Question:
Two lengths, a and b, are measured to be 51 ± 1 cm and 49 ± 1 cm respectively. In which of the following quantities is the percentage uncertainty the largest?
A. a + b
B. a – b
C. a x b
D. a : b
Answer: B
% uncertainty of a = (1 : 51) x 100% = 2.0 %
% uncertainty of b = (1 : 49) x 100% = 2.0 %
A. a + b = (51 + 49) ± (1 + 1) = 100 ± 2 cm
% uncertainty of (a + b) = (2 : 100) x 100 % = 2 %
B. a - b = (51 - 49) ± (1 + 1) = 2 ± 2 cm
% uncertainty of (a - b) = (2 : 2) x 100% = 100 %
C. & D.
When we multiply or divide quantities we add their fractional or percentage uncertainties, so:
% uncertainty of (a x b) and (a : b) = 2.0 % + 2.0 % = 4.0 %
Errors
Core 1. Measurements and uncertainties
1.2 Uncertainties and errors
Uncertainties in measurement
No experimental quantity can be absolutely accurate when measured – it is always subject to some degree of uncertainty. We will look at the reasons for this in this section.
There are two types of error that contribute to our uncertainty about a reading – systematic and random.
Systematic errors
These types of errors are due to the system being used to make the measurement. This may be due to faulty apparatus. For example, a scale may be incorrectly calibrated either during manufacture of the equipment, or because it has changed over a period of time. Rulers warp and, as a result, the divisions are no longer symmetrical. A timer can run slowly if its quartz crystal becomes damaged (not
because the battery voltage has fallen – when the timer simply stops).
The zero setting on apparatus can drift, due to usage, so that it no longer reads zero when it should – this is called a zero error.
Often it is not possible to spot a systematic error and experimenters have to accept the reading on their instruments, or else spend significant effort in making sure that they are re-calibrated by checking the scale against a standard scale. Repeating a reading never removes the systematic error. The real problem with systematic errors is that it is only possible to check them by performing the same task with another apparatus. If the two sets give the same results, the likelihood is that they are both performing well; however, if there is disagreement in the results a third set may be needed to resolve any difference.
In general we deal with zero errors as well as we can and then move on with our experimentation. When systematic errors are small, a measurement is said to be accurate.
Uncertainty when using a 300 mm ruler may be quoted to ±0.5 mm or ±1 mm depending on your
view of how precisely you can gauge the reading. To be on the safe side you might wish to use the
larger uncertainty and then you will be sure that the reading lies within your bounds.You should make sure you observe the scale from directly above and at right angles to the plane of the ruler in order to avoid parallax errors.
In each of uncertainty is quoted to the same precision (number of decimal places) as the reading – it is essential to do this as the number of decimal places is always indicative of precision. When we write an energy value as being 8 J we are implying that it is (8 ± 1) J and if we write it as 8.0 J it implies a precision of ±0.1 J.
Random errors
Random errors can occur in any measurement, but crop up most frequently when the experimenter has to estimate the last significant figure when reading a scale. If an instrument is insensitive then it may be difficult to judge whether a reading would have changed in different circumstances. For a single reading the uncertainty could well be better than the smallest scale division available. But, since you are determining the maximum possible range of values, it is a sensible precaution to use this larger precision. Dealing with digital scales is a problem – the likelihood is that you have really no idea how precisely the scales are calibrated. Choosing the least significant digit on the scale may severely underestimate the uncertainty but, unless you know the manufacturer’s data regarding calibration, it is probably the best you can do.
When measuring a time manually it is inappropriate to use the precision of the timer as the uncertainty in a reading, since your reaction time is likely to be far greater than this. For example, if you timed twenty oscillations of a pendulum to take 16.27 s this should be recorded as being (16.3 ± 0.1) s. This is because your reaction time dominates the precision of the timer. If you know that your reaction time is greater than 0.1 s then you should quote that value instead of 0.1 s.
The best way of handling random errors is to take a series of repeat readings and find the average of each set of data. Half the range of the values will give a value that is a good approximation to the statistical value that more advanced error analysis provides. The range is the largest value minus the smallest value.
Readings with small random errors are said to be precise (this does not mean they are accurate, however).
Source: IB DP Physics Course Companion by David Homer and Michael Bowen-Jones.
1.2 Uncertainties and errors
 |
| http://theendlessfurther.com/wp-content/uploads/2016/12/Uncertainty.jpg |
Uncertainties in measurement
No experimental quantity can be absolutely accurate when measured – it is always subject to some degree of uncertainty. We will look at the reasons for this in this section.
There are two types of error that contribute to our uncertainty about a reading – systematic and random.
Systematic errors
These types of errors are due to the system being used to make the measurement. This may be due to faulty apparatus. For example, a scale may be incorrectly calibrated either during manufacture of the equipment, or because it has changed over a period of time. Rulers warp and, as a result, the divisions are no longer symmetrical. A timer can run slowly if its quartz crystal becomes damaged (not
because the battery voltage has fallen – when the timer simply stops).
The zero setting on apparatus can drift, due to usage, so that it no longer reads zero when it should – this is called a zero error.
Often it is not possible to spot a systematic error and experimenters have to accept the reading on their instruments, or else spend significant effort in making sure that they are re-calibrated by checking the scale against a standard scale. Repeating a reading never removes the systematic error. The real problem with systematic errors is that it is only possible to check them by performing the same task with another apparatus. If the two sets give the same results, the likelihood is that they are both performing well; however, if there is disagreement in the results a third set may be needed to resolve any difference.
In general we deal with zero errors as well as we can and then move on with our experimentation. When systematic errors are small, a measurement is said to be accurate.
Uncertainty when using a 300 mm ruler may be quoted to ±0.5 mm or ±1 mm depending on your
view of how precisely you can gauge the reading. To be on the safe side you might wish to use the
larger uncertainty and then you will be sure that the reading lies within your bounds.You should make sure you observe the scale from directly above and at right angles to the plane of the ruler in order to avoid parallax errors.
In each of uncertainty is quoted to the same precision (number of decimal places) as the reading – it is essential to do this as the number of decimal places is always indicative of precision. When we write an energy value as being 8 J we are implying that it is (8 ± 1) J and if we write it as 8.0 J it implies a precision of ±0.1 J.
Random errors
Random errors can occur in any measurement, but crop up most frequently when the experimenter has to estimate the last significant figure when reading a scale. If an instrument is insensitive then it may be difficult to judge whether a reading would have changed in different circumstances. For a single reading the uncertainty could well be better than the smallest scale division available. But, since you are determining the maximum possible range of values, it is a sensible precaution to use this larger precision. Dealing with digital scales is a problem – the likelihood is that you have really no idea how precisely the scales are calibrated. Choosing the least significant digit on the scale may severely underestimate the uncertainty but, unless you know the manufacturer’s data regarding calibration, it is probably the best you can do.
When measuring a time manually it is inappropriate to use the precision of the timer as the uncertainty in a reading, since your reaction time is likely to be far greater than this. For example, if you timed twenty oscillations of a pendulum to take 16.27 s this should be recorded as being (16.3 ± 0.1) s. This is because your reaction time dominates the precision of the timer. If you know that your reaction time is greater than 0.1 s then you should quote that value instead of 0.1 s.
The best way of handling random errors is to take a series of repeat readings and find the average of each set of data. Half the range of the values will give a value that is a good approximation to the statistical value that more advanced error analysis provides. The range is the largest value minus the smallest value.
Readings with small random errors are said to be precise (this does not mean they are accurate, however).
Source: IB DP Physics Course Companion by David Homer and Michael Bowen-Jones.
Scientific notation and prefixes
Core 1. Measurements and uncertainties
1.1 – Measurements in physics
Scientific notation
One of the fascinations for physicists is dealing with the very large (e.g. the universe) and the very small (e.g. electrons). Many physical constants (quantities that do not change) are also very large or very small. This presents a problem: how can writing many digits be avoided? The answer is to use scientific notation.
The speed of light has a value of 299 792 458 m s^−1. This can be rounded to three significant figures as 300 000 000 m s^−1. There are a lot of zeros in this and it would be easy to miss one out or add another. In scientific notation this number is written as 3.00 × 10^8 m s^−1 (to three significant figures).
Let us analyse writing another large number in scientific notation. The mass of the Sun to four significant figures is 1 989 000 000 000 000 000 000 000 000 000 kg (that is 1989 and twenty-seven zeros). To convert this into scientific notation we write it as 1.989 and then we imagine moving the decimal point 30 places to the left (remember we can write as many trailing zeros as we like to a decimal number without changing it). This brings our number back to the original number and so it gives the mass of the Sun as 1.989 × 10^30 kg.
Apart from avoiding making mistakes, there is a second reason why scientific notation is preferable to writing numbers in longhand. This is when we are dealing with several numbers in an equation. In writing the value of the speed of light as 3.00 × 10^8 m s^−1, 3.00 is called the “coefficient” of the number and it will always be a number between 1 and 10. The 10 is called the “base” and the 8 is the “exponent”.
There are some simple rules to apply:
Metric multipliers (prefixes)
Scientists have a second way of abbreviating units: by using metric multipliers (usually called “prefixes”). An SI prefix is a name or associated symbol that is written before a unit to indicate the appropriate power of 10. So instead of writing 2.5 × 10^12 J we could alternatively write this as 2.5 TJ (terajoule). Check this following figure.
Source:
1.1 – Measurements in physics
 |
| http://figures.boundless-cdn.com/11165/full/scientific-notation2.png |
Scientific notation
One of the fascinations for physicists is dealing with the very large (e.g. the universe) and the very small (e.g. electrons). Many physical constants (quantities that do not change) are also very large or very small. This presents a problem: how can writing many digits be avoided? The answer is to use scientific notation.
The speed of light has a value of 299 792 458 m s^−1. This can be rounded to three significant figures as 300 000 000 m s^−1. There are a lot of zeros in this and it would be easy to miss one out or add another. In scientific notation this number is written as 3.00 × 10^8 m s^−1 (to three significant figures).
Let us analyse writing another large number in scientific notation. The mass of the Sun to four significant figures is 1 989 000 000 000 000 000 000 000 000 000 kg (that is 1989 and twenty-seven zeros). To convert this into scientific notation we write it as 1.989 and then we imagine moving the decimal point 30 places to the left (remember we can write as many trailing zeros as we like to a decimal number without changing it). This brings our number back to the original number and so it gives the mass of the Sun as 1.989 × 10^30 kg.
Apart from avoiding making mistakes, there is a second reason why scientific notation is preferable to writing numbers in longhand. This is when we are dealing with several numbers in an equation. In writing the value of the speed of light as 3.00 × 10^8 m s^−1, 3.00 is called the “coefficient” of the number and it will always be a number between 1 and 10. The 10 is called the “base” and the 8 is the “exponent”.
There are some simple rules to apply:
- When adding or subtracting numbers the exponent must be the same or made to be the same.
- When multiplying numbers we add the exponents.
- When dividing numbers we subtract one exponent from the other.
- When raising a number to a power we raise the coefficient to the power and multiply the exponent by the power.
Metric multipliers (prefixes)
Scientists have a second way of abbreviating units: by using metric multipliers (usually called “prefixes”). An SI prefix is a name or associated symbol that is written before a unit to indicate the appropriate power of 10. So instead of writing 2.5 × 10^12 J we could alternatively write this as 2.5 TJ (terajoule). Check this following figure.
 |
| http://www.icrf.nl/Portals/106/SI_prefixes.jpg |
Significant figures
Core 1. Measurements and uncertainties
1.1 – Measurements in physics
Significant figures
Calculators usually give you many digits in an answer. How do you decide how many digits to write down for the final answer?
Scientists use a method of rounding to a certain number of significant figures (often abbreviated to s.f.). “Significant” here means meaningful. Some rules for using significant figures are:
1.1 – Measurements in physics
Significant figures
Calculators usually give you many digits in an answer. How do you decide how many digits to write down for the final answer?
Scientists use a method of rounding to a certain number of significant figures (often abbreviated to s.f.). “Significant” here means meaningful. Some rules for using significant figures are:
- A digit that is not a zero will always be significant – 345 is three significant figures (3 s.f.).
- Zeros that occur sandwiched between non-zero digits are always significant – 3405 (4 s.f.); 10.3405 (6 s.f.).
- Non-sandwiched zeros that occur to the left of a non-zero digit are not significant – 0.345 (3 s.f); 0.034 (2 s.f.).
- Zeros that occur to the right of the decimal point are significant, provided that they are to the right of a non-zero digit – 1.034 (4 s.f.); 1.00 (3 s.f.); 0.34500 (5 s.f.); 0.003 (1 s.f.).
- When there is no decimal point, trailing zeros are not significant (to make them significant there needs to be a decimal point) – 400 (1 s.f.); 400. (3 s.f.) – but this is rarely written.
Quantities and units
Core 1. Measurements and uncertainties
1.1 – Measurements in physics
Quantities and units
Fundamental quantities are those quantities that are considered to be so basic that all other quantities need to be expressed in terms of them. In the density equation ρ = m/V only mass is chosen to be fundamental (volume being the product of three lengths), density and volume are said to be derived quantities.
Note that the symbols in the equation are all written in italic (sloping) fonts – this is how we can be sure that the symbols represent quantities. Units are always written in Roman (upright) font because they sometimes share the same symbol with a quantity. So “m” represents the quantity “mass”
but “m” represents the unit “metre”. We will use this convention throughout the IB course.
In SI (Système international d’unités) there are seven fundamental units and only six of these will use on the Diploma course (the seventh, the candela, is included for completeness). The fundamental quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The units for these quantities have exact definitions and are precisely reproducible, given the right equipment. This means that any quantity can, in theory, be compared with the fundamental measurement to ensure that a measurement of that quantity is accurate. In practice, most measurements are made against more easily achieved standards so, for example, length will usually be compared with a standard metre rather than the distance travelled by light in a vacuum. They are:
metre (m): the length of the path travelled by light in a vacuum during a time interval of 1/299 792 458 of a second.
kilogram (kg): mass equal to the mass of the international prototype of the kilogram kept at the Bureau International des Poids et Mesures at Sèvres, near Paris.
second (s): the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
ampere (A): that constant current which, if maintained in two straight parallel conductors of infinite length, negligible circular cross-section, and placed 1 m apart in vacuum, would produce between
these conductors a force equal to 2 × 10^–7 newtons per metre of length.
kelvin (K): the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
mole (mol): the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon–12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
candela (cd): the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
All quantities that are not fundamental are known as derived and these can always be expressed in terms of the fundamental quantities through a relevant equation.
The units used for fundamental quantities are unsurprisingly known as fundamental units and those for derived quantities are known as derived units. It is a straightforward approach to be able to express the unit of any quantity in terms of its fundamental units, provided you know the equation relating the quantities. Nineteen fundamental quantities have their own unit but it is also valid, if cumbersome, to express this in terms of fundamental units. For example, the SI unit of pressure is the pascal (Pa), which is expressed in fundamental units as m^−1 kg s^−2.
Notice that when we write the unit newton in full, we use a lower case n but we use a capital N for the symbol for the unit – unfortunately some word processors have default setting to correct this so take care! All units written in full should start with a lower case letter, but those that have been derived in honour of a scientist will have a symbol that is a capital letter. In this way there is no confusion between the scientist and the unit: “Newton” refers to Sir Isaac Newton but “newton” means the unit. Sometimes units are abbreviations of the scientist’s surname, so amp (which is a shortened form of ampère anyway) is named after Ampère, the volt after Volta, the farad, Faraday, etc.
The unit of force is the newton (N). This is a derived unit and can be expressed in terms of fundamental units as kg m s^−2. The reason for this is that force can be defined as being the product of mass and acceleration or F = ma. Mass is a fundamental quantity but acceleration is not. This is such a common unit that it has its own name, the newton, (N ≡ kg m s^−2 – a mathematical way of expressing that the two units are identical). So if you are in an examination and forget the unit of force you could always write kg m s^−2 (if you have time to work it out!).
Source:
1.1 – Measurements in physics
 |
| https://physics.nist.gov/cuu/Units/units.html |
Quantities and units
Fundamental quantities are those quantities that are considered to be so basic that all other quantities need to be expressed in terms of them. In the density equation ρ = m/V only mass is chosen to be fundamental (volume being the product of three lengths), density and volume are said to be derived quantities.
Note that the symbols in the equation are all written in italic (sloping) fonts – this is how we can be sure that the symbols represent quantities. Units are always written in Roman (upright) font because they sometimes share the same symbol with a quantity. So “m” represents the quantity “mass”
but “m” represents the unit “metre”. We will use this convention throughout the IB course.
In SI (Système international d’unités) there are seven fundamental units and only six of these will use on the Diploma course (the seventh, the candela, is included for completeness). The fundamental quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The units for these quantities have exact definitions and are precisely reproducible, given the right equipment. This means that any quantity can, in theory, be compared with the fundamental measurement to ensure that a measurement of that quantity is accurate. In practice, most measurements are made against more easily achieved standards so, for example, length will usually be compared with a standard metre rather than the distance travelled by light in a vacuum. They are:
metre (m): the length of the path travelled by light in a vacuum during a time interval of 1/299 792 458 of a second.
kilogram (kg): mass equal to the mass of the international prototype of the kilogram kept at the Bureau International des Poids et Mesures at Sèvres, near Paris.
second (s): the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
ampere (A): that constant current which, if maintained in two straight parallel conductors of infinite length, negligible circular cross-section, and placed 1 m apart in vacuum, would produce between
these conductors a force equal to 2 × 10^–7 newtons per metre of length.
kelvin (K): the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
mole (mol): the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon–12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
candela (cd): the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
All quantities that are not fundamental are known as derived and these can always be expressed in terms of the fundamental quantities through a relevant equation.
The units used for fundamental quantities are unsurprisingly known as fundamental units and those for derived quantities are known as derived units. It is a straightforward approach to be able to express the unit of any quantity in terms of its fundamental units, provided you know the equation relating the quantities. Nineteen fundamental quantities have their own unit but it is also valid, if cumbersome, to express this in terms of fundamental units. For example, the SI unit of pressure is the pascal (Pa), which is expressed in fundamental units as m^−1 kg s^−2.
Notice that when we write the unit newton in full, we use a lower case n but we use a capital N for the symbol for the unit – unfortunately some word processors have default setting to correct this so take care! All units written in full should start with a lower case letter, but those that have been derived in honour of a scientist will have a symbol that is a capital letter. In this way there is no confusion between the scientist and the unit: “Newton” refers to Sir Isaac Newton but “newton” means the unit. Sometimes units are abbreviations of the scientist’s surname, so amp (which is a shortened form of ampère anyway) is named after Ampère, the volt after Volta, the farad, Faraday, etc.
The unit of force is the newton (N). This is a derived unit and can be expressed in terms of fundamental units as kg m s^−2. The reason for this is that force can be defined as being the product of mass and acceleration or F = ma. Mass is a fundamental quantity but acceleration is not. This is such a common unit that it has its own name, the newton, (N ≡ kg m s^−2 – a mathematical way of expressing that the two units are identical). So if you are in an examination and forget the unit of force you could always write kg m s^−2 (if you have time to work it out!).
Source:
Diploma Programme - HL Syllabus
- Measurements and uncertainties
- Mechanics
- Thermal physics
- Waves
- Electricity and magnetism
- Circular motion and gravitation
- Atomic, nuclear and particle physics
- Energy production
Additional Higher Level (AHL)
- Wave phenomena
- Fields
- Electromagnetic induction
- Quantum and nuclear physics
Option Topics
- Relativity
- Engineering physics
- Imaging
- Astrophysics
Assessment
- Paper 1: 20%
- Paper 2: 36%
- Paper 3: 24%
- Internal assessment: 20%
Diploma Programme - SL Syllabus
Core Topics
- Measurements and uncertainties
- Mechanics
- Thermal physics
- Waves
- Electricity and magnetism
- Circular motion and gravitation
- Atomic, nuclear and particle physics
- Energy production
Option Topics
- Relativity
- Engineering physics
- Imaging
- Astrophysics
Assessment
- Paper 1: 20%
- Paper 2: 40%
- Paper 3: 20%
- Internal assessment: 20%
Diploma Programme - Introduction
The Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop intercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate a range of points of view.
Distinction between SL and HL
Group 4 students at standard level (SL) and higher level (HL) undertake a common core syllabus, a common internal assessment (IA) scheme and have some overlapping elements in the option studied. They are presented with a syllabus that encourages the development of certain skills, attributes and attitudes, as described in the “Assessment objectives” section of the guide.
While the skills and activities of group 4 science subjects are common to students at both SL and HL,
students at HL are required to study some topics in greater depth, in the additional higher level (AHL) material and in the common options. The distinction between SL and HL is one of breadth and depth.
Prior learning
Past experience shows that students will be able to study a group 4 science subject at SL successfully with no background in, or previous knowledge of, science. Their approach to learning, characterized by the IB learner profile attributes, will be significant here.
However, for most students considering the study of a group 4 subject at HL, while there is no intention to restrict access to group 4 subjects, some previous exposure to formal science education would be necessary. Specific topic details are not specified but students who have undertaken the IB Middle Years Programme (MYP) or studied an equivalent national science qualification or a school-based science course would be well prepared for an HL subject.
Source: Physics guide First assessment 2016
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