IB Physics Unit 1. Measurements and uncertainties – Measurement & uncertainties Notes

UNITS AND MEASUREMENTS

PHYSICAL QUANTITY

All the quantities in terms of which laws of physics are described and which can be measured directly or indirectly are called physical quantities. For example mass, length, time, speed, force etc.

TYPES OF PHYSICAL QUANTITY

  1. Fundamental quantities : The physical quantities which do not depend upon other physical quantities are called fundamental or base physical quantities. e.g. mass, length, time temperature electric current, luminous intensity and amount of substance.
  2. Derived quantities : The physical quantities which depend on fundamental quantities are called derived quantities
    e.g. speed, acceleration, force, etc.

UNIT

The process of measurement is a comparison process.
Unit is the standard quantity used for comparison.
The chosen standard for measurement of a physical quantity, which has the same nature as that of the quantity is called the unit of that quantity.
Choice of a unit (Characteristics of a unit):
  • It should be suitable in size (suitable to use)
  • It should be accurately defined (so that everybody understands the unit in same way)
  • It should be easily reproducible.
  • It should not change with time.
  • It should not change with change in physical conditions i.e., temperature, pressure, moisture etc.
  • It should be universally acceptable.
Every measured quantity (its magnitude) comprises of a number and a unit. Ex: In the measurement of time, say
If Q is the magnitude of the quantity (which does not depend on the selection of unit) then
Q = n u = n1 u1 = n2 u2
Where u1 and u2 are the units and n1 and n2 are the numerical values in two different system of units.

FUNDAMENTAL (OR BASE) AND DERIVED UNITS

Fundamental units are those, which are independent of unit of other physical quantity and cannot be further resolved into any other units or the units of fundamental physical quantities are called fundamental or base units. e.g., kilogram, metre, second etc,
All units other than fundamental are derived units (which are dependent on fundamental units) e.g., unit of speed (ms–1) which depends on unit of length (metre) and unit of time (second), unit of momentum (Kgms–1) depends on unit of mass, length and time etc.

SYSTEM OF UNITS

A system of units is a complete set of fundamental and derived units for all physical quantities.

DIFFERENT TYPES OF SYSTEM OF UNITS

F.P.S. (Foot – Pound – Second) system. (British engineering system of units.): In this system the unit of length is foot, mass is pound and time is second.

 

C.G.S. (Centimetre – Gram – Second) system. (Gaussian system of units): In this system the unit of length is centimetre, mass is gram and time is second.

 

M.K.S (Metre – Kilogram – Second) system. This system is related to mechanics only. In this system the unit of length is metre, mass is kilogram and time is second.

 

S.I.  (International system) units. (Introduced in 1971) Different countries use different set of units. To avoid complexity, by international agreement, seven physical quantities have been chosen as fundamental or base physical quantities and two as supplementary. These quantities are

MERITS OF S.I. UNITS

  • SI is a coherent system of units: This means that all derived units are obtained by multiplication and division without introducing any numerical factor.
  • SI is a rational system of units: This is because it assigns only one unit to a particular physical quantity.
  • SI is an absolute system of units: There is no gravitational unit in this system.
  • SI system is applicable to all branches of science.

CONVENTIONS OF WRITING OF UNITS AND THEIR SYMBOLS

  • Unit is never written with capital initial letter.
  • For a unit named after scientist the symbol is a capital letter otherwise not.
  • The unit or symbol is never written in plural form.
  • Punctuations marks are not written after the symbol.

DEFINITIONS OF FUNDAMENTAL UNITS

  • Metre : One metre is equal to 1650763.73 wavelength in vacuum of the radiation corresponding to transition between the levels 2p10 and 5d5 of the krypton – 86 atom
or
The distance travelled by light in vacuum in second is called 1 metre.
  • Kilogram : The mass of cylinder (of height and diameter 39 cm) made of Platinum-iridium alloy kept at International Bureau of weights and measures in paris is defined as 1kg.
  • Second : It is the duration of 9,192,631,770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of Caesium (133) atom.
  • Ampere : It is the current which when flows through two infinitely long straight conductors of negligible cross-section placed at a distance of one metre in air or vacuum produces a force of 2 × 10–7 N/m between them.
  • Candela : It is the luminous intensity in a perpendicular direction, of a surface of 1/600,000 square metre of a black body at the temperature of freezing platinum under a pressure of 1.013 × 105 N/m2.
  • Kelvin : It is the 1/273.16 part of thermodynamic temperature of triple point of water.
  • Mole : It is the amount of substance which contains as many elementary entities as there are in 0.012 kg of Carbon-12.

S.I. PREFIXES

The magnitudes of physical quantities vary over a wide range. For example, the atomic radius, is equal to 10–10m, radius of earth is 6.4×106 m and the mass of electron is 9.1×10–31 kg. The internationally recommended standard prefixes for certain powers of 10 are given in the table:

SOME IMPORTANT PRACTICAL UNITS

  • For large distance (macro-cosm)
Astronomical unit: It is the average distance of the centre of the sun from the centre of the earth.
1 A.U. = 1.496 × 1011m
Light year: It is the distance travelled by the light in vacuum in one year.
1 ly = 9.46 × 1015m
Parsec: One parsec is the distance at which an arc 1A.U. long subtends an angle of one second.
1 parsec = 3.1 × 1016m
  • For small distance (micro-cosm)
1 micron = 10–6m
1 nanometre = 10–9m
1 angstorm = 10–10m
1 fermi = 10–15 m
  • For small area
    1 barn = 10–28m2
  • For heavy mass 
    1 ton = 1000kg
    1 quintal = 100kg
    1 slug = 14.57kg
    1 C.S.L (chandrasekhar limit) = 1.4 times the mass of the sun
  • For small mass 
    1 amu = 1.67 x 10–27kg
    1 pound = 453.6g = 0.4536 kg
  • For small time 
    1 shake = 10–8s
  • For large time
Lunar month: It is the time taken by the earth to complete one rotation about its axis with respect to sun.
1L.M.  = 27.3 days.
Solar day: It is the time taken by the earth to complete one rotation about its axis with respect to sun.
Sedrial day: It is the time taken by earth to complete one rotation on its axis with respect to distant star.
  • For measuring pressure
1 bar = 1atm pressure = 105N/m2 = 760mmHg
1torr = 1 mmHg
1 poiseuille = 10 Poise

DIMENSIONS

The powers to which the fundamental units of mass, length and time must be raised to represent the physical quantity are called the dimensions of that physical quantity.
For example :
Force = mass × acceleration
 = mass ×  = [MLT–2]
Hence the dimensions of force are 1 in mass 1 in length and (– 2) in time.

DIMENSIONAL FORMULA

Unit of a physical quantity expressed in terms of M, L and T is called dimensional formula. It shows how and which of the fundamental quantities represent the dimensions.
For example, the dimensional formula of work is [ML2T–2]

DIMENSIONAL EQUATION

When we equate the dimensional formula with the physical quantity, we get the dimensional equation.
For example : Work = [ML2T–2]

CLASSIFICATION OF PHYSICAL QUANTITIES (ON THE BASIS OF DIMENSIONS)

DIMENSIONAL FORMULA OF SOME IMPORTANT PHYSICAL QUANTITIES

 

SHORT CUTS / TIME SAVING TECHNIQUES
  • To find dimensions of a typical physical quantity which is involved in a number of formulae, try to use that formula which is easiest for you. For example if you want to find the dimensional formula of magnetic induction then you can use the following formulae
Out of these the easiest is probably the third one.
  • If you have to find the dimensional formula of a combination of physical quantities, then instead of finding the dimensional formula of each, try to correlate the combination of physical quantities with a standard formula. For example, if you have to find the dimension of CV2, then try to use formula where E is energy of a capacitor.
  • velocity of light in vacuum
  • Dimensions of the following are same
[ML2T–2]
  • Dimensions of the following are same
Force = Impulse / time
= q v B = q E
= Thrust
= weight  = energy gradient   [MLT–2]
  • The dimension of RC = is same as that of time
  • Dimensions of the following are same
Velocity =     [M°LT–1]
  • Dimensions of the following are same
Frequency   [M°L°T–1]
  • Dimensions of the following are same
(E) Modulus of elasticity = Y (Young’s modulus)
               = B (Bulk modulus)
   = η (Modulus of rigidity)
    = Stress
  = Pressure = [ML–1T–2]
  • Dimensions of the following are same
Acceleration, retardation, centripetal acceleration, centrifugal acceleration, gravitational intensity/strength.       [M°LT–2]
  • Dimensions of the following are same
Water equivalent, thermal capacity, entropy, Boltzmann’s constant. [ML2T–2K–1]

 

KEEP IN MEMORY
The dimensional formula of
  • all trigonometric ratio is [M0L0T0]
  • x in ex is [M0L0T0]
  • ex is [M0L0T0]
  • x in log x is [M0L0T0]
  • log x is [M0L0T0]

DIMENSIONAL ANALYSIS AND ITS APPLICATIONS

Principle of Homogeneity : Only those physical quantities can be added /subtracted/equated /compared which have the same dimensions.

 

USES OF DIMENSIONS
  1. Conversion of one system of unit into another
Example : Convert a pressure of 106 dyne/cm2 in S.I units.
Sol. We know that 1N = 105 dyne ⇒ 1 dyne = 10–5 N
Also 1m = 100 cm ⇒ 1cm = 10–2 m
Now, the pressure 106 dyne/cm2 in SI unit is
  1. Checking the accuracy of various formulae
Example : Check the correctness of the following equation dimensionally
where F = force, η  = coefficient of viscosity, A = area, gradient w.r.t distance, θ = angle of contact
Sol. L.H.S = force = [MLT–2]
R.H.S =
The equation is dimensionally correct.
  1. Derivation of formula
Example : The air bubble formed by explosion inside water performed oscillation with time period T which is directly proportional to Pa db Ec where P is pressure, d is density and E is the energy due to explosion. Find the values of a, b and c.
Sol.  Let us assume that the required expression for time period is T = K Pa db Ec
where K is a dimensionless constant.
Writing dimensions on both sides,
Equating the powers,
a +  b + c = 0 ….(1)
– a  – 3b + 2c = 0 ….(2)
– 2a – 2c = 1 ….(3)
Solving these equations,  we get,
a = , b = , c = .

LIMITATIONS OF DIMENSIONAL ANALYSIS

  • No information about the dimensionless constant is obtained during dimensional analysis
  • Formula cannot be found if a physical quantity is dependent on more than three physical quantities.
  • Formula containing trigonometrical /exponential function cannot be found.
  • If an equation is dimensionally correct it may or may not be absolutely correct.

SIGNIFICANT FIGURES

The number of digits, which are known reliably in our measurement, and one digit that is uncertain are termed as significant figures.

RULES TO DETERMINE THE NUMBERS OF SIGNIFICANT FIGURES

  1. All non-zero digits are significant. 235.75 has five significant figures.
  2. All zeroes between two non-zero digits are significant. 2016.008 has seven significant figures.
  3. All zeroes occurring between the decimal point and the non-zero digits are not significant provided there is only a zero to left of the decimal point.  0.00652 has three significant figures.
  4. All zeroes written to the right of a non-zero digit in a number written without a decimal point are not significant. This rule does not work if zero is a result of measurement. 54000 has two significant figures whereas 54000m has five significant figures.
  5. All zeroes occurring to the right of a non-zero digit in a number written with a decimal point are significant. 32.2000 has six significant figures.
  6. When a number is written in the exponential form, the exponential term does not contribute towards the significant figures. 2.465 × 105 has four significant figures.

 

KEEP IN MEMORY
  • The significant figures depend upon the least count of the instrument.
  • The number of significant figure does not depend on the units chosen.

ROUNDING OFF

  • If digit to be dropped is less than 5 then preceding digit should be left unchanged.
  • If digit to be dropped is more than 5 then one should raise preceding digit by one.
  • If the digit to be dropped is 5 followed by a digit other than zero then the preceding digit is increased by one.
  • If the digit to be dropped is 5 then the preceding digit is not changed if it is even.
  • If digit to be dropped is 5 then the preceding digit is increased by one if it is odd.

ARITHMETICAL OPERATIONS WITH SIGNIFICANT FIGURES AND ROUNDING OFF

  • For addition or subtraction, write the numbers one below the other with all the decimal points in one line. Now locate the first column from the left that has a doubtful digit. All digits right to this column are dropped from all the numbers and rounding is done to this column.  Addition subtraction is then done.
Example : Find the sum of 23.623 and 8.7 to correct significant figures.
Sol.  Step-1 :- 23.623 + 8.7 
Step-2 :-  23.6 + 8.7=32.3
  • In multiplication and division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity, which has minimum number of significant digits.
The insignificant digits are dropped from the result if they appear after the decimal point. They are replaced by zeroes if they appear to the left of the decimal point. The least significant digit is rounded off.
Example : 107.88 (5. S. F.) × 0.610 (3 S. F.) =  65.8068 ≅ 65.8

ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENTS

Accuracy and Precision are two terms that have very different meanings in experimental physics. We need to be able to distinguish between an accurate measurement and a precise measurement. An accurate measurement is one in which the results of the experiment are in agreement with the ‘accepted’ value.
Note:- This only applies to experiments where this is the goal like measuring the speed of light. A precise measurement is one that we can make to a large number of decimal places.

 

The following diagrams illustrate the meaning of terms accuracy and precision
In the above figure : The centre of the target represents the accepted value. The closer to the centre, the more accurate the experiment. The extent of the scatter of the data is a measure of the precision.
A – Precise and accurate
B – Accurate but imprecise
C – Precise but not accurate
D – Not accurate nor precise
When successive measurements of the same quantity are repeated there are different values obtained. In experimental physics it is vital to be able to measure and quantify this uncertainty. (The words “error” and “uncertainty” are often used interchangeably by physicists – this is not ideal – but get used to it!)

 

Error in measurements is the difference of actual or true value and measured value.
Error = True value – Measured value

 

KEEP IN MEMORY
  1. Accuracy depends on the least count of the instrument used for measurement.
  2. In the addition and subtraction operation, the result contains the minimum number of decimal places of the figures being used
  3. In the multiplication and division operation, the result contains the minimum number of significant figures.
  4. Least count (L.C.) of vernier callipers =  one MSD – one VSD
where MSD = mains scale division, VSD = vernier scale division
  1. Least count of screw gauge  (or spherometer)
    where pitch is the ratio of number of  divisions moved on linear scale and number of rotations given to circular scale.
  2. Pure number or unmeasured value do not have significant numbers
  3. Change in the position of decimal does not change the number of significant figures. Similarly the change in the units of measured value does not change the significant figures.

COMMON ERRORS IN MEASUREMENTS

It is not possible to measure the 100% correct value of any physical quantity, even after measuring it so many times. There always exists some uncertainty, which is usually referred to as experimental error.

EXPERIMENTAL ERRORS

Random error : It is the error that has an equal chance of being positive or negative.
It occurs irregularly and at random in magnitude and direction. It can be caused
  • by the lack of perfection of observer
  • if the measuring instrument is not perfectly sensitive.

 

Systematic error :  It tends to occur in one direction either positive or negative. It occurs due to
  • measuring instrument having a zero error.
  • an instrument being incorrectly calibrated (such as slow- running-stop clock)
  • the observer persistently carrying out a mistimed action (e.g., in starting and stopping a clock)
For measuring a particular physical quantity, we take a number of readings. Let the readings be X1, X2…………,Xn. Then the mean value is found as follows

METHODS OF EXPRESSING ERROR

Absolute error :  It is the difference between the mean value and the measured value of the physical quantity.
|ΔX1|  = |Xmean–X1|  
…………………………….
…………………………….
|ΔXn|  = |Xmean–Xn|  
Mean absolute error:
ΔXmean or =

 

Relative error : It is the ratio of the mean absolute error and the value of the quantity being measured.

 

Percentage error : It is the relative error expressed in percent
Percentage error

TO FIND THE MAXIMUM ERROR IN COMPOUND QUANTITIES

SUM AND DIFFERENCE
We have to find the sum or difference of two values given as (a ± Δa) and (b ± Δb), we do it as follows
X ± ΔX = (a ± Δa) + (b ± Δb) = (a + b) ± (Δa + Δb)
⇒ X = a + b and ΔX = Δa + Δb in case of sum
And X = (a – b) and ΔX = Δa + Δb in case of difference.

 

PRODUCT AND QUOTIENT
We add the fractional or percentage errors in case of finding product or quotient.
If P = ab then
If  then

 

POWER OF A QUANTITY
If  x = an then
Example :   
For, If and
Then × 100 = (2 × 3 + 4)% = 10%
Similarly :

 

KEEP IN MEMORY
  1. More the accuracy, smaller is the error.
  2. Absolute error |ΔX| is always positive.
  3. |ΔX| has the same dimensions as that of X.
  4. If the least count of measuring instrument is not given and the measured value is given the least error in the measurement can be found by taking the last digit to be 1 and rest digit to be zero. For e.g. if the measured value of mass m = 2.03 kg then.
  5. If a number of physical quantities are involved in an expression then the one with higher power contributes more in errors and therefore should be measured more accurately.
  6. Relative error is a dimensionless quantity.
  7. We are always interested in calculating the maximum possible error.

Measurement and uncertainties

1.2.1 State the fundamental units in the SI system.

Many different types of measurements are made in physics. In order to provide a clear and concise set of data, a specific system of units is used across all sciences. This system is called the International System of Units (SI from the French “Système International d’unités”).

The SI system is composed of seven fundamental units:

Fundamental units:

  • Length: metre (m)

  • Mass: kilogram (kg)

  • Time: second (s)

  • Electrical current: ampere (A) 

  • Temperature: kelvin (K)

  • Amount of substance: mole (mol)

  • Luminous intensity: candela (cd) (Not required for the IB!)

Figure 1.2.1 – The fundamental SI units
QuantityUnit nameUnit symbol
masskilogramkg
timeseconds
lengthmeterm
temperaturekelvinK
Electric currentampereA
Amount of substancemolemol
Luminous intensitycandelacd

Note that the last unit, candela, is not used in the IB diploma program.

Metric multipliers: correspond to a power of ten, e.g. kilo (k) – 10³.

  • All available on the Physics Data Booklet! (page 5)

PhysicsGrandezas.png

Scientific notation: convenient way of expressing numbers that are too small or too big.

  • Notation: m x 10n, where 1 ≤ m < 10 and n is an integer (positive or negative).

  • Example: 213 000 000 = 2.13 x 108

Order of magnitude: approximation of a number to the nearest power of ten.

  • Example: 0.0000945 ≅ 10-4

Significant figures (s.f.) 

The number of digits that should be used to express a certain number, which shows how precise the information is. In any calculation or experimentation, the final answer should be expressed with the same number of s.f. as the value with least s.f. used.

Rules to count the number of significant figures:

  • Non-zero numbers are always significant. Example: 1234 – four s.f.

  • “Sandwiched” zeros are always significant. Example: 5403 – four s.f.

  •  Zeros to the left are never significant. Example: 0.0004578 – four s.f.

  • Zeros to the right are only significant if there is a point. Example: 1403.00000 – nine s.f.

Rules to round a number:

  • If the number following the last significant digit is less than five, the digit remains equal, e.g. 678.4 (4 s.f.) rounded to 3 s.f. = 678.

  • If the number following the last significant digit is greater than five, the digit rounds up (i.e. +1), e.g. 678.6 (4 s.f.) rounded to 3 s.f. = 679.

  • If the number following the last significant digit is five and it is only followed by zeros:

    • The last significant digit (number before five) remains equal if it is even, e.g. 3.2500 (5 s.f.) rounded to 2 s.f. = 3.2.

    • The last significant digit (number before five) rounds up if it is odd, e.g. 3.3500 (5 s.f.) rounded to 2 s.f. = 3.4.

1.2.2 Distinguish between fundamental and derived units and give examples of derived units.

In order to express certain quantities we combine the SI base units to form new ones. For example, if we wanted to express a quantity of speed which is distance/time we write m/s (or, more correctly m s-1). For some quantities, we combine the same unit twice or more, for example, to measure area which is length x width we write m2.

Certain combinations or SI units can be rather long and hard to read, for this reason, some of these combinations have been given a new unit and symbol in order to simplify the reading of data.
For example: power, which is the rate of using energy, is written as kg m2 s-3. This combination is used so often that a new unit has been derived from it called the watt (symbol: W).

Below is a table containing some of the SI derived units you will often encounter:

Table 1.2.2 – SI derived units
SI derived unitSymbolSI base unitAlternative unit
newtonNkg m s-2
jouleJkg m2 s-2N m
hertzHzs-1
wattWkg m2 s-3J s-1
voltVkg m2 s-3 A-1W A-1
ohmΩkg m2 s-3 A-2V A-1
pascalPakg m-1 s-2N m-2

1.2.3 Convert between different units of quantities.

Often, we need to convert between different units. For example, if we were trying to calculate the cost of heating a litre of water we would need to convert between joules (J) and kilowatt hours (kW h), as the energy required to heat water is given in joules and the cost of the electricity used to heat the water is a certain price per kW h.

If we look at table 1.2.2, we can see that one watt is equal to a joule per second. This makes it easy to convert from joules to watt hours: there are 60 second in a minutes and 60 minutes in an hour, therefor, 1 W h = 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h being a prefix standing for kilo which is 1000).

1.2.4 State units in the accepted SI format.

There are several ways to write most derived units. For example: meters per second can be written as m/s or m s-1. It is important to note that only the latter, m s-1, is accepted as a valid format. Therefor, you should always write meters per second (speed) as m s-1 and meters per second per second (acceleration) as m s-2. Note that this applies to all units, not just the two stated above.

1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.

When expressing large or small quantities we often use prefixes in front of the unit. For example, instead of writing 10000 V we write 10 kV, where k stands for kilo, which is 1000. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000).

When expressing the units in words rather than symbols we say 10 kilowatts and 1 milliwatt.

A table of prefixes is given on page 2 of the physics data booklet.

1.2 Uncertainties and Errors

Errors (or uncertainties) in experimentation

All measurements are an estimate of the real value, since they are always subject to errors:​

  • Systematic error: biases measurements in the same direction, e.g. always +0.1 cm.

    • Cause (e.g.): Not adequately calibrated equipment.​

    • Cause (e.g.): Ignoring the effects of friction (given that it is constant).

  • Random error: biases measurements in all directions,​ yielding a wide spread of values.

    • Cause (e.g.): Using a stopwatch manually – some measurements (of time) will be above the real time and some measurements will be below the real time.

    • Cause (e.g.): Changing external circumstances, e.g. alternating atmospheric conditions.

    • Solution: Gathering a wide range of values and then taking the average.

1.2.6 Describe and give examples of random and systematic errors.

Random errors
A random error, is an error which affects a reading at random.
Sources of random errors include:

  • The observer being less than perfect
  • The readability of the equipment
  • External effects on the observed item

Systematic errors

A systematic error, is an error which occurs at each reading.
Sources of systematic errors include:

  • The observer being less than perfect in the same way every time
  • An instrument with a zero offset error
  • An instrument that is improperly calibrated

Accuracy​ and precision:

  • Accurate measurement: Low systematic error – average close to real value.

  • Precise measurement: Low random error – values close to each other.

AccuracyPrecision.png

 

1.2.7 Distinguish between precision and accuracy.

Precision
A measurement is said to be accurate if it has little systematic errors.

Accuracy
A measurement is said to be precise if it has little random errors.

A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error).

Errors (or uncertainties) in measurements

Estimation of random errors ​in instruments: +- uncertainty. The uncertainty should always have the same number of decimal places as the value measured, and normally only 1 s.f.​​

  • Digital instrument (e.g. stopwatch): Smallest possible width of graduation, i.e. smallest division that the instrument can read.

    • Example: Image to the right (stopwatch)Stopwatch.jpg

      • Smallest width of graduation:​ 0.01 s

      • Uncertainty: +- 0.01 s

      • Value:  8.78 +- 0.01 s​​

  • Analogical instrument (e.g. a ruler): Half the smallest possible width of graduation​​​

  • Ruler 1:

    • Smallest width of graduation:​ 1 cmRuler1&2.png

    • Uncertainty: +- 0.5 cm

    • Value: 12.5 +- 0.5 cm

  • Ruler 2: more precise than Ruler A.

    • Smallest width of graduation:​ 0.1 cm

    • Uncertainty: +- 0.05 cm

    • Value:12.50 +- 0.05 cm

Errors (or uncertainties) in calculations

Consider the following value: L1: 8.3 +- 0.1 cm.

  • Absolute uncertainty (∆x): has the same units as the value, e.g. for L1: +- 0.1 cm

  • Fractional uncertainty: division between the absolute uncertainty and the value itself, e.g. for L1: 0.1/8.3 = 0.012

  • Percentage uncertainty: the product of the fractional uncertainty by 100%, e.g. for L1: 0.012 x 100% = 1.2%

Now consider the following value as well: L2: 7.4 +- 0.5 cm.

Propagation of uncertainties:

  • Addition or Subtraction: Addition or subtraction of the values and addition of the absolute uncertainties.

    • Example: L1 – L2:​ (8.3 +- 0.1) – (7.4+- 0.5) = 0.9 +- 0.6 cm.

  • Multiplication or division: Multiplication or division of the values and ​addition of the fractional uncertainties or percentage uncertainties.

    • Example: ​L1 x L2: (8.3 +- 1.2%) x (7.4 +- 6.8%) = 61 +- 8.0% cm².

  • Power and roots:​ Value raised to a certain power and multiplication of the fractional uncertainty or percentage uncertainty by the value of the power.

    • Example: L1³:​ (8.3 +- 1.2%)3 = 570 +- 3.6% cm³.

Errors (or uncertainties) in graphs

Error box:

  • Uncertainties of one value in a graph is commonly represented by error bars.

  • Error of y-value may differ from the error of x-value.

 

Best-fit line: line that goes through all error bars (it does not have to be a straight line!)

  • Direct proportionality (in the form y = ax) only if best-fit line is a straight line that passes through the origin (0,0).​​​

  • More than 2 points are needed to confirm a relationship​ between two variables (e.g. x and y)

  • Gradient: found by using two points at least half the line’s length away from each other: gradient = rise/run = ∆x/∆y

  • Uncertainty in the gradient:

    ∆gradient = gradientMAX – gradientMIN/2

  • Uncertainty in the y-intercept:

    ∆y – intercept = y – interceptMAX – y – interceptMIN/2

1.2.8 Explain how the effects of random errors may be reduced.

The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their affect on the data.

1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.

The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value.

Example:
Find the speed of a car that travels 11.21 meters in 1.23 seconds.

11.21 x 1.13 = 13.7883

The answer contains 6 significant figures. However, since the value for time (1.23 s) is only 3 s.f. we write the answer as 13.7 m s-1.

The number of significant figures in any answer should reflect the number of significant figures in the given data.

1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.

Absolute uncertainties
When marking the absolute uncertainty in a piece of data, we simply add ± 1 of the smallest significant figure.

Example:

13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1

Fractional uncertainties
To calculate the fractional uncertainty of a piece of data we simply divide the uncertainty by the value of the data.

Example:

1.2 s ± 0.1

Fractional uncertainty:

0.1 / 1.2 = 0.0625

Percentage uncertainties
To calculate the percentage uncertainty of a piece of data we simply multiply the fractional uncertainty by 100.

Example:

1.2 s ± 0.1

Percentage uncertainty:

0.1 / 1.2 x 100 = 6.25 %

1.2.11 Determine the uncertainties in results.

Simply displaying the uncertainty in data is not enough, we need to include it in any calculations we do with the data.

Addition and subtraction
When performing additions and subtractions we simply need to add together the absolute uncertainties.

Example:

Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.01

1.2 + 12.01 + 7.21 = 20.42
0.1 + 0.01 + 0.01 = 0.12
20.42 ± 0.12

Multiplication, division and powers
When performing multiplications and divisions, or, dealing with powers, we simply add together the percentage uncertainties.

Example:

Multiply the values 1.2 ± 0.1, 12.01 ± 0.01

1.2 x 12.01 = 14
0.1 / 1.2 x 100 = 8.33 %
0.01 / 12.01 X 100 = 0.083%
8.33 + 0.083 = 8.413 %

14 ± 8.413 %

Other functions
For other functions, such as trigonometric ones, we calculate the mean, highest and lowest value to determine the uncertainty range. To do this, we calculate a result using the given values as normal, with added error margin and subtracted error margin. We then check the difference between the best value and the ones with added and subtracted error margin and use the largest difference as the error margin in the result.

Example:

Calculate the area of a field if it’s length is 12 ± 1 m and width is 7 ± 0.2 m.

Best value for area:
12 x 7 = 84 m2

Highest value for area:
13 x 7.2 = 93.6 m2

Lowest value for area:
11 x 6.8 = 74.8 m2

If we round the values we get an area of:
84 ± 10 m2

1.2.12 Identify uncertainties as error bars in graphs.

When representing data as a graph, we represent uncertainty in the data points by adding error bars. We can see the uncertainty range by checking the length of the error bars in each direction. Error bars can be seen in figure 1.2.1 below:

Figure 1.2.1 – A graph with error bars

1.2.13 State random uncertainty as an uncertainty range (±) and represent it graphically as an “error bar”.

In IB physics, error bars only need to be used when the uncertainty in one or both of the plotted quantities are significant. Error bars are not required for trigonometric and logarithmic functions.

To add error bars to a point on a graph, we simply take the uncertainty range (expressed as “± value” in the data) and draw lines of a corresponding size above and below or on each side of the point depending on the axis the value corresponds to.

Example:

Plot the following data onto a graph taking into account the uncertainty.

 
Time ± 0.2 sDistance ± 2 m
3.413
5.136
764

Table 1.2.1 – Distance vs Time data

Figure 1.2.2 – Distance vs. time graph with error bars

In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Therefor, we often skip certain points and only add error bars to specific ones. We can use the list of rules below to save time:

  • Add error bars only to the first and last points
  • Only add error bars to the point with the worst uncertainty
  • Add error bars to all points but use the uncertainty of the worst point
  • Only add error bars to the axis with the worst uncertainty

1.2.14 Determine the uncertainties in the gradient and intercepts of a straight- line graph.

Gradient
To calculate the uncertainty in the gradient, we simply add error bars to the first and last point, and then draw a straight line passing through the lowest error bar of the one points and the highest in the other and vice versa. This gives two lines, one with the steepest possible gradient and one with the shallowest, we then calculate the gradient of each line and compare it to the best value. This is demonstrated in figure 1.2.3 below:

Figure 1.2.3 – Gradient uncertainty in a graph

Intercept
To calculate the uncertainty in the intercept, we do the same thing as when calculating the uncertainty in gradient. This time however, we check the lowest, highest and best value for the intercept. This is demonstrated in figure 1.2.4 below:

Figure 1.2.4 – Intercept uncertainty in a graph

Note that in the two figures above the error bars have been exaggerated to improve readability.

1.2.1 State the fundamental units in the SI system.
Mass
Length
Time
Substance amount
Electrical current
Thermodynamic temperature
Luminosity
Kilogram
Meter
Second
Mole
Ampere
Kelvin
Candela
kg
m
s
mol
A
K
cd
1.2.2 Distinguish between fundamental and derived units and give examples of derived units.
Fundamental units are the original SI units, while derived units are new units created from fundamental units.
Energy
Force
Frequency
Pressure
Power
Voltage
Resistance
Joule
Newton
Hertz
Pascal
Watt
Volt
Ohm
J
N
Hz
Pa
W
V
Ω
kg × m^2 / s^2
kg × m / s^2
1 / s
kg / m / s^2
kg × m^2 / s^3
kg × m^2 / s^3 / A
kg × m^2 / s^3 / A^2
1.2.3 Convert between different units of quantities.
Questions should always be double-checked in order to make sure that the answer corresponds with the requested units. Common conversions include joules to kilowatts per hour and joules to electron volts. Sometimes numbers need to be multiplied by constants — for example, the electron volt is equal to 1.60 × 10^-19 joules.
Practice questions
In an unidentified question, an answer is given in joules per second. The question requires a kilowatts per hour value. Which conversions should be made?
Answer: The joules per second value should be multiplied by 3600 to get watts per hour. This value can then be divided by 1000 to get kilowatts per hour.
1.2.4 State units in the accepted SI format.
The IB requires that students write “m × s^-1” instead of “m/s” for all units. This is mentioned in the syllabus:

Picture

 
1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.

Prefix

exa
peta
tera
giga
mega
kilo
hecto
deka

deci
centi
milli
micro
nano
pico
femto
atto

Symbol

E
P
T
G
M
k
h
da

d
c
m
µ
n
p
f
a

Scientific Notation

10^18
10^15
10^12
10^9
10^6
10^3
10^2
10^1
0
10^-1
10^-2
10^-3
10^-6
10^-9
10^-12
10^-15
10^-18

(This is given in your formula booklet.)
1.2.6 Describe and give examples of random and systematic errors.
Random errors are errors that occur irregularly and cannot be attributed to a consistent failure in method or equipment. Systematic errors are continuous errors that have a source, such as a broken instrument.
Random
  • Human error (ie. parallax)
  • Misreading
  • External effects (ie. vibrations and air convection)
Systematic
  • Instrument parallax error
  • An instrument with a zero offset error
  • An instrument being calibrated incorrectly
1.2.7 Distinguish between precision and accuracy.

Picture

 
1.2.8 Explain how the effects of random errors may be reduced.
Random errors can be reduced through multiple trials of the same experiment. This allows for a better elimination of outliers due to random errors. Repeated experimentation will not change systematic errors as they persist throughout every trial and continuously produce imprecise results.
1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.
When giving answers to questions on the exam, answers should always be given with the same amount of significant figures as the smallest amount in the question. For example, if the question contained the numbers 200.345, 600 and 0.2, the answer should be given to one significant figure because 0.2 has the smallest amount of significant figures. Usually, the significant figure amount will remain constant throughout a question. Also note that this is only required when multiplication and division is involved. If the question only asks for the addition of 0.2 and 0.45, the answer should be given as 0.65 regardless of the fact that 0.2 has one significant figure.
1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.

Uncertainties are presented as the tenth of the smallest decimal of a given number. For example:

13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1

These are called absolute uncertainties. Uncertainties can also be presented as fractional uncertainties, where the tenth of the smallest decimal is presented as a fraction of the original number. For example:

1.2 s ± 0.1
0.1 / 1.2 = 0.0625

Percentage uncertainties are fractional uncertainties converted to percentages:

1.2 s ± 0.1
0.1 / 1.2 x 100 = 6.25 %

1.2.11 Determine the uncertainties in results.

Adding/subtracting numbers with uncertainties

  • The uncertainties are simply added or subtracted.
  • For example: “2 ± 0.1 + 3 ± 0.3 = 5 ± 0.4”

Multiplying/dividing numbers with uncertainties

  • The uncertainties need to be converted to fractions or percentages before being added.
  • For example: “2 ± 0.1 × 3 ± 0.3 = 5 ± (0.1/2 + 0.3/3) = 5 ± 0.105”
Functions with uncertainties
  • The value should be calculated with the highest possible components and lowest possible components then compared to find the uncertainty of the answer.
  • For example: “Calculate the area of a field with a length of 12 ± 1 m and width of 7 ± 0.2 m.”
  • The ‘best value’ would be to take 12 and 7 and calculate the area. The highest value uses 13 and 7.2 and the lowest value uses 11 and 6.8. When we work these areas out, we get 84, 93.6 and 74.8. When rounding to the appropriate number of significant figures, we get 84, 94 and 75. This translates to approximately 84 ± 10 m^2.
1.2.12 Identify uncertainties as error bars in graphs.

Picture

 
  • The length of the error bars represents the uncertainty range for each value.
1.2.13 State random uncertainty as an uncertainty range (±) and represent it graphically as an error bar.
  • The uncertainty range is the “± …” value (ie. 2 ± 0.1). This value can be directly drawn onto a graph at each point as an error bar, as shown in the image above.
  • When drawing graphs for the IA, error bars only need to be included if the uncertainty is significant. If they are not included, the student should provide an explanation. If there is a large amount of data, only the largest data point, smallest data point and a few points in-between require error bars.
  • Error bars are not required for logarithmic or trigonometric functions.
1.2.14 Determine the uncertainties in the gradient and intercepts of a straight-line graph.
Calculating the uncertainty range of the best-fit line/gradient requires the “largest” (steepest) gradient and “smallest” (shallowest) gradient. The steepest gradient is drawn from the lowest possible value (bottom error bar) to the highest possible value (top error bar). The shallowest gradient is drawn from the largest value to the smallest value. These two gradients are then calculated and compared to the best-fit line. If, for example, the best-fit has a gradient of 9.0, the gradient of the steepest line is 9.2 and the gradient of the shallowest line is 8.8, the best-fit line can be written as 9.0 ± 0.2. The following graph shows how the gradients are drawn:

Picture

 
The same method can be applied to the y-intercept. In this case, we check the highest and lowest possible value of the intercept in relation to the steepest and shallowest gradients. In the graph below, the highest intercept is around 23 while the lowest is around 11. If the ‘best’ intercept is around 17, the overall y-intercept can be written as 17 ± 6.
Picture

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