Year+13+Measurements+and+Uncertainties

= = = Measurement and Uncertainties =

Uncertainties
Whenever a measurement is taken there is always a small uncertainty due to several factors. When reading a scale the uncertainty is usually ± half of the smallest scale. For Example: If a ruler has a smallest scale in mm the error ( Δx ) would be ±0.5mm.

Absolute Uncertainty
When writing an absolute uncertainty we write the measured value, then state the uncertainty in the same units. For Example: 46 ± 0.5cm 23 ± 1kg Note: The position of the units is **after** the uncertainty is written The uncertainly is written as one significant figure.

Relative (or Percentage) Uncertainty
If we measured 25 ± 0.5mm (absolute uncertainty) we could express the error as a percentage of absolute uncertainty. Δx = x 100 = 2% The relative uncertainty is 25 mm ± 2%. Note: The position of the units is **before** the percentage uncertainty is stated. The relative uncertainty can be stated to more than one significant figure.

Systematic Errors or Uncertainties
Systematic errors can also occur. These are often due to a problem with the measuring equipment like a zero error measurement. Other systematic errors also include parallax errors, friction losses, and reaction time.

Random Errors or Uncertainties
Random errors can occur when taking a measurement using electronic equipment if the value being read is fluctuating like a voltage or current. The skill of the operator can also cause fluctuations.

Uncertainties in Averages
In Physics it is often useful to use an average to improve the accuracy of measurements. To determine the uncertainly when an average has been calculated: · Find the range of all the values (eliminate any obvious incorrect values first) · Then half it to get your uncertainty For Example: Measured values: 23s, 21s, 22s, 22s average = 22s range 23 – 21 = 2s Δx = 1s Average 22 ± 1s

Uncertainties in Multiple Measurements
Another way improve the accuracy and decrease the uncertainty is to take multiple measurements. For example: You have 20 identical paper clips. Put the 20 together and weigh them all together. Measurement 25.2 ± 0.1g One paperclip would weigh ±  g  1.255 ± 0.005 g

Uncertainties in Calculations
When we undertake calculations with measurements with uncertainties, we may change the uncertainty.

2.3 ± 0.1mm plus 6.3 ± 0.2mm = 8.6 ± 0.3mm 240 V ± 10 % times 0.20 A ± 5% (P = V x I) (240 x 0.2) W ± (10% + 5%) 48W ± 15% **Squaring and Square rooting** - is the same as multiplying a number by itself so add % error. For a Square root you need to half the % error.
 * Addition and Subtraction** – Add the absolute uncertainty
 * Multiplication and Division** – Add the relative or percentage uncertainty
 * Multiplying or dividing by a constant ­**– This is the same as multiplying by a measurement with a zero percentage uncertainty, so the uncertainty does not change.

Common Relationships
When one value changes due to a direct relationship between another we say one value is proportional to the other. For Example : Voltage is proportional to Current, so V α I α – “proportional” Other common relationships Y α x y α x2 y α √x y α     a

Plotting Graphs
When plotting graphs, remember the following: · Dependant variable (the one you don’t control) goes on y – axis · Independent variable (the one you control) goes on the x – axis · Use **pencil** to mark a **cross** for each point. · Choose your scale carefully · Label the axes including units · Title the graph to describe the relationship


 * ||  ||   || **Read – Study Guide – pages 23 - 28** ||
 * ||  ||   || **Read – Study Guide – pages 23 - 28** ||

Error Bars
When we plot our data we must show the uncertainties or errors for each plotted point. This is done by drawing error bars.



Δx and Δy will not normally be the same size.

Line of Best Fit
After we have plotted all points and drawn the error bars we draw the line of best fit. This is done by “eye”. It should pass through all error bars and have about the same number of points above the line as below, and the points should be of similar distance above and below the line. Next find the gradient m, by working out the rise over run for two points towards each end of the line. The point where the line crosses the y axis will give us c. So the equation of the line will give us y = mx + c

Finding Δm and Δc
This gradient and intercept of the line of best fit also has an uncertainly which we can determine. Plot a new “line of best fit” which has the steepest possible gradient that still fits within the error bars. Find the gradient m’ and the intercept c’ of the new line. To find the uncertainty of the of the gradient Δm = m’ – m Then find the uncertainty of the intercept Δc = c – c’