Transcript Slide 1

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
• use appropriate apparatus and methods to make accurate and reliable
measurements
• tabulate and process measurement data
• use equations and carry out appropriate calculations
• plot and use appropriate graphs to establish or verify relationships between
variables
• relate the gradient and the intercepts of straight line graphs to appropriate
linear equations.
• distinguish between systematic and random errors
• make reasonable estimates of the errors in all measurements
• use data, graphs and other evidence from experiments to draw conclusions
• use the most significant error estimates to assess the reliability of conclusions
drawn
SI Base Units
Physical Quantity
Name
Symbol
mass
length
time
m
x
t
electric current
temperature interval
amount of substance
luminous intensity
I
ΔT
n
I
Unit
Name
kilogram
metre
second
ampere
kelvin
mole
candela
‘SI’ comes from the French ‘Le Système International d'Unités’
Symbol cases are significant (e.g. t = time; T = temperature)
Symbol
kg
m
s
A
K
mol
cd
Derived units (examples)
Consist of one or more base units multiplied or divided together
quantity
symbol
unit
area
A
m2
volume
V
m3
density
D or ρ
kg m-3
velocity
u or v
m s-1
momentum
p
kg m s-1
acceleration
a
m s-2
force
F
kg m s-2
work
W
kg m2 s-2
Special derived units (examples)
All named after scientists and/or philosophers to simplify notation
physical quantity
unit
name
symbol (s)
name
symbol
base SI form
force
F
newton
N
kg m s-2
work & energy
W&E
joule
J
kg m2 s-2
power
P
watt
W
kg m2 s-3
pressure
p
pascal
Pa
kg m-1 s-2
electric charge
q or Q
coulomb
C
As
p.d. (voltage)
V
volt
V
kg m2 A-1 s-3
resistance
R
ohm
Ω
kg m2 A-2 s-3
frequency
f
hertz
Hz
s-1
Note – Special derived unit symbols all begin with an upper case letter
Some Greek characters used in physics
character
name
use
character
name
use
α
alpha
radioactivity
μ
mu
micro
& muons
β
beta
radioactivity
ν
nu
neutrinos
γ
gamma
radioactivity
π
pi
3.142…
& pi mesons
δ Δ
delta
very small &
finite changes
ρ
rho
density &
resistivity
ε
epsilon
emf of cells
σ Σ
sigma
summation
Κ
kappa
K mesons
τ
tau
tau lepton
θ
theta
angles
φ
phi
work function
λ Λ
lambda
wavelength
& lambda
particle
ω Ω
omega
angular speed
& resistance
Larger multiples
multiple
prefix
symbol
example
x 1000
kilo
k
km
x 1000 000
mega
M
MΩ
x 109
giga
G
GW
x 1012
tera
T
THz
x 1015
peta
P
Ps
x 1018
exa
E
Em
also, but rarely used: deca = x 10, hecto = x 100
Smaller multiples
multiple
prefix
symbol
example
÷ 10
deci
d
dB
÷ 100
centi
c
cm
÷ 1000
milli
m
mA
÷ 1000 000
micro
μ
μV
x 10-9
nano
n
nC
x 10-12
pico
p
pF
x 10-15
femto
f
fm
x 10-18
atto
a
as
Powers of 10 presentation
Answers :
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
There are 5000 mA in 5A
There are 8000 pV in 8 nanovolts
There are 500 μm in 0.05 cm
There are 6 000 000 g in 6 000 kg
There are 4 fm in 4 000 am
There are 5.0 x 107 kHz in 50 GHz
There are 3.6 x 106 ms in 1 hour
There are 0.030 MΩ in 30 k Ω
There are 4.0 x 1028 pC in 40 PC
There are 60 pA in 0.060 nA
Mathematical signs – complete:
sign
meaning
sign
√
>
less than
much greater than
«
≥
mean value
< x2 >
√<x2>
root mean square value
proportional to
less than or equal to
approximately equal to
≠
≡
meaning
equivalent to
finite change
∂
∑
∞
extremely small change
sum of
Mathematical signs – answers:
sign
>
<
»
«
≥
≤
≈
≠
≡
meaning
greater than
less than
much greater than
much less than
greater than or equal to
less than or equal to
approximately equal to
not equal to
equivalent to
sign
√
<x>
< x2 >
√<x2>
α
∆
∂
∑
∞
meaning
square root
mean value
mean square value
root mean square value
proportional to
finite change
extremely small change
sum of
infinity
Significant figures
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)
Complete the table below:
raw number
to 3 s.f.
5672
5.67 x 103
to 0 s.f.
104
2 x 104
18649
0.045632
to 1 s.f.
0.0456
or 4.56 x 10-2
0.05
or 5 x 10-2
10-2
0.00200
or 2.00 x 10-3
0.002
or 2 x 10-3
10-3
900
0.00200308
Answers:
raw number
to 3 s.f.
to 1 s.f.
to 0 s.f.
5672
5.67 x 103
6 x 103
104
18649
1.86 x 104
2 x 104
104
0.045632
0.0456
or 4.56 x 10-2
0.05
or 5 x 10-2
10-2
900
900
9 x 102
103
0.00200308
0.00200
or 2.00 x 10-3
0.002
or 2 x 10-3
10-3
Results tables
Headings should be clear
Physical quantities should have units
All measurements should be recorded (not just the ‘average’)
Reliability and validity of measurements
Reliable
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
Valid
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 = red
Range and mean value of measurements
Range
Mean value < x >
This equal to the difference
between the highest and lowest
reading
This is calculated by adding the
readings together and dividing
by the number of readings
Readings: 45g; 44g; 44g; 47g;
46g; 45g
Readings: 45g; 44g; 44g; 47g;
46g; 45g
Range: = 47g – 44g
= 3g
Mean value of mass <m> =
(45+44+44+47+46+45) / 6
<m> = 45.2 g
Systematic and random errors
Suppose a measurement
should be 567cm
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.
Accuracy and precision of measurements
Accurate
Accurate measurements are
obtained using a good technique
with correctly calibrated
instruments so that there is no
systematic error.
Precise
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
inaccurate
Uncertainty or probable error
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
Uncertainty in a single reading
OR when measurements do not vary
• The probable error is
equal to the precision
in reading the
instrument
• For the scale opposite
this would be
± 0.1 without the
magnifying glass
± 0.02 perhaps with the
magnifying glass
Percentage uncertainty
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 %
Combining uncertainties
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
Powers
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
The equation of a
straight line graph
For any straight line:
y = mx + c
where:
m = gradient
= (yP – yR) / (xR – xQ)
and
c = y-intercept
Direct proportion
The graph below shows
how the extension of a
wire, ∆L varies with the
tension, T applied to the
wire.
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:
– a straight line
– AND passes through the origin
The general equation of the straight line
in this case is: y = mx, with, c = 0
Linear relationships - 1
The graph below shows how
the velocity of a body changes
when it undergoes constant
acceleration, a from an initial
velocity u.
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
Linear relationships - 2
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)
Linear relationships - 3
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
Calculating the y-intercept
The graph opposite shows two
quantities that are linearly related but it
does not show the y-intercept.
P
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)
16
10
6
8
12
Q
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
Answers
1.
Quantity P is related to quantity Q by the equation:
P = 5Q + 7. If a graph of P against Q was plotted what
would be the gradient and y-intercept?
m = + 5; c = + 7
2.
Quantity J is related to quantity K by the equation:
J - 6 = K/3. If a graph of J against K was plotted what
would be the gradient and y-intercept?
m = + 0.33; c = + 6
3.
Quantity W is related to quantity V by the equation:
V + 4W = 3. If a graph of W against V was plotted what
would be the gradient and x-intercept?
m = - 0.25; x-intercept = + 3; (c = + 0.75)
Analogue Micrometer
The micrometer is reading 4.06 ± 0.01 mm
Analogue Vernier Callipers
The callipers reading is 3.95 ± 0.01 cm
NTNU Vernier Applet
Further Reading
Breithaupt chapter 14.3; pages 221 & 222
Internet Links
• Unit Conversion - 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.
Core Notes from Breithaupt pages 219 to 239
Notes from Breithaupt pages 232 & 236
1. Copy table 1 on page 232
2. What is the difference between a base unit and
a derived unit? Give five examples of derived
units.
3. Convert (a) 52 kg into g; (b) 4 m2 into cm2; (c)
6 m3 into mm3 ; (d) 3 kg m-3 into g cm-3
4. How many (a) mg in 1 Mg; (b) Gm in 1 TM; (c)
μs in 1 ks; (d) fV in 1 nV; am in 1 pm?
5. Copy and learn table 2 on page 236
6. Try the summary questions on pages 233 &
237
Notes from Breithaupt pages 219 to 220, 223 to 225 & 233
1.
2.
3.
4.
5.
6.
Define in the context of recording measurements, and give
examples of, what is meant by: (a) reliable; (b) valid;
(c) range; (d) mean value; (e) systematic error; (f) random
error; (g) zero error; (h) uncertainty; (i) accuracy;
(j) precision and (k) linearity
What determines the precision in (a) a single reading and
(b) multiple readings?
Define percentage uncertainty.
Two measurements P = 2.0 ± 0.1 and Q = 4.0 ± 0.4 are
obtained. Determine the uncertainty (probable error) in:
(a) P + Q; (b) Q – P; (c) P x Q; (d) Q / P; (e) P3; (f) √Q.
Measure the area of a piece of A4 paper and state the
probable error (or uncertainty) in your answer.
State the number 1230.0456 to (a) 6 sf, (b) 3 sf and (c) 0 sf.
Notes from Breithaupt pages 238 & 239
1. Copy figure 2 on page 238 and define the
terms of the equation of a straight line graph.
2. Copy figure 1 on page 238 and explain how it
shows the direct proportionality relationship
between the two quantities.
3. Draw figures 3, 4 & 5 and explain how these
graphs relate to the equation y = mx + c.
4. How can straight line graphs be used to solve
simultaneous equations?
5. Try the summary questions on page 239