# Units & Dimensions Q&A

1. What is a physical quantity?

Ans : Any quantity which is measurable is called physical quantity.

2. Explain the term Fundamental Physical quantity.

Ans: The physical quantity which is independent or which can not be derived from any other physical quantity is called fundamental physical quantity. EX: Mass, Length and Time.

3.Explain the term Derived physical quantity.Give examples.

Ans :The physical quantity which is dependent on other physical quantity or which is derived from other physical quantity is called derived physical quantity. Ex : Area, Electric charge, Magnetic field strength, power etc.

4.How many fundamental quantities are there in C.G.S; F.P.S and M.K.S systems? What are they?

Ans : There are 3 fundamentals quantities in C.G.S; F.P.S and M.K.S systems, they are mass, length and time.

5.How many fundamental quantities are there in S.I systems? What are they?

Ans : In S.I system 7 fundamental quantities are there,they are i) Mass ii)Length iii)Time iv)Electric current v)Intensity of light vi) Thermodynamic temperature vii) Quantity of matter.

6.How many supplementary quantities are there in S.I system? What are they?

Ans : In S.I system there are 2 supplementary quantities, they are i) Plane Angle ii) Solid Angle.

7. What are the units of length in C.G.S ; F.P.S and M.K.S systems.

Ans : The units of length  are cm,foot and meter respectively in C.G.S ; F.P.S and M.K.S systems .

8. what are the units of fundamental quantities in S.I system?

Ans : Mass → kg ; Length → m ; Time → sec ; Electric current → Amp Thermodynamic temperature → kelvin ;

Intensity of light → candela ; Quantity of matter → mole .

9.what are the units of supplementary quantities in S.I system?

Ans : Plane angle → radian ; Solid angle → steradian .

10. Name the physical quantities whose dimensional formula is $M^0L^1T^0$ ?

Ans : The physical quantities are i)Distance or length ii) displacement iii)wave length

11. Name the physical quantities whose dimensional formula is $M^0L^1T^{-1}$ ?

Ans : The physical quantities are i) speed ii) velocity

12.Name the physical quantities whose dimensional formula is $M^1L^2T^{-2}K^{-1}$ ?

Ans : The physical quantities are i)Thermal capacity ii) Entropy

13. Name the physical quantities whose dimensional formula is $M^1L^1T^{-1}$ ?

Ans : The physical quantities are i)Momentum ii) impulse .

14.Name the physical quantities whose dimensional formula is $M^1L^1T^{-2}$ ?

Ans : The physical quantities are i)force ii ) Tension iii) weight .

15.Name the physical quantities whose dimensional formula is $M^1L^2T^{-2}$ ?

Ans : The physical quantities are i) Work ii) Energy iii) Heat iv)Moment of force Iv) Torque .

16.Name the physical quantities whose dimensional formula is $M^1L^{-1}T^{-2}$?

Ans : The physical quantities are i) pressure ii ) stress iii) Young’s modulus iv) Rigidity modulus v) Bulk modulus .

17.Name the physical quantities whose dimensional formula is $M^0L^0T^{-1}$ ?

Ans : The physical quantities are i) frequency ii) Decay constant iii)Angular velocity .

18 . Name the physical quantities whose dimensional formula is $M^1L^2T^{-1}$ ?

Ans : The physical quantities are i )angular momentum ii )Plank constant .

19. Name the physical quantities whose dimensional formula is $M^1L^0T^{-2}$ ?

Ans : The physical quantities are i )Force constant ii )surface tension .

20. Which physical quantity has negative dimensions in mass ?

Ans : Gravitational constant (G) .

21. State few constants which have dimensions ?

Ans : i) Plnak’s constant (h) ii)Velocity of light in vacuum (c) iii)Permeability of free space ($\mu_0$) iv) Permittivity of free space ($\epsilon_0$) v)Universal gravitational constant (G) vi) Universal gas constant (R)

vii)Boltzmann constant (k) .

22 .which physical quantities have the unit henry ?

Ans : self Inductance and Mutual Inductance have the unit henry .

23. What are the dimensions of  electric conductivity in mass , length and current.

Ans : Electric conductivity has -1,-3 and 2 dimensions  in mass,length and current respectively.

24. What is the unit of electric conductivity in C.G.S and S.I systems?

Ans : It has no unit in C.G.S system ; its unit in S.I system is Siemen/meter or S/m.

25.What are the uses of Dimensional methods?

Ans : To convert units from one system to another. ii )To check the correctness of equations connecting physical quantities iii )To derive the expressions connecting physical quantities.

26. Which is the physical quantity whose S.I unit is Am ?

Ans: Magnetic pole strength.

27. V/m or N/Coulomb are the units of ……. Physical quantity.

Ans : These are the units of Electric field strength.

28.Name five physical quantities which neither have dimensions nor units.

Ans : Refractive Index , specific gravity,susceptibility,dielectric constant,  coefficient of friction.

29. If  V = Xt+Y ; V is the velocity , t is time.What are the dimensional formulas of X and Y ?

Ans : According to principle of homogeneity of dimensions, the dimensions  of M,L and T in every term should be same.

Therefore $M^0L^1T^{-1}$ = X $M^0L^0T^1$ → X = $\frac{M^0L^1T^{-1}}{ M^0L^0T^1}$ ; X → $L^1T^{-2}$ and Y→ $L^1T^{-1}$

30.Which physical quantities does not possess dimensions in mass ?

Ans :Area,volume, velocity, acceleration,angular displacement, angular velocity, angular acceleration.

# Dimensions & Dimensional formulae.

Dimensions: Dimensions of a physical quantity are,the powers to which the fundamental units  are raised to get one unit of the physical quantity.

The fundamental quantities are expressed with following symbols while writing dimensional formulas of derived physical quantities.

Mass →[M] ; Length→[L]; Time→[T]; Electric current →[I] ; Thermodynamic temperature →[K] ;Intensity of light →[cd] ; Quantity of matter →[mol] .

Dimensional Formula :Dimensional formula of a derived physical quantity  is the “expression showing powers to which different fundamental units are raised”.

Ex : Dimensional formula of Force  F →[$M^1L^1T^{-2}$]

Dimensional equation:When the dimensional formula of  a physical quantity is expressed in the form of  an equation by writing the physical quantity on the left hand side and the dimensional formula on the right hand side,then the resultant equation is called Dimensional equation.

Ex: Dimensional equation of Energy is E = [$M^1L^2T^{-2}$] .

Question : How can you derive Dimensional formula of a derived physical quantity.

Ans : We can derive dimensional formula of any derived physical quantity in two ways

i)Using the formula of the physical quantity : Ex: let us derive dimensional formula of Force .

Force F→ma ; substitute the dimensional formula of mass m →[M] ; acceleration →[$LT^{-2}$]

we get F → [M][$L T^{-2}$]; F →[$M^1L^1T^{-2}$] .

ii) Using the units of the derived physical quantity. Ex: let us derive the dimensional formula of momentum.

Unit of Momentum ( p ) → [$kg-m sec^{-1}$] ;

kg is unit of mass → [M] ; is unit of length → [L] ; sec is the unit of time →[T]

Substitute these dimensional formulas in above equation we get p →[$M^1L^1T^{-1}$].

Quantities having no units, can not possess dimensions: Trigonometric ratios, logarithmic functions, exponential functions, coefficient of friction, strain, poisson’s ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities neither possess units nor dimensional formulas.

Quantities having units, but no dimensions : Plane angle,angular displacement, solid angle.These  physical quantities possess units but they does not possess dimensional formulas.

Quantities having both units & dimensions : The following quantities  are examples of such quantities.

Area, Volume,Density, Speed, Velocity, Acceleration, Force, Energy etc.

Physical Constants : These are two types

i) Dimension less constants (value of these constants will be same in all systems of units): Numbers, pi, exponential functions are dimension less constants.

ii)Dimensional constants(value of these constants will be different in different systems of units): Universal gravitational constant (G),plank’s constant (h), Boltzmann’s constant (k), Universal gas constant (R), Permittivity of free space($\in_0$) , Permeability of free space ($\mu_0$),Velocity of light (c).

Principle of Homogeneity of dimensions: The term on both sides of a dimensional equation should have same dimensions.This is called principle of Homogeneity of dimensions. (or) Every term on both sides of a dimensional equation should have same dimensions.This is called principle of homogeneity of dimensions.

Uses of Dimensional equations : dimensional equations are used i) to convert units from one system to another,

ii)to check the correctness of the dimensional equations iii)to derive the expressions connecting different physical quantities..

Limitations of dimensional method: The limitations of dimensional metthod,s are

i)The value of dimensionless constants can not be calculated using dimensional methods,

ii) We can not analyze the equations containing trigonometrical, exponential and logarithmic functions using method of dimensions.

iii)If a physical quantity is sum or difference of two or more than two physical quantities, such physical quantities can not be derived with dimensional methods,

iv)If  any equation having dimensional constants like, G, R etc can not be derived using dimensional methods,

v)If any equations is involving more than three fundamental quantities in it, such expressions can not be derived using dimensional methods.