Physical Measurement
Measurement and uncertainties1.2.1 State the fundamental units in the SI system.Many different types of measurements are made in physics. In order to provide a clear and concise set of data, a specific system of units is used across all sciences. This system is called the International System of Units (SI from the French "Système International d'unités"). The SI system is composed of seven fundamental units:
Note that the last unit, candela, is not used in the IB diploma program. 1.2.2 Distinguish between fundamental and derived units and give examples of derived units.In order to express certain quantities we combine the SI base units to form new ones. For example, if we wanted to express a quantity of speed which is distance/time we write m/s (or, more correctly m s^{1}). For some quantities, we combine the same unit twice or more, for example, to measure area which is length x width we write m^{2}. Certain combinations or SI units can be rather long and hard to read, for this reason, some of these combinations have been given a new unit and symbol in order to simplify the reading of data. Below is a table containing some of the SI derived units you will often encounter:
1.2.3 Convert between different units of quantities. Often, we need to convert between different units. For example, if we were trying to calculate the cost of heating a litre of water we would need to convert between joules (J) and kilowatt hours (kW h), as the energy required to heat water is given in joules and the cost of the electricity used to heat the water is a certain price per kW h. If we look at table 1.2.2, we can see that one watt is equal to a joule per second. This makes it easy to convert from joules to watt hours: there are 60 second in a minutes and 60 minutes in an hour, therefor, 1 W h = 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h being a prefix standing for kilo which is 1000). 1.2.4 State units in the accepted SI format.There are several ways to write most derived units. For example: meters per second can be written as m/s or m s^{1}. It is important to note that only the latter, m s^{1}, is accepted as a valid format. Therefor, you should always write meters per second (speed) as m s^{1 }and meters per second per second (acceleration) as m s^{2}. Note that this applies to all units, not just the two stated above. 1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.When expressing large or small quantities we often use prefixes in front of the unit. For example, instead of writing 10000 V we write 10 kV, where k stands for kilo, which is 1000. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000). When expressing the units in words rather than symbols we say 10 kilowatts and 1 milliwatt. A table of prefixes is given on page 2 of the physics data booklet. 1.2.6 Describe and give examples of random and systematic errors.Random errors
Systematic errors A systematic error, is an error which occurs at each reading.
1.2.7 Distinguish between precision and accuracy.Precision Accuracy A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error). 1.2.8 Explain how the effects of random errors may be reduced.The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their affect on the data.1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value. Example: 11.21 x 1.13 = 13.7883 The answer contains 6 significant figures. However, since the value for time (1.23 s) is only 3 s.f. we write the answer as 13.7 m s^{1}. The number of significant figures in any answer should reflect the number of significant figures in the given data. 1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.Absolute uncertainties Example: 13.21 m ± 0.01 Fractional uncertainties Example: 1.2 s ± 0.1 Percentage uncertainties Example: 1.2 s ± 0.1 Percentage uncertainty: 0.1 / 1.2 x 100 = 6.25 % 1.2.11 Determine the uncertainties in results.Simply displaying the uncertainty in data is not enough, we need to include it in any calculations we do with the data. Addition and subtraction Example: Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.01 1.2 + 12.01 + 7.21 = 20.42 Multiplication, division and powers Example: Multiply the values 1.2 ± 0.1, 12.01 ± 0.01 1.2 x 12.01 = 14 14 ± 8.413 % Other functions Example: Calculate the area of a field if it's length is 12 ± 1 m and width is 7 ± 0.2 m. Best value for area: Highest value for area: Lowest value for area: If we round the values we get an area of: 1.2.12 Identify uncertainties as error bars in graphs.When representing data as a graph, we represent uncertainty in the data points by adding error bars. We can see the uncertainty range by checking the length of the error bars in each direction. Error bars can be seen in figure 1.2.1 below:
1.2.13 State random uncertainty as an uncertainty range (±) and represent it graphically as an error bar.In IB physics, erro bars are only need to be used when the uncertainty in one or both of the plotted quantities are significant. Error bars are not required for trigonometric and logarithmic functions. To add error bars to a point on a graph, we simply take the uncertainty range (expressed as "± value" in the data) and draw lines of a corresponding size above and below or on each side of the point depending on the axis the value corresponds to. Example: Plot the following data onto a graph taking into account the uncertainty.
In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Therefor, we often skip certain points and only add error bars to specific ones. We can use the list of rules below to save time:
1.2.14 Determine the uncertainties in the gradient and intercepts of straight line graphs.Gradient
Intercept
Note that in the two figures above the error bars have been exaggerated to improve readability.
Vectors and Scalars1.3.1 Distinguish between vector and scalar quantities, and give examples of each.When expressing a quantity we give it a number and a unit (for example, 12 kg), this expresses the magnitude of the quantity. Some quantities also have direction, a quantity that has both a magnitude and direction is called a vector. On the other hand, a quantity that has only a magnitude is called a scalar quantity. Vectors are represented in print as bold and italicised characters (for example F). Below is a table listing some vector and scalar quantities:
Note that some quantities appear to be the same, such as velocity and speed, both representing distance over time, the difference is that velocity has a direction whilst speed does not. 1.3.2 Determine the sum or difference of two vectors by a graphical method.The difference of two vectors
The sum of two vectors
Adjacent vectors
Alternatively, we can use trigonometry for a faster and more accurate result. This is demonstrated in figure 1.3.4 below:
Scalar multiplication
Scalar multiplication is demonstrated in figure 1.3.5 below:
1.3.3 Resolve vectors into perpendicular components along chosen axes.When working with adjacent vectors that do not form a 90° angle, it is often useful to brake certain vectors into component vectors so that they are concurrent with the other vectors. To do this, we draw two vectors, one horizontal and the other vertical to our plane of reference. We then use trigonometry to work out the magnitude of each new vector and figure out the resulting force. This is shown in figure 1.3.6 and 1.3.7: Figure 1.3.6 shows a diagram of the forces acting on a block being pushed along a smooth surface:
Figure 1.3.7 shows the same diagram but with the surface and pushing forces broken down into their components:
Sometimes the plane of reference will not be parallel to the page, such and example is shown in figure 1.3.8 below:

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