MEASUREMENTS AND THEIR ERRORS Flashcards
SI UNIT MASS
kg
SI UNIT LENGTH
m
SI UNIT TIME
s
SI UNIT TEMPERATURE
K
SI UNIT AMOUNT OF SUBSTANCE
mol
SI UNIT ELECTRICAL CURRENT
A
Tera T
10^12
Giga G
10^9
Mega M
10^6
Kilo k
10^3
Centi c
10^-2
Milli m
10^-3
Micro µ
10^-6
Nano n
10^-9
Pico p
10^-12
Femto f
10^-15
1eV in J
1.6x10^-19J
Precision
Precise measurements are consistent, they fluctuate slightly about a mean value - this doesn’t indicate the value is accurate.
Repeatability
If the original experimenter can redo the experiment with the same equipment and method then get the same results it is repeatable.
Reproducability
If the experiment is redone by a different person or with different techniques and equipment and the same results are found, it is reproducible.
Resolution
The smallest change in the quantity being measured that gives a recognisable change in reading.
Accuracy
A measurement close to the true value is accurate.
Random Errors
Random errors affect PRECISION, meaning they cause differences in measurements which causes a spread about the mean. You cannot get rid of all random errors.
An example of random error is electronic noise in the circuit of an electrical instrument.
Reducing Random Errors
Take at least 3 repeats and calculate a mean, this method also allows anomalies to be identified.
Use computers/data loggers/cameras to reduce human error and enable smaller intervals.
Use appropriate equipment, e.g a micrometer has higher resolution (0.1 mm) than a ruler (1 mm).
Systematic Errors
Systematic errors affect ACCURACY and occur due to the apparatus or faults in the experimental method. Systematic errors cause all results to be too high or too low by the same amount each time.
An example is a balance that isn’t zeroed correctly (zero error) or reading a scale at a different
angle (this is a parallax error).
Reducing Random Errors
● Calibrate apparatus by measuring a known value (e.g. weigh 1 kg on a mass balance), if the reading is inaccurate then the systematic error is easily identified.
● In radiation experiments correct for background radiation by measuring it beforehand and excluding it from final results.
● Read the meniscus (the central curve on the surface of a liquid) at eye level (to reduce parallax error) and use controls in experiments.
Parallax Error
A systematic error, reading a scale at a different
angle. Read the meniscus (the central curve on the surface of a liquid) at eye level.
Zero Error
A systematic error, a balance that isn’t zeroed correctly.
Uncertainty
The uncertainty of a measurement is the bounds in which the accurate value can be expected to lie e.g. 20°C ± 2°C , the true value could be within 18-22°C
Absolute Uncertainty
uncertainty given as a fixed quantity e.g. 7 +/- 0.6 V
Fractional Uncertainty
uncertainty as a fraction of the measurement e.g. 7 V +/- 3/35
Percentage Uncertainty
uncertainty as a percentage of the measurement e.g. 7 +/- 8.6%V
Reducing percentage and fractional uncertainty
you can measure larger quantities
Readings
Readings are when one value is found e.g. reading a thermometer
Measurements
measurements are when the difference between 2 readings is found, e.g. a ruler (as both the starting point and end point are judged).
Uncertainty in a Reading
The uncertainty in a reading is ± half the smallest division,
e.g. for a thermometer the smallest division is 1°C so the uncertainty is ±0.5°C
Uncertainty in a Measurement
The uncertainty in a measurement is at least ±1 smallest division: e.g. a ruler, must include both the uncertainty for the start and end value, as each end has ±0.5mm, they are added so the uncertainty in the measurement is ±1mm.
Uncertainty in Digital Readings
Digital readings and given values will either have the uncertainty quoted or assumed to be ± the
last significant digit e.g. 3.2 ± 0.1 V, the resolution of an instrument affects its uncertainty.
Uncertainty in Repeated Data
For repeated data the uncertainty is half the range (largest - smallest value), show as: mean ± (range/2)
Reducing Uncertainty
By fixing one end of a ruler as only the uncertainty in one reading is included.
By measuring multiple instances:
e.g. to find the time for 1 swing of a pendulum by measuring the time for 10 giving e.g. 6.2 ± 0.1 s, time for 1 swing is 0.62 ± 0.01s (the uncertainty is also divided by 10).
Sig.fig in uncertainties
Uncertainties should be given to the same number of significant figures as the data.
UNCERTAINTIES: Adding/Subtracting data
ADD ABSOLUTE UNCERTAINTIES
A thermometer with an uncertainty of 0.5 K shows the temperature of water falling from 298-0.5 K to 273-0.5K, what is the difference in temperature?
298-273 = 25K 0.5 + 0.5 = 1K (add absolute uncertainties) difference = 25 1 K
UNCERTAINTIES: Multiplying / dividing data
ADD PERCENTAGE UNCERTAINTIES
E.g. a force of 91 3 N is applied to a mass of 7 +/- 0.2 kg, what is the acceleration of the mass?
F=ma A=f/m
91/7=13ms^-2 (3/91x100)+(0.2/7x100)=6.2%
13x0.062=0.8
13+/-0.8ms^-2
UNCERTAINTIES: Raising to a power
MULTIPLY PERCENTAGE UNCERTAINTY BY POWER
The radius of a circle is 5 +/- 0.3 cm, what is the percentage uncertainty in the area of the circle?
Area = π x 25 = 78.5 cm2
Area = πr
%uncertain in radius=0.3/5x100=6%
%in area (r^2)= 6*2=12%
78.5+/-12%cm^2
Percentage Uncertainty
(uncertainty/value)x100
Uncertainties on Graphs
Uncertainties are shown as error bars on graphs,
e.g. if the uncertainty is 5mm then have 5 squares of error bar on either side of the point. A line of best fit on a graph should go through all error bars (excluding anomalous points).
Uncertainty in a Gradient
The uncertainty in a gradient can be found by lines of best and worst fit, this is especially useful when the gradient represents a value such as the acceleration due to gravity:
● Draw a steepest and shallowest line of worst fit, it must go through all the error bars.
● Calculate the gradient of the line of best and worst fit, the uncertainty is the difference between the best and worst gradients.
Percentage Uncertainties on Graphs Equation
(best gradient-worst gradient)/best gradient x 100
(best y intercept - worst y interecpt)/best y intercept x100
Orders of Magnitude
Orders of magnitude - Powers of ten which describe the size of an object, and which can also be
used to compare the sizes of objects.
E.g: The diameter of nuclei have an order of magnitude of around 10^−14
100 m is two orders of magnitude greater than 1m.
How is Estmation Used
Estimation is a skill physicists must use in order to approximate the values of physical quantities, in order to make comparisons, or to check if a value they’ve calculated is reasonable.
