Symbol | Pronounced | Meaning | Example |
Delta | The change in | A movement ofmeans | |
Delta | The small change in | A movement ofmeans | |
Deltaover delta | The average rate of change ofover the time period fromto | Average speed | |
Deltaover delta | The average rate of change ofover the small time period fromto | Average speed | |
Dee over dee | The instantaneous speed at a time | The speed |
Identify which symbols in the equation are variables and which are constants.
The symbols that correspond toandmust be variables and the symbols that correspond toandmust be constants.
If a variable is cubed, square rooted or the reciprocal, log or exponential is taken, the result is still a variable and may still be used to label one of the axes.
Any function of the readings may be used to label the axes, since the result is still a variable.
Sometimes the physical quantities use the same symbols as in our notation e.g.is used to denote the speed of light. Do not get these confused.
For the equationabove, taking natural logs results in a straight line.
For the equationabove, plottingagainstresults in a straight line with gradient -1 andinterceptsince
]]>To find the units for speed, simply substitute into the above expression the units for distance and time:
Obviously there are many possible derived units, and it is often convenient to refer to a particular derived unit by a label. For example the unit of force, kg m/s^2 , is a derived unit. This particular derived unit is often called the newton.
The system of SI units is consistent. As long as the quantities that are substituted into an equation are expressed in SI units, the answer will be expressed in SI units, but sometime SI units are inconvenient. In astronomy, the distances are so large that expressed in SI units they are always accompanied by many powers of ten. Instead, often astronomical units (1 AU is the distance between the Earth and the Sun), light years (the distance travelled by light in one year) or the parsec are used (1 parsec is the distance to the nearest star such that 1 AU would subtend an angle of 1 arcsecond orof a degree).. Some units – the hour, gram or mile for example – are in common use even though they are not SI units. All of these need to be converted into SI units before they can be used in most equations.
Some derived units are given in the table below.
SI Derived Unit | SI Base Unit | Alternative Derived Unit |
Newton () | ||
Pascal () | Same | |
Hertz () | ||
Joule () | ||
Coulomb () | ||
Volt () | ||
Ohm () | ||
Weber () | ||
Tesla () | ||
Becquerel () | ||
Gray () | ||
Sievert () | ||
Watt () |
Example:
For more practical purposes it helps to know the masses of many objects in terms of SI units. Some are given below.
1 kg – A packet of sugar or a litre of water. A person might weigh from 50 kg upwards.
1 m – The distance between a persons outstretched hands.
1 s - Duration of a heartbeat.
1 amp – The current consumption of a typical computer.
25 degrees celsius – Roughly room temperature. Actually the SI unit of temperature is Kelvin (K), with 1 degree celsius = 1 degree kelvin.
1 mol – the mass of one mol of a substance is typically from a few grams to a few tens of grams. 1 mol of Carbon 12 is about the number of carbon atoms in a pencil lead.
1 m/s – Walking speed. A car might move at 30 m/s on the motorway.
1 N – About the weight of an apple.
1.5 V – A typical small battery voltage.
10 J – The energy used to lift a bag of sugar to waist height. The typical person uses about 10000 J per day.
500,000 Pa – A typical pressure exerted on the ground by a person standing up.
]]>Normally the uncertainty range to to reading errors is given as below.
Device |
Example |
Uncertainty |
Analogue Scale |
Rulers and meters with moving pointers |
half the smallest scale division |
Digital Scale |
Top pan balances and digital meters |
half the smallest scale division |
We can estimate the uncertainty given several measurements. For example, given the five measurements 2.01, 1.82, 1.97, 2.16 and 1.94 seconds respectively, the average is 1.98 seconds. The difference between the average and the smallest reading (1.98-1.82=0.16 seconds) and the difference between the average and the largest reading (2.16-1.98=0.18 seconds)is found and the largest of these values is taken to be the uncertainty. It this case the uncertainty is 0.18 seconds and the uncertainty range isseconds.
]]>The intercept is a point on an axis where the graph crosses.
If the graph is a line and the y – intercept is zero then the quantity represented on the y axis is said to be directly proportional to the quantity represented on the x – axis.
The gradient of a straight line graph is the increase in the y – value as we move along the graph divided by the increase in the x – value. For a straight line graph the gradient is constant. For example, the gradient below isNotice the triangle drawn is as large as possible. The graph illustrates Hooke's Law.
It must be noted that the gradient has units derived from the units on the axes (which must be included). The units of the gradient in the above example are
Often the area under the graph represents a useful quantity. The area under the graph above represents the energy needed to stretch the spring. The area above is a triangle so is equal toOften the area can only be estimated, by counting squares for example.
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The last model is not a good fit to the data.
]]>The graph should have a descriptive title and if needed, a key.
The scales should be suitable and regular, either linear or log for example. There should be no sudden jumps in values.
Consider whether or not to include the origin. The data should take up most of the graph, and if including the origin makes this impossible, it should be excluded. Often the y intercept needs to be found, and in this case the y axis needs to be included at least in part.
The axes are labelled with the correct quantity and units e.g. Current (A).
The points are labelled clearly. Vertical and horizontal lines are better than crosses or dots. If error bars are to be drawn then vertical and horizontal bars are the only option.
A best fit trend line is added (or even a curve). The line (or curve) should not just join the dots. If it is a line it should go through the centre of the points, with as many points below the line as above, if possible and should be drawn with a ruler. If it is a curve it should be smooth.
The points should be randomly above and below the line (or curve). If this cannot be done with a line, a curve may be a better fit, as below.
Outliers or extreme values, that do not fit the line or curve are identified.