3 Coursework Measurement. Breithaupt pages 219 to 239. AQA AS Specification. Candidates will be able to: choose measuring instruments according to their sensitivity and precision identify the dependent and independent variables in an investigation and the control variables
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Breithaupt pages 219 to 239
‘SI’ comes from the French ‘Le Système International d'Unités’
Symbol cases are significant (e.g. t = time; T = temperature)
Note – Special derived unit symbols all begin with an upper case letter
also, but rarely used: deca = x 10, hecto = x 100
Powers of 10 presentation
Consider the number 3250.040
It is quoted to SEVEN significant figures
3250.04 is SIX s.f.
3250.0 is FIVE s.f.
3250 is FOUR s.f. (NOT THREE!)
325 x 101 is THREE s.f. (as also is 3.25 x 103)
33 x 102 is TWO s.f. (as also is 3.3 x 103)
3 x 103 is ONE s.f. (3000 is FOUR s.f.)
103 is ZERO s.f. (Only the order of magnitude)
Headings should be clear
Physical quantities should have units
All measurements should be recorded (not just the ‘average’)
Measurements are reliable if consistent values are obtained each time the same measurement is repeated.
Reliable: 45g; 44g; 44g; 47g; 46g
Unreliable: 45g; 44g; 67g; 47g; 12g; 45g
Measurements are valid if they are of the required data OR can be used to obtain a required result
For an experiment to measure the resistance of a lamp:
Valid: current through lamp = 5A; p.d. across lamp = 10V
Invalid: temperature of lamp = 40oC; colour of lamp = redReliability and validity of measurements
This equal to the difference between the highest and lowest reading
Readings: 45g; 44g; 44g; 47g; 46g; 45g
Range: = 47g – 44g
Mean value < x >
This is calculated by adding the readings together and dividing by the number of readings
Readings: 45g; 44g; 44g; 47g; 46g; 45g
Mean value of mass <m> = (45+44+44+47+46+45) / 6
<m> = 45.2 gRange and mean value of measurements
Example of measurements showing systematic error: 585cm; 583cm; 584cm; 586cm
Systematic errors are often caused by poor measurement technique or incorrectly calibrated instruments.
Calculating a mean value will not eliminate systematic error.
Zero error can occur when an instrument does not read zero when it should do so. If not corrected for, zero error will cause systematic error. The measurement examples opposite may have been caused by a zero error of about + 18 cm.
Example of measurements showing random error only: 566cm; 568cm; 564cm; 567cm
Random error is unavoidable but can be minimalised by using a consistent measurement technique and the best possible measuring instruments.
Calculating a mean value will reduce the effect of random error.Systematic and random errors
Accurate measurements are obtained using a good technique with correctly calibrated instruments so that there is no systematic error.
Precise measurements are those that have the maximum possible significant figures. They are as exact as possible.
The precision of a measuring instrument is equal to the smallest possible non-zero reading it can yield.
The precision of a measurement obtained from a range of readings is equal to half the range.
Example: If a measurement should be 3452g
Then 3400g is accurate but not precise
whereas 4563g is precise but inaccurateAccuracy and precision of measurements
The uncertainty (or probable error) in the mean value of a measurement is half the range expressed as a ± value
Example: If mean mass is 45.2g and the range is 3g then:
The probable error (uncertainty) is ±1.5g
Uncertainty is normally quoted to ONE significant figure (rounding up) and so the uncertainty is now ± 2g
The mass might now be quoted as 45.2 ± 2g
As the mass can vary between potentially 43g and 47g it would be better to quote the mass to only two significant figures
So mass = 45 ± 2g is the best final statement
NOTE: The uncertainty will determine the number of significant figures to quote for a measurement
± 0.1 without the magnifying glass
± 0.02 perhaps with the magnifying glass
It is often useful to express the probable error as a percentage
percentage uncertainty = probable error x 100% measurement
Example: Calculate the % uncertainty the mass measurement 45 ± 2g
percentage uncertainty = 2g x 100% 45g
= 4.44 %
Addition or subtraction
Add probable errors together, examples:
(56 ± 4m) + (22 ± 2m) = 78 ± 6m
(76 ± 3kg) - (32 ± 2kg) = 44 ± 5kg
Multiplication or division
Add percentage uncertainties together, examples:
(50 ± 5m) x (20 ± 1m) = (50 ± 10%) x (20 ± 5%) = 1000 ± 15% = 1000 ± 150 m2
(40 ± 2m) ÷ (2.0 ± 0.2s) = (40 ± 5%) ÷ (2.0 ± 10%) = 20 ± 15% = 20 ± 1.5 ms-1
Multiply the percentage uncertainty by the power, examples:
(20 ± 1m)2 = (20 ± 5%)2 = (202± (2 x 5%)) = (400 ± 10%) = 400 ± 40 m2
√(25 ± 5 m2) = √(25 ± 20%) = √(25 ± (0.5 x 20%)) = (5 ± 10%) = 5 ± 0.5 m
For any straight line:
y = mx + c
m = gradient
= (yP – yR) / (xR – xQ)
c = y-intercept
The graph below shows how the extension of a wire, ∆L varies with the tension, T applied to the wire.Direct proportion
Physical quantities are directly proportional to each other if when one of them is multiplied by a certain factor the other changes by the same amount.
For example if the extension, ∆L in a wire is doubled so is the tension, T
A graph of two quantities that are proportional to each will be:
The general equation of the straight line in this case is: y = mx, with, c = 0
The graph below shows how the velocity of a body changes when it undergoes constant acceleration, a from an initial velocity u.Linear relationships - 1
Physical quantities are linearly related to each other if when one of them is plotted on a graph against the other, the graph is a straight line.
In the case opposite, the velocity, v of the body is linearly related to time, t. The velocity is NOT proportional to the time as the graph line does not pass through the origin.
The quantities are related by the equation: v = u + at. When rearranged this becomes: v = at + u.
This has form: y = mx + c
In this case m = gradient = a
c = y-intercept = u
The potential difference, V of a power supply is linearly related to the current, I drawn from the supply.
The equation relating these quantities is: V = ε – r I
This has the form: y = mx + c
In this case:
m = gradient = - r (cell resistance)
c = y-intercept = ε(emf)
The maximum kinetic energy, EKmax, of electrons emitted from a metal by photoelectric emission is linearly related to the frequency, f of incoming electromagnetic radiation.
The equation relating these quantities is: EKmax= hf – φ
This has the form: y = mx + c
In this case:
m = gradient = h (Planck constant)
c = y-intercept = – φ(work function)
The x-intercept occurs when y = 0
At this point, y = mx + c becomes:
0 = mx + c
x = x-intercept = - c / m
In the above case, the x-intercept, when EKmax= 0
is = φ / h
P when it undergoes constant acceleration,
8Calculating the y-intercept
The graph opposite shows two quantities that are linearly related but it does not show the y-intercept.
To calculate this intercept:
1. Measure the gradient, m
In this case, m = 1.5
2. Choose an x-y co-ordinate from any point on the straight line. e.g. (12, 16)
3. Substitute these into: y = mx +c, with (P ≡ y and Q ≡ x)
In this case 16 = (1.5 x 12) + c
16 = 18 + c
c = 16 - 18
c = y-intercept = - 2
m = + 5; c = + 7
m = + 0.33; c = + 6
m = - 0.25; x-intercept = + 3; (c = + 0.75)
The micrometer is reading 4.06 ± 0.01 mm
The callipers reading is 3.95 ± 0.01 cm
NTNU Vernier Applet
Breithaupt chapter 14.3; pages 221 & 222
Unit Conversion when it undergoes constant acceleration, - meant for KS3 - Fendt
Hidden Pairs Game on Units - by KT - Microsoft WORD
Fifty-Fifty Game on Converting Milli, Kilo & Mega - by KT - Microsoft WORD
Hidden Pairs Game on Milli, Kilo & Mega - by KT - Microsoft WORD
Hidden Pairs Game on Prefixes- by KT - Microsoft WORD
Sequential Puzzle on Energy Size - by KT - Microsoft WORD
Sequential Puzzle on Milli, Kilo & Mega order - by KT - Microsoft WORD
Powers of 10 - Goes from 10E-16 to 10E+23 - Science Optics & You
A Sense of Scale - falstad
Use of vernier callipers - NTNU
Equation Grapher - PhET - Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. y=bx ) to see how they add to generate the polynomial curve.Internet Links