digits in a is the number of significant figures in the number.

1.1.2.2 Metric multipliers

Small or large quantities can be expressed in terms of units that are related to the basic ones by powers of 10. Thus, a nanometre (nm) is 10⁻⁹ m. See more in Table 1.1 (Tsokos book page 2)/ Physics data booklet (page 3).

1.1.3 Significant figures

The number of digits used to express a number carries information about how precisely the number is known. How to find the number of significant figures in a number is illustrated in Table 1.5 (Tsokos book page 5).

In multiplication or division (or in raising a number to a power or taking a root), the result must have as many significant figures as the least precisely known number entering the calculation. 23 × 578 =13294 ≈ 1.3 ×10⁴

In adding and subtracting, the number of decimal digits in the answer must be equal to the least number of decimal places in the numbers added or subtracted. 3.21+4.1=7.32 ≈7.3

1.1.4 Orders of magnitude

Expressing a quantity as a plain power of 10 gives what is called the order of magnitude of that quantity. Tables 1.2, 1.3 and 1.4  (Tsokos book page 2) give examples of distances, masses and times, given as orders of magnitude.

1.1.5 Estimation

Based on the orders of magnitude, we can estimate and compare values. Thus, the mass of the universe has an order of magnitude of 10⁵³ kg and the mass of the Milky Way galaxy has an order of magnitude of 10⁴¹ kg. The ratio of the two masses is then simply 10¹².

1.2 – Uncertainties and errors

Physics is an experimental science. No measurement will ever be completely accurate, however, and so the result of the experiment will be presented with an experimental error.

1.2.1 Random and systematic errors

1.2.1.1 Random errors

The presence of random uncertainty is revealed when repeated measurements of the same quantity show a spread of values, some too large some too small. Unlike systematic errors, which are always biased to be in the same direction, random uncertainties are unbiased.

The uncertainty in reading an instrument is ± half of the smallest width of the graduations on the instrument.

For digital instruments, we may take the reading error to be the smallest division that the instrument can read.

Repeat measurements can reduce the random errors.

A measurement is said to be precise if the random uncertainty is small. This means in practice that when the measurement was repeated many times, the individual values were

close to each other in Figure 1.5 (Tsokos book page 10)

1.2.1.1 Systematic errors

A systematic error biases measurements in the same direction, the measurements are always too large or too small. In the graph, if theoretically it should be a straight line through

the origin, but you actually do the experiment, you will find that you do get a straight line, but not through the origin, that means there is a systematic error. in Figure 1.2 (Tsokos book page 8)

A measurement is said to be accurate if the systematic error in the measurement is small in Figure 1.5 (Tsokos book page 10)

1.2.2 Absolute, fractional and percentage uncertainties

The ± errors is called the absolute uncertainty in the measurement.

The ratio of absolute uncertainty to mean value is called the fractional uncertainty.

Multiplying the fractional uncertainty by 100% gives the percentage uncertainty.

In general, if a = a₀ ± ∆a, we have:

• absolute uncertainty = ∆a

• fractional uncertainty = ∆a/a₀

• percentage uncertainty = (∆a/a₀) × 100%

In addition and subtraction, we always add the absolute

uncertainties, neversubtract.

In multiplication and division, the fractional uncertainty of the result is the sum of the fractional uncertainties of the quantities involved

In powers and roots, the fractional uncertainty of the result is the fractional uncertainty of the quantity multiplied by the absolute value of the power.

For all of equations,check “Sub-topic 1.2 – Uncertainties and errors” (Physics Data Booklet page 5)

Exam tip: The final absolute uncertainty must be expressed to one significant figure.

1.2.3 Error bars

In physics, it is slightly more involved because the point consists of measured or calculated values and so is subject to uncertainty. So the point (x0±∆x, y0±∆y) is plotted as shown in Figure 1.6 (Tsokos book page 16). The uncertainties are represented by error bars.

1.2.4 Uncertainty of gradient and intercepts

In a physics experiment we usually try to plot quantities that will give straight-line graphs. To ‘go through the error bars’ can draw a best-fit line.

The gradient (slope) of the best-f t line is found by using two points on the best-f t line as far from each other as possible.

The idea is to draw lines of maximum and minimum gradient in such a way that they go through all the error bars (not just the ‘first’ and the ‘last’ points). Figure 1.9 (Tsokos book page 19) shows the best-f t line (in blue) and the lines of maximum and minimum gradient.

The uncertainty in the vertical and horizontal intercepts can be calculated.

1.3 – Vectors and scalars

1.3.1 Vector and scalar quantities

Quantities in physics are either scalars (i.e. they just have magnitude) or vectors (i.e. they have magnitude and direction). Such as time, distance, mass, speed and temperature  are called scalar quantities. And  velocity and acceleration which need to know the direction  are called scalar quantities.

1.3.2 Combination and resolution of vectors

Suppose that we use perpendicular axes x and y and draw vectors on this x–y plane.

The x- and y-components of A are called A_x and A_y. They are given by: “Sub-topic 1.3 – Vectors and scalars” (Physics Data Booklet page 5): A_x = Acosθ, A_y = Asinθ

The combination of vectors can be calculated by their x- and y-components combination. Then the final vector is reconstructing from its components