Measurements and Mathematics

Measurement and Significant Figures

When we calculate the value from our results the answer should we written to the same number of significant figures as the original results. For example, is the two sides of a rectangle are measured as 24.2 cm and 18.3 cm, then the are of the triangle is 24.2 cm x 18.3 cm = 442.86 cm²

Since the sides of the triangle were given at three significant figures, then the answer should also be to three significant figures. Therefore, the answer is 443 cm²

Reading Scales

Many readings in physics are taken from a scale on an instrument e.g. thermometers, ammeters, voltmeters etc. When reading a scale you make an estimate when the pointer is not actually on a mark on the scale. In the example below of an ammeter, the result would be taken as 1.34 A.

Accuracy of Results

For an experiment to be useful we must obtain accurate results. There are certain steps that can be taken to increase the certainty of our results. These are:

1. Take the same reading more than once can calculate an average value.
2. Measure a large number of a quantity and calculate the value for one. For example, if we have to find the thickness of a sheet of paper, we can measure the thickness of 300 sheets. We then divide our result by 300 to find the thickness of one sheet.
3. We can select and instrument which is appropriate to the reading. If a current of about 0.4 A is being measured we use an ammeter with a range of 0 to 1 A, not 0 to 5 A.
4. We take care to avoid parallax error. Always try to read scales from directly over the mark.

 Large and Small Numbers

When we have very large and small numbers there are useful alternative ways to write them.

 Standard Form

In standard form we write numbers in two parts as follows:

You should be able to multiply and divide numbers in standard form:

 Prefixes

Prefixes are also used to represent very large and small numbers. The following examples show there meaning.

 Graphs

A common way to present results is to draw graphs. Graphs often provide us with extra information and helps our understanding.

When plotting a graph the axes are labelled with the quantities involved, their symbols and the units of the quantities. A convenient scale must be chosen in order for the results to use up most of the graph.

A small cross or a circled dot can be used to plot the points. Results must be plotted as accurately as possible and should not be rounded off.

A line of best fit must be drawn to join the plotted points together. This line goes as close as possible to as many points as possible. The points should also be 'balanced' about the line with equal numbers of points below and above the line.

A the results are arranged in a curve, a smooth curve must be drawn.

 Common Graphs

If the graph is a straight line passing through the origin, it means that the two quantities are proportional to each other.

If the graph is a straight line but not passing through the origin, the quantities are linearly related to each other but not proportional to each other.

 Gradient and intercepts of a graph

Two important quantities of a straight-line graph are its gradient and its intercepts on the axes.

The intercept is the point where the line cuts the axis. The y-intercept is the point where the line cuts the y-axis and the x-intercept is the point where the x-axis. In the example below, the y-intercept is 4 and the x-intercept is -8.

The gradient is found using the formula below: