Measurements and physical quantities (Topic 1A)
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Physical quantities are properties of systems that can be measured and quantified. They have a numerical value which may or may not have an associated unit.
Examples of physical quantities without a unit are:
- Refractive index
- Efficiency
- Strain
- Magnification
Examples of physical quantities with a unit are:
- Length ()
- Mass ()
- Energy ()
- Electric field strength ()
The International System of Units (SI) is the standard set of units used by physicists and other scientists for measurements.
SI consists of seven quantities and their corresponding units, known as the SI base units, which are shown in the table below.
The SI base unit symbols are written in lowercase letters, except the symbols named after a person: kelvin has the symbol and the ampere has the symbol
There are many other physical quantities that can be measured, other than those corresponding to the SI base units.
Speed, acceleration, and force are examples of derived units. These quantities are known as derived quantities and are measured in derived units, which can be determined by substituting the base units into the equation that relates the derived quantity to the base quantities.
An example of this can be seen from the speed equation:
Where:
- is speed,
- is the distance travelled in a straight line, and
- is the time taken.
Substituting the base units for length and time leads to:
Many derived units are used so frequently in measurements that they have been given specific names. A few of the most useful derived units are shown in the table below.
It is useful to note that it is much more convenient to use these abbreviated units rather than write out the full SI units during calculations. All of the above units are capitalised since they are named after people.
Physical equations must have the same units on either side: in other words, they must be homogeneous. The homogeneity of a physical equation can be checked by substituting SI base units into either side of the equation to ensure that both sides result in the same combination of SI units.
An example of this can be demonstrated with an equation of motion for uniform acceleration:
Velocity on the LHS of the equation has SI base units of For on the RHS, SI base units must be substituted for initial velocity , acceleration and time
It is important to note that the numerical coefficient of 2 is dropped, as pure numbers are dimensionless. The equation is homogeneous:
Prefixes are used to represent decimal submultiples or multiples of SI units in compact form.
For example:,
- can instead be written as and
- is equal to
A list of the most useful prefixes required for your exams is shown in the table below.
All the prefixes with a multiplication factor greater than one are in uppercase, with the exception of kilo, whereas all the prefixes with a multiplication factor less than one are in lowercase.
When recording data in a table, the independent variable data points are listed in the first column, and the dependent variable data points are listed in the second column (for quantitative measurements).
Both column headings should include the SI units for the variable in brackets.
The table above shows the measurements of an object’s speed in a straight line against time.
Note that milliseconds are used for time so that the standard form is not required in the table.
The number of significant figures used for each variable depends on the precision of the measuring equipment.
When data from tables is plotted as a graph, the independent variable is plotted on the Y axis and the dependent variable on the X axis.
The graph on the left is less useful since the data points would ideally fill more than half the graph in both the X and Y directions.
In the graph on the right, the scale of the Y axis has been adjusted using a false origin (an origin that is not zero) to better highlight the relevant data points and the line of best fit.
Physical quantities can be estimated by using known values for everyday objects.
For example, the mass of an average human can be taken as and the average mass of a car is around
When making estimation calculations, approximations of different physical quantities can also be used. Examples of this include:
- gravitational field strength at the Earth’s surface being taken as for estimations,
- the diffraction of light; the refractive index of air can be approximated as while its true value is
Calculations can be made using estimated values. These calculations do not require exact precision, and one significant figure is often sufficient. Finding only the order of magnitude ( etc.) of a physical quantity is often enough in some contexts, such as finding the approximate value in engineering solutions or drafting an experiment.
Known constants can also be approximated in calculations to provide numbers which are easier to work with. This technique can be used to quickly validate an answer obtained from a full calculation is approximately the right order of magnitude.
An example of an estimation calculation is determining the average weight of a person on Earth:
Where:
- is the person’s mass, and
- is the gravitational field strength.
Using and leads to:
Question walkthrough
Estimating Gravity on Mars from Force and Mass
Estimates the acceleration due to gravity on Mars by rounding an astronaut's mass and the force felt to convenient values before dividing.
The error of a measurement is the difference between an individual measurement and the true value of the quantity being measured. There are two types of errors, random and systematic.
Random error results in a random fluctuation of the measured value about the true value over repeated measurements.
Examples of causes of random error include:
- Fluctuations in the external conditions, such as electronic noise in an electrical component.
- Reading the measuring instrument differently each time, such as the level of the line on a thermometer.
Random error is often unavoidable, but its effect can be reduced by using more precise measuring instruments or taking many repeated measurements and averaging them to find the mean value.
Systematic error results in a skewing of the measured data by a given amount related to a flaw in the measurement process. Systematic errors can be reduced by calibrating the measurement apparatus or by comparing the results of different measurement techniques if possible.
Examples of systematic error include:
- Zero error – caused by the measuring instrument not being calibrated correctly, such as a weight scale showing a non-zero reading when no object is placed on it.
- Scale error – when measurements are consistently different from the true value by a certain proportion. For example, a weight scale may measure higher than the true value.
Random errors, where all results are affected by fluctuating amounts, affect the precision of a measurement, which describes the closeness of independent test results made under the same conditions. Precision depends only on the distribution of the random errors about the true value and does not depend on the true value itself.
Systematic errors, however, skew all results by a consistent amount and impact the accuracy of a measurement. Accuracy reflects how closely an individual test result aligns with the true value.
Sometimes, an accepted reference value can be used as the true value; however, the true value is usually unknown and must be measured.
The way that systematic and random errors affect accuracy and precision can be visualised by imagining a person throwing darts at a dartboard.
In the left image, the darts land near the bullseye with a random spread. This is an example of random error. The dart throws are both precise and accurate.
Systematic error in dart throwing in the right image causes a consistent deviation from the intended target, causing darts to consistently land in a particular area off-centre, rather than being randomly scattered. It’s often caused by a flawed technique, such as an improper grip, a poor follow-through, or an incorrect stance, which biases the throws in a predictable direction.
The darts will have a similar spread, but their average position will be displaced from the centre of the bullseye. This is an example of systematic error. The dart throws are precise but not accurate.
Uncertainty is an estimate attached to a measurement which characterises the range of values which should contain the true value.
- When using analogue measurement tools with a graduated scale that can be read, the uncertainty is taken as half of the smallest graduation. For example, a ruler with divisions of has an absolute uncertainty of
- When using a digital apparatus, the uncertainty is equal to the smallest graduation. For example, a one decimal place ammeter has an absolute uncertainty of
When multiple measurements are combined, the uncertainty in the final result will be a combination of the uncertainties in each measurement. So, when measurements are added or subtracted, the absolute uncertainties are summed.
An example of absolute uncertainty is in finding the tensile strain of a metal bar by measuring the change in length before and after the application of force.
If the initial length is and the final length is then the change in length is:
The uncertainty is:
So the change in length is:
When measurements are multiplied or divided, the percentage uncertainties must be added.
An example of percentage uncertainty is the calculated resistance of a resistor. It is found by measuring the current, for an applied voltage, .
The percentage uncertainty in and respectively is:
Ohm’s law can be rearranged to:
This expression shows that the percentage uncertainty in is found by adding the percentage uncertainties in and . Therefore, the percentage uncertainty in is:
Raising to a power is a special case of multiplication. The percentage uncertainty is multiplied by the power to which the value is being raised.
For example, consider measuring the power supplied to a resistor of resistance with current flowing through it. The percentage error in is and the percentage error in is
Electrical power, is given by:
Therefore, the power through the resistor is:
The percentage uncertainty in the power is equal to twice the percentage uncertainty in since it is raised to the power of two, added to the percentage uncertainty in which is:
Therefore, the uncertainty in the power is:
The calculated power is written as:
Measurements can be made more accurate and precise by recording many data points over a range of values.
In this example, the resistance of a resistor can be measured by recording the current as a function of the applied voltage over a range of voltages and using the gradient to determine the resistance .
The red line is the line of best fit through the data. The line of best fit is drawn going roughly through the middle of the data points, mitigating any anomalous results. Ohm’s law can be written as:
The resistance of the resistor, can be found from the gradient of the line of best fit, by:
The gradient of the line of best fit can be found by identifying two points on the line and and using:
For the graph above, this leads to and hence
Each data point, based on a measured value, has an associated uncertainty. Error bars are a way to visually represent this uncertainty. They are drawn stretching above and below the data point by the absolute uncertainty in each direction. If there are no horizontal or vertical error bars for a data point, then the error is negligible in the X axis or Y axis measurement.
Error bars can be used to find anomalous data points. If an error bar does not pass through the line of best fit and is a significant distance away from it, the data point is likely an anomaly.
Error bars can be used to estimate the uncertainty in the gradient of the line of best fit, which can be done by drawing the worst lines of fit. These lines are drawn from the bottom of the first error bar to the top of the last error bar and vice versa.
The difference between the gradient of the line of best fit and each line of worst fit is calculated. The greatest of the two differences is taken as the absolute uncertainty in the gradient.
The gradient of the line of best fit is and the lines of worst fit have gradients and Each of the worst line gradients has a difference of to the line of best fit gradient, so this is the absolute uncertainty and the gradient of the line of best fit is:
The percentage uncertainty is:
Since the resistance only depends on the gradient and no other variables, it has the same percentage uncertainty. Therefore, the absolute uncertainty in the resistance is:
The calculated resistance is written as:
It is useful to know how to determine the difference between experimental values and accepted values to confirm that the experiment was performed correctly. The percentage difference between an experimental and accepted value is equal to:
An example of the difference between experimental values and accepted values is in the true and experimental values of a resistor. If it were known that the true resistance of a resistor was and the measured experimental value was then the percentage difference between the true value and the experimental value would have been: