Units and measurements is a chapter that carries good weightage and helps students improve their marks. Along with that, this chapter should be on the priority list for students who are trying to regain their understanding of physics. This blog talks about:
- Units
- Physical quantities
- Dimensional formula
- Homogeneity formula
What are units?
Measurement of any physical quantity involves comparison with a certain basic, internationally accepted reference standard; this is called a unit.
What is measurement?
Comparison of any physical quantity with its standard unit is called measurement.
System of units
It is the complete set of units, applicable for both fundamental and derived kinds of physical quantities. The common system of units which is used in mechanics is given below:
- CGS System: In this system, the unit of length is centimetre, the unit of mass is gram, and the unit of time is second.
- FPS System: In this system, the unit of length is foot, the unit of mass is pound, and the unit of time is second.
- MKS System: In this system, the unit of length is metre, the unit of mass is kilogram, and the unit of time is second.
- SI System: This system contains seven fundamental units and two supplementary fundamental units.
International System of Units
The International System of Units, or SI units, defines standard units for measurement of all physical quantities.
| Property | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | Ampere | A |
| Temperature | Kelvin | K |
| Amount of Substance | mole | mol |
| Luminous Intensity | candela | cd |
Definition of Basic and Supplementary Units
Basic Units
- Metre (m): One metre is the distance travelled by light in a vacuum during a time interval of (1/299792458) seconds.
- Kilogram (kg): It is the mass of a platinum-iridium cylinder kept at the National Bureau of Weights and Measures.
- Second (s): The second is the time taken by the light of a specified wavelength emitted by a caesium-133 atom to execute 9192631770 vibrations.
- Ampere (A): One ampere is that current which, when passed through two straight parallel conductors of infinite length and of negligible cross-section kept at a distance of 1 metre apart in the vacuum, produces between them a force equal to 2 x 10-7 newton per metre length.
- Kelvin (K): It is the fraction 1/273.6 of the thermodynamic temperature of the triple point of water.
- Candela (cd): A candela is defined as 1/60 th of the luminous intensity of 1 square centimetre of a perfect black body maintained at the freezing temperature of platinum (1773 0C).
- Mole (mol): One mole is the amount of substance that contains elementary units equal to the number of atoms in 0.012 kg of carbon-12.
Supplementary Units
- Radian (rad): The radian is the angle subtended at the centre of the circle by the arc whose length is equal to the radius of the circle.
- Steradian (Sr): The steradian is the solid angle subtended at the centre of a sphere by a spherical surface of an area equal to the square of its radius.
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Join Xylem TuitionPhysical quantities
A quantity that can be measured and that explains and expresses various physical phenomena as laws is called a physical quantity. A physical quantity is represented completely by its magnitude and unit.
Physical quantity = Magnitude * Unit
Fundamental and Derived Units
- Fundamental or base quantities are basic physical measurements that do not depend on other values. They cannot be broken down or written in terms of other physical quantities.
- Derived units are additional units that can be represented in terms of basic units. For instance, derived units include energy, velocity, and so forth.
Supplementary fundamental units
They are used to measure plane angle and solid angle. Two types of supplementary fundamental units are:
- Radian
- Staradian
| SUPPLEMENTARY QUANTITY | SUPPLEMENTARY UNIT | SYMBOL |
|---|---|---|
| Plane angle | radian | rad |
| Solid angle | steradian | sr |
Dimensional Formula
The dimensional formula of any physical quantity is the formula that tells which of the fundamental units have been used for the measurement of that physical quantity.
Applications of Dimensional Analysis
Dimensional analysis is a fundamental aspect of measurement and is applied in real life physics.
We make use of dimensional analysis for three prominent reasons:
- To check the consistency of a dimensional equation.
- To derive the relation between physical quantities in physical phenomena.
- To change units from one system to another.
LIMITATIONS OF DIMENSIONAL ANALYSIS
(i) It supplies no information about dimensionless constants and the nature (vector and scalar) of physical quantities.
(ii) This method is applicable only if the relation is of product type. It fails in the case of exponential and trigonometric relations.
(iii) It does not predict the numerical correctness of the formula.
| Physical Quantity | Unit | Dimensional Formula |
|---|---|---|
| Acceleration | m s⁻² | M⁰L¹T⁻² |
| Angular Acceleration | rad s⁻² | M⁰L⁰T⁻² |
| Angular Velocity | rad s⁻¹ | M⁰L⁰T⁻¹ |
| Area | m² | M⁰L²T⁰ |
| Density | kg m⁻³ | M¹L⁻³T⁰ |
| Displacement | m | M⁰L¹T⁰ |
| Electric Current | Ampere (or) C/sec | QT⁻¹ |
| EMF/Voltage/Potential | Volt (or) kg m²/sec² C | ML²T⁻²Q⁻¹ |
| Force | Newton (or) kg m/sec² | M¹L¹T⁻² |
| Frequency | Hertz (or) sec⁻¹ | T⁻¹ |
| Impulse | Newton sec (or) kg m/sec | M¹L¹T⁻¹ |
| Inertia | kg m² | M¹L²T⁰ |
| Moment of Inertia | kg m² | M¹L²T⁰ |
| Permeability | henry/m (or) kg m/C² | MLQ⁻² |
| Permittivity | Farad/m (or) sec² C²/kg m³ | T²Q²M⁻¹L⁻³ |
| Power | Watt (W) (or) Joule/sec | M¹L²T⁻³ |
| Pressure | Newton/m (or) kg m⁻¹/sec² | M¹L⁻¹T⁻² |
| Momentum | kg m s⁻¹ | M¹L¹T⁻¹ |
| Resistance/Impedance | Ohm (or) kg m²/sec C² | ML²T⁻¹Q⁻² |
| Volume | m³ | M⁰L³T⁰ |
| Wavelength | m | L¹ |
| Velocity | m s⁻¹ | M⁰L¹T⁻¹ |
| Work/Energy/Heat | joule (or) kg m²/sec² | M¹L²T⁻² |
Homogeneity Principle of Dimensional Analysis
The principle of Homogeneity states that the dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another.
Significant figures
Significant figures (SF) in a measurement are the figures or digits that are known with certainty plus one that is uncertain
Rules to find out the number of significant figures
- Rule I: All the non-zero digits are significant; e.g., 347 has 3 SF
- Rule II: All the zeros between two non-zero digits are significant. e.g.,1002 has 4 SF.
- Rule III: All the zeros to the left of the first non-zero digit are not significant. e.g., 00456 has 3 SF.
- Rule IV: If a number is less than 1, the zeros to the right of the decimal point but to the left of the first non-zero digit are not considered significant. For example, the number 0.00608 has 3 SF.
- V Rule: The trailing zeros (zeros to the right of the last non-zero digit) in a number with a decimal point are significant. e.g., 15.60 has 4 SF
- VI Rule: The trailing zeros in a number without a decimal point may not be significant; e.g., 010100 has 3 SF.
- VII Rule: When the number is expressed in exponential form, the exponential term does not affect the number of S.F. For example, 3.45 × 10⁴ has 3 SF.
MEASUREMENT ERRORS
The difference between the true value and the measured value of a quantity is known as the error of measurement.
Classification of Errors
Errors are usually classified as follows:
Systematic or controllable errors:
Systematic errors are the errors whose causes are known. They can be either positive or negative.
Systematic errors can further be classified into three categories:
(i) Instrumental errors: These errors are due to imperfect design or manufacturing defects or misuse of the measuring instrument. These can be reduced by using more accurate instruments.
(ii) Environmental errors: These errors are due to changes in external environmental conditions such as temperature, pressure, humidity, dust, vibrations, or magnetic and electrostatic fields.
(iii) Observational errors: These errors arise due to improper setting of the apparatus or carelessness in taking observations. This can be reduced by properly setting up the instrument before use.
(iv) Personal Error: This type of error may arise if the experimenter is not careful enough while performing the experiment and noting down the readings.
(v) Random errors: These errors occur due to unknown causes. Therefore, they occur irregularly and are variable in magnitude and sign. Since the causes of these errors are not known precisely, they can not be eliminated.
FAQ'S
The seven fundamental SI units are metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).
Fundamental units are basic units that do not depend on other units, while derived units are expressed in terms of fundamental units.
Supplementary fundamental units are used to measure angles. Radian (rad) is the unit of plane angle, while steradian (sr) is the unit of solid angle.
The dimensional formula of velocity is M⁰L¹T⁻¹.
Significant figures are the digits known with certainty plus one uncertain digit, while measurement error is the difference between the true value and the measured value.
Systematic errors include instrumental, environmental, observational, and personal errors.
Random errors occur due to unknown causes and vary irregularly in magnitude and sign.
Dimensional analysis cannot determine dimensionless constants, distinguish scalar and vector quantities, or verify the numerical correctness of a formula.
Relative error is the ratio of the absolute error to the measured value.
Percentage error is the relative error expressed as a percentage.