Units and Dimensions

Physical Quantities
All the quantities in terms of which laws of physics are described, and whose measurement is necessary are called physical quantities.
Units
The standard scale which is used to measure value of any physical quantity is called units.
Fundamental Units
Those physical quantities which are independent to each other are called fundamental quantities and their units are called fundamental units.
S.No.
 Fundamental Quantities
 Fundamental Units
 Symbol
1.
Mass
 kilogram
Kg
2.
Length
 metre
m
3.

Time
 second
sec or s
4.
Temperature
 kelvin
K
5.
Electric current
 ampere
A
6.
Luminous intensity
 candela
cd
7.
Amount of substance
 mole

mol
Supplementary Fundamental Units
Radian and steradian are two supplementary fundamental units. It measures plane angle and solid angle respectively.
S.No.
 Supplementary Fundamental Quantities
 Supplementary Unit
 Symbol
1.
Plane angle
  radian
 rad
2.
Solid angle
steradian
Sr
Derived Units
Those physical quantities which are derived from fundamental quantities are called derived quantities and their units are called derived units.e.g., velocity, acceleration, force, work etc.
Definitions of Fundamental Units
The seven fundamental units of SI have been defined as under.
Fundamental Units
Definition
1 kilogram
A cylindrical prototype mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. It is mass of 5.0188 x 1025 atoms of carbon-12.
1 metre
1 metre is the distance that contains 1650763.73 wavelength of orange-red light of Kr-86.
1 second
1 second is the time in which cesium atom vibrates 9192631770 times in an atomic clock.
1 kelvin
1 kelvin is the (1/273.16) part of the thermodynamics temperature of the triple point of water.
1 candela
1 candela is (1/60) luminous intensity of an ideal source by an area of cm’ when source is at melting point of platinum (1760°C).
1 ampere
1 ampere is the electric current which it maintained in two straight parallel conductor of infinite length and of negligible cross-section area placed one metre apart in vacuum will produce between them a force 2 x 10-7 N per metre length.
1 mole
1 mole is the amount of substance of a system which contains a many elementary entities (may be atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope 6C12.
Systems of Units
A system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:
CGS System
FPS System
MKS System
SI System
  • unit of length :centimetre
  • unit of mass :gram
  • unit of time :second
  • unit of length : foot
  • unit of mass :pound
  • unit of time :second
  • unit of length : metre
  • unit of mass :kilogram
  • unit of time :second

This system contain seven fundamental units and two supplementary fundamental units as given in the above table.
Dimensions
Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula.Symbolically,dimensional formula for a physical quantity,say X is represented by putting it in the square bracket [ X ] (to be read as dimensional formula of X).
e.g. Force has dimensional formula given by
[Force] = [ M L T−2 ]
i.e. force has dimensions,1 in mass,1 in length and -2 in time.
Note: Whenever the dimensions of a physical quantity is equated with its dimensional formula,we get a dimensional equation.
Dimensional Formula of Some Physical Quantities
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Specific energy
Derived quantity
SI units
Dimensional Formula
Plane angle
radian (rad)
1
Solid angle
steradian (sr)
1
Absorbed dose rate
Gy s−1
L2 T−3
Acceleration
m s−2
L T−2
Angular acceleration
rad s−2
T−2
Angular speed
rad s−1
T−1
Angular momentum
kg m2 s−1
M L2 T−1
Area
m2
L2
Area density
kg m−2
M L−2
Capacitance
farad (F = A2 s4 kg−1m−2)
I2 T4 M−1 L−2
Catalytic activity
katal (kat = mol s−1)
N T−1
Catalytic activity concentration
kat m−3
N L−3 T−1
Chemical potential
J mol−1
M L2 T−2 N−1

Molar concentration
mol m−3
N L−3
Current density
A m−2
I L−2
Dose equivalent
sievert (Sv = m2 s−2)
L2 T−2
Dynamic Viscosity
Pa s
M L−1 T−1
Electric Charge
coulomb (C = A s)
I T
Electric charge density
C m−3
I T L−3
Electric displacement
C m−2
I T L−2
Electric field strength
V m−1
M L T−3 I−1
Electrical conductance
siemens (S = A2 s3 kg−1 m−2)
L−2 M−1 T3 I2
Electric potential
volt (V = kg m2 A−1 s−3)
L2 M T−3 I−1
Electrical resistance

ohm (Ω = kg m2 A−2 s−3)
L2 M T−3 I−2
Energy
joule (J = kg m2 s−2)
M L2 T−2
Energy density
J m−3
M L−1 T−2
Entropy
J K−1
M L2 T−2 Θ−1
Force
newton (N = kg m s−2)
M L T−2
Impulse
kg m s−1
M L T−1
Frequency
hertz (Hz =s−1)
T−1
Half-life
s
T
Heat
J
M L2 T−2
Heat capacity
J K−1
M L2 T−2 Θ−1
Heat flux density
W m−2
M T−3
Illuminance
lux (lx = cd sr m−2)
J L−2
Impedance
ohm (Ω = kg m2 A−2 s−3)
L2 M T−3 I−2
Index of refraction
1
Inductance
henry (H = kg m2 A−2 s−2)
M L2 T−2 I−2
Irradiance
W m−2
M T−2
Linear density
M L−1
Luminous flux
lumen (lm = cd sr)
J
Magnetic field strength
A m−1
I L−1
Magnetic flux
weber (Wb = kg m2 A−1 s−2)
M L2 T−2 I−1
Magnetic flux density
tesla (T = kg A−1 s−2)
M T−2 I−1
Magnetization
A m−1
I L−1
Mass fraction
kg/kg
1
(Mass) Density (volume density)
kg m−3
M L−3
Mean lifetime
s
T
Molar energy
J mol−1
M L2 T−2 N−1
Molar entropy
J K−1 mol−1
M L2 T−2 Θ−1 N−1
Molar heat capacity
J K−1 mol−1
M L2 T−2 N−1
Moment of inertia
kg m2
M L2
Momentum

N s
M L T−1
Permeability
H m−1
M L−1 I−2
Permittivity
F m−1
I2 M−1 L−3 T4
Power
watt (W)
M L2 T−3
Pressure
pascal (Pa = kg m−1 s−2)
M L−1 T−2
(Radioactive) Activity
becquerel (Bq = s−1)
T−1
(Radioactive) Dose
gray (unit) (Gy = m2 s−2)
L2 T−2
Radiance
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M T−3
Radiant intensity
W sr−1
M L2 T−3
Reaction rate
mol m−3 s−1
N L−3 T−1
Speed
m s−1
L T−1
J kg−1
L2 T−2
Specific heat capacity
J kg−1 K−1
L2 T−2 Θ−1
Specific volume
m3 kg−1
L3 M−1
Spin
kg m2 s−1
M L2 T−1
Stress
Pa
M L−1 T−2
Surface tension
N m−1 or J m−2
M T−2
Thermal conductivity
W m−1 K−1
M L T−3 Θ−1
Torque
N m
M L2 T−2
Velocity
m s−1
L T−1
Volume
m3
L3
Wavelength
m
L
Wavenumber
m−1
L−1
Weight
newton (N = kg m s−2)
M L T−2
Work
joule (J = kg m2 s−2)
M L2 T−2
Homogeneity Principle
If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.
Mathematically,we can write
[LHS of equation] = [RHS of equation]
Applications of Dimensional analysis:
  1. To check the accuracy of physical equations.
  2. To find the physical relationship of a physical quantity

    with other physical quantities knowing its dependency on others.

  3. To change a physical quantity from one system of units to another system of units.
Significant Figures
The number of figures required to specify a given measurement are called the significant figures.Though the last digit of the measurement is always doubtful,yet it is included in the number of significant figures.As an example if we measure the length of an object as 5046 cm,then it has 3 significant figures.
Rules for Finding Significant Figures
  1. All non-zeros digits are significant figures, e.g., 5269 m has 4 significant figures.
  2. All zeros occuring between non-zero digits are significant figures, e.g., 20005 has 5 significant figures.
  3. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3 significant figures.
  4. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures.
  5. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures.
Rules of rounding off significant figures
  1. If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5.
  2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5.
  3. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6.
  4. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6.
Significant Figures in algebric Operations
(i) In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g.,
If l1= 4.326 m and l2 = 1.50 m
Then, l1 + l2 = (4.326 + 1.50) m = 5.826 m
As l2 has measured upto two decimal places, therefore
l1 + l2 = 5.83 m
(ii) In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values.
e.g., If length ,l = 12.5 m and breadth ,b = 4.125 m.
Then, area A = l x b = 12.5 x 4.125 = 51.5625 m²
As l has only 3 significant figures, therefore
A= 51.6 m²

Error

The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error.
1. Absolute Error
The difference between the true(actual) value and the measured value of a quantity is called absolute error.
If a1 , a2, a3 ,…, an are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (am) of the quantity.
The absolute error in measured values is given by
Δa1 = am – a1
Δa2 = am – a1
.
.
.
Δam = Δam – Δan
2. Mean Absolute Error
The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error.
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3. Relative Error: The ratio of mean absolute error to the actual value is called relative error.
LError5
4. Percentage Error: The relative error expressed in percentage is called percentage error.
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Propagation of Error

(i) Error in Addition or Subtraction
Let y = a ± b
If the measured values of two quantities a and b are (a ± Δa and (b ± Δb), then maximum absolute error in their addition or subtraction
Δy = ±(Δa + Δb)
The percentage of error in the value of y = daum_equation_1423395776702
(ii) Error in Multiplication or Division
Let y = a × b or y = (a/b)
If the measured values of a and b are (a ± Δa) and (b ± Δb), then maximum relative error
daum_equation_1423395977772
percentage error in value of y is given by
daum_equation_1423396064963
percentage error in value of y = percentage error in value of a + percentage error in value of b
iii) Error in power functions:
daum_equation_1423396594329
percentage error in value of y = m × (percentage error in value of A) + n × ( percentage error in value of B ) + p × ( percentage error in value of C )
Note:constant k does not effect the value of error.

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