IB Physics Topic 1
Measurement and Uncertainties
1.1 Measurements in Physics
A. Quantities vs. Units
Quantities : things that are measureable
( time, length, current, etc.)
IB uses italics for quantities:
v = Ds/Dt
Units: what you measure quantities in:
( seconds, hours, feet, meters,
amperes)
IB uses Roman upright : s, hr, ft, m, A
1.1 Measurements in Physics
B. Fundamental (or Base) Quantities and
Units in the SI System
Quantity
Unit Name
Symbol
length
meter
m
time
second
s
mass
kilogram
kg
current
ampere
A
temperature
kelvin
K
amount
mole
mol
luminous intensity
candela
cd
1.1 Measurements in Physics
Note: units written out in full always use the lower case:
meter, newton, pascal, ampere, second
When the symbol for the unit is written, it may
be upper case if named for a person:
m, N, P, A, s
1.1 Measurements in Physics
C. Derived Quantities and Units in the SI
System - made of Base Units
Example:
Volume comes in cubic meters.
The volume of a cube, 2.0m on each edge is
8.0 m3
Example:
Density comes in kg per cubic meters.
The density of a 16.0 kg cube, 2.0m on each edge is :
r = m/V = 16.0 kg/ 8.0 m3 = 2.0 kg m-3
Example:
Velocity comes meters per second.
The velocity of a car that moves 10.0m in 2.00 seconds is:
v = Ds/Dt = 10.0m / 2.00 s = 5.00 m s-1
1.1 Measurements in Physics
Note: instead of m/s, IB uses :
instead of kg/ m3 IB uses:
m s-1
kg m-3
Example: Acceleration is derived from: a = Dv/Dt
What are the derived units?
Units of velocity are m s-1 and of time are s. Therefore the units of
acceleration are : m s-1 / s =
m s-2
Example: A newton is the SI force unit derived from:
F = ma
What are the derived units?
kg m s-2
1.1 Measurements in Physics
D.
Significant Figures
Counting Significant Figures
For a decimal number: Find the 1st non-zero digit
and count all the way to the right. Trailing zeros
are significant.
002.00
( 3 s.f.)
0.02050
( 4 s.f.)
1.1 Measurements in Physics
D.
Significant Figures - Examples
Round Each Number to 3 s.f.
a. 2.04
b. 1.005
c. 0.002455
a. 2.04
b. 1.01
c.
.00246
1.1 Measurements in Physics
D.
Significant Figures
Counting Significant Figures
For a non – decimal number, trailing zeros are not
significant:
2450
( 3 s.f.)
10205
( 5 s.f.)
1000
(1 s.f.)
1.1 Measurements in Physics
E.
Scientific Notation
The speed of light is 3.00 x 108 m s-1
This number is in Scientific Notion:
The Coefficient is 3.00
The Base is 10
The Exponent is 8
1.1 Measurements in Physics
E.
Division:
Exponent Rules
xa / xb = xa-b
Example: 12 x 105 / 4 x 103 = (12/4) x 105-3
= 3 x 102
Multiplication:
xa . xb = xa+b
Example: (12 x 105 )( 4 x 103) = (12.4) x 105+3
= 48 x 108
1.1 Measurements in Physics
E.
Exponent Rules
Addition or Subtraction:
( Must have same exponent! )
Axa ± Bxa = (A ± B)xa
Example: (12 x 105 ) + (4 x 106 )
= (1.2 x 106 ) + (4 x 106 )
= 5.2 x 106
SI Prefixes
SI Prefixes
Example: Put into scientific notation:
a. 300mm
b. 368mm c. 200MV
a. 300mm = 300 x 10-3m = 3.00 x 10-1 m
b. 368mm = 368 x 10-6m = 3.68 x 10-4 m
c. 200MV = 200 x 106V = 2.00 x 108 V
SI Prefixes
Example: Change .000000056m into
mm.
.000000056m = .056 x 10-6m = .056mm
Example: Divide 10m by 200mm.
10m/200mm = (10m)/(200x10-3m) = (10)/(.200)
= 50
1.1 Measurements in Physics
F.
Order of Magnitude
A number rounded to the nearest power of 10 is
called an “order of magnitude”.
Example: What’s the order of magnitude for a 3.8 gram
sheet of paper?
3.8 grams = 3.8 x 10-3 kg rounds to 10-3 kg
( 3.8 is closer to 1 than 10), so :
The order of magnitude is 10-3 kg.
IB Physics Topic 1.2
Uncertainties and errors
Systematic Error
- Due to the system used to make the
measurements examples: Improperly calibrated instrument,
zero error, damaged instrument
- Cannot be corrected by repeat measurements.
- Causes poor accuracy
IB Physics Topic 1.2
Uncertainties and errors
Random Error
- Due to the estimating a scale readingThe precision of an analog scale is usually
± ½ of the smallest division but this may
not always be true. You look at each
situation and make a reasonable
estimate.
The precision of a digital readout is ±
one of the last digit place shown
Example
Meter stick
Precision = ± .05 cm
(some may use ± .1cm)
Digital caliper
Precision = ± .01 mm
Example
The precision is ± .05 cm (1/2 of .1)
The reading is 3.45cm
The measurement is expressed as:
L = 3.45 ± .05cm
1.2
Precision from several
measurements
To reduce the random error
- take several measurements.
- find the average and round off to the same
decimal place as the precision of the
instrument
- find ½ of the range
Report your measurement M as:
M = average ± (1/2 of the range)
Example
The following lengths were measured with a meter stick:
12.30cm, 12.40cm, 12.20cm, 12.35cm, 12.40cm
1st: Get the average
(12.33 cm)
2nd: Get ½ of the range .5( 12.40-12.20) = .10cm
The reported measurement is:
12.33 ± .10cm
Reporting a Measurement
Summary
The precision of a scale is ½ of the smallest division.
For a single reading:
Value = quantity ± precision
For several readings ( best method) :
Value = average ± (1/2 of the range)
Each part of the measured Value must go to the
same number of decimal places.
1.2
Uncertainties
Consider this reported length:
Length = 12.5 ± .2 cm
We can generalize this to:
Measured Value = Quantity ± Uncertainty
The Absolute Uncertainty is the absolute value
of ±.2cm = | ±.2cm | = .2cm
The Fractional Uncertainty = Uncertainty/Quanatity
= .2/12.5 = .008
The Percent Uncertainty = 100 x Fractional Uncertainty =
.008 x 100 = .8%
Example
For the following temperature readings find
the absolute, fractional, and percent uncertainties.
12.5o, 12.7o, 12.4o
Average = (12.5 + 12.7 + 12.4)/3 = 12.5 (Round to 10ths )
Uncertainty = ½ of range = .5( 12.7-12.4) = .15 , rounds to .2
Value = average ± (1/2 of the range) = 12.5 ± .2o
Absolute uncertainty = .2o
Fractional uncertainty = .2 / 12.5 = .016
Percent uncertainty = 100 x .016 = 1.6%
1.2 Propagation of Uncertainties
Value = Quantity ± Uncertainty
= a ± Da
Addition or Subtraction of 2 Values:
If V1 = a ± Da and
V2 = b ± Db
then V1 + V2 = (a + b) ± (Da +Db)
1.2 Propagation of Uncertainties
Example: Combine two masses if
m1 = 120 ± 5g and m1 = 150 ± 5g
mtotal = (120+150) ± (5+5)g = 270 ± 10g
Example: Subtract 35 ± .5mm from
55 ± .5mm
(55-35) ± (.5+.5)mm = 20 ± 1mm
1.2 Propagation of Uncertainties
Multiplication or Division of 2 Values:
If V1 = a ± Da and
V2 = b ± Db
then V1 x V2 = (ab) ± (bDa +aDb)
OR…………………….
%Uncertainty of the product =
%Uncertainty of a + %Uncertainty of b
Dp/p = Da/a +Db/b
Area Example
A rectangle’s sides are measured as L1 = 10 ± 2cm and L2 = 12 ± 2cm.
Find the area and the uncertainty in the area.
Take a = 10 and Da = 2, b = 12 and Db = 2, then
Area = (ab) ± (bDa +aDb) = 120 ± (12(2) + 10(2)) cm
= 120 ± 44cm
Alternately : DA/A = Da/a +Db/b = 2/10 + 2/12 = 11/30
A = 10(12) = 120cm
and DA = A (11/30) = 44cm
Area = 120 ± 44cm
Example
A solid cylinder has a measured mass of 2.00 kg with a
2.5% uncertainty and a measured volume of .00100 m3
with a 5% uncertainty. Find the density and the
uncertainty of the density.
%uncertainty in the density = % uncertainty of the mass
+ % uncertainty of the volume = 7.5% = .075
Density = m/v = 2.00kg/ .00100 m3 = 2.00 x 103 kg/m3
Density Value = 2.00 x 103 ± .075(2.00 x 103) kg/m3
=
2.00 x 103 ± .15 x 103 kg m-3
1.2 Propagation of Uncertainties
Quantity Raised to a Power:
If a = bn
Da /a = n |Db /b|
%Uncertainty of the product =
%Uncertainty of the base x the power
1.2 Exponent Example
The potential energy, U, of a compressed spring
is given by: U=1/2kx 2 where k = 500 ± 10 N m-1
And x = .25 ± .02 m. Find the quantity and
uncertainty of U.
DU/U = Dk/k + 2Dx/x = 10/500 + 2(.02/.25)
= .02 + .16 = .18 or 18%
Then U=1/2kx 2 = .5(500)(.252) = 6.25 J
And DU = .18U = .18(6.25) = 1.13 J
U = 6.25 ± 1.13 J
Data and Graphing
A student wants to determine the relationship between the distance
a spring stretches and the force used to do the stretching. She does
this by hanging known weights on the spring and measuring the stretch.
What is the independent variable?
That’s what she controls – the weight F / N
How much data should she take?
The independent variable should be varied 5 times – use 5 different weights.
This should be done 3 times – do 3 different trials.
Data
Trial 1
F ± .020N s ± .01 cm
Trial 2
Trial 3
Average Stretch
s ± .01 cm
s ± .01 cm
s ± 1.25 cm
0.098
9.90
9.50
10.20
9.87
0.196
21.00
19.00
18.50
19.50
0.294
31.00
29.00
30.00
30.00
0.392
39.50
37.50
39.00
38.67
0.490
48.00
47.50
48.00
47.83
Note: 5 variations of the independent variable ( F) and 3 trials.
We want to plot F on the y- axis and the average value of s on the x-axis.
Where did she get ± 1.25cm for the uncertainty in the average value?
Data and Graphs
The uncertainty in the average value is ½ of the greatest
range of the dependent variable.
Trial 1
Trial 2 Trial 3 1/2 Range
Average Stretch
F±
s ± .01 s s ± .01
.020N
cm
cm
s ± .01
cm
cm
s ± 1.25 cm
0.098
9.90
9.50
10.20
0.35
9.87
0.196
21.00
19.00
18.50
1.25
19.50
0.294
31.00
29.00
30.00
1.00
30.00
0.392
39.50
37.50
39.00
1.00
38.67
0.490
48.00
47.50
48.00
0.25
47.83
Points to Plot
Average Stretch/ cm
Force/N
Ds ± 1.25 cm
DF ± .020N
9.87
3.430
19.50
12.250
30.00
9.800
38.67
9.800
47.83
2.450
Horizontal Error Bars are 2Ds = 2.50cm
and vertical are 2DF = .040N
Graph with Best Linear Fit
0.600
0.500
0.400
F = 0.010s - 0.006
0.300
Force/N
0.200
0.100
0.000
-10.00
-0.100
0.00
10.00
20.00
30.00
Stretch /cm
40.00
50.00
60.00
Finding the Uncertainty in the
Slope and y-intercept
Find the two extreme lines that
Still pass through the error bars
and find their equations.
The uncertainty in the slope is ½ of the range:
Dm = (.011-.009)/2 =.001
The uncertainty in the intercept is ½ the range: Db =(.03-(-.03))/2 = .03
Final Results
0.600
F = 0.010s - 0.006
0.500
0.400
0.300
Force/N
0.200
m = .010 ±.001 N cm-1
b = .006 ± .030 N
0.100
0.000
-10.00
0.00
-0.100
10.00
20.00
30.00
Stretch /cm
40.00
50.00
60.00