# Dimensions

Dimension

The power of fundamental quantity involved in any physical quantity is called dimension of that physical quantity. The representation of physical quantity in terms of power of fundamental quantities involved in it is called dimensional formula of that physical quantity. Three fundamental quantities mass, length, and time are represented by [M], [L] and [T] respectively. All other derived quantities are also expressed in terms of these representations.

E.g. Force = ma = kgm/s2 = [MLT–2]

The dimension of mass is 1, length is 1 and time is –2 in force and the representation [MLT–2] is called dimensional formula of force.

#### Uses of dimensional equations:

• To check the correctness of physical equation:

Any physical equation will be correct only when the dimensions of all terms on left side become equal to the dimensions of all terms on right side. The physical equation which is dimensionally correct may be physically incorrect but the equation which is dimensionally in correct is always incorrect.

• To convert the physical quantity from one system to another system:

Dimensional method is used to convert physical quantity from one system to another system if the unit of measurement in those systems are known. It is based on the value of physical quantity remain same whatever be the system of measurement. i.e.

n1u1 = n2u2

Where n1 and n2 are numerical values in two system and u1 and u2 are the units in two system

It can be written as

${\mathrm{n}}_{1}\left[{\mathrm{M}}_{1}^{\mathrm{a}}{\mathrm{L}}_{1}^{\mathrm{b}}{\mathrm{T}}_{1}^{\mathrm{c}}\right]\phantom{\rule{1em}{0ex}}={\mathrm{n}}_{2}\left[{\mathrm{M}}_{2}^{\mathrm{a}}{\mathrm{L}}_{2}^{\mathrm{b}}{\mathrm{T}}_{2}^{\mathrm{c}}\right]$
scriptlevel="0">M2scriptlevel="0">m1]a[script
level="0">L2scriptlevel="0">L1]b[scriptlevel="0">T2scriptlevel="0">T1]c
• To derive the physical equation:

Dimensional method is used to derive the physical equation if the dependence of a physical quantity on other quantities is known.

Any quantity 'X' depends on other quantities P, Q and R then

$\propto$ Pa Qb Rc

Or, X = k Pa Qb Rc, where k is constant without dimension

#### Limitations of dimensional analysis

• It does not give any information about numerical constants used in physical equation.
• Dimensional method can't be used to derive the physical equation which depends on more than three quantities.
• Dimensional method can't be used to derive the physical equation having trigonometrical function, logarithmic function, exponential function etc.
• Dimensional method can't be used to derive the physical equation containing more than two terms in one side of equation
• Dimensional method does not give any information whether the quantity is vector or scalar.

#### Principle of homogeneity of dimension:

The dimensional equation of each term in a physical equation must be equal if the physical equation is correct. In a physical equation A = B + C, the dimensional equation of A = Dimensional equation of B = Dimensional equation of C.

 S.N. Physical quantity Formula Dimen-sional formula Unit 1. Density massvolume [ML–3T°] Kgm–3 2. Specific gravity Density of bodyDensity of water at 4∘ C [M°L°T°] - 3. Linear momentum mv [MLT-1] Kgm s–1 4. Impulse F × t [MLT–1] Ns 5. Pressure F/A [ML–1T–2] Nm–2 6. Universal gravita-tional constant G=Fr2m1m2 [M-1L3T-2] Nm2/Kg2 7. Work F × d [ML2T–2] Kgm2/s2 8. Moment of force F × r [ML2T–2] Nm 9. Power Wt [ML2T–3] w 10. Surface tension Fl [ML°T–2] Nm–1 11. Surface energy Energy [ML2T–2] J 12. Force constant K=Fx [MT–2] Nm–1 13. Thrust Force [MLT–2] N 14. Stress FA [ML-1T-2] N/m2 15. Strain eL [M°L°T°] - 16. Modulus of elasticity stressstrain [ML–1T–2] N/m2 17. Radius of gyration Length [M°L1T°] m 18. Moment of inertia Mr2 [ML2T°] Kgm2 19. Angle θ=lr [M°L°T°] radian 20. Angular velocity ω=ωt [M°L°T–1] rad/s 21. Angular acceleration α=ωt [M°L°T–2] rad/s2 22. Angular momentum L=Iω [ML2T–1] Kgm2/s 23. Torque τ=Iα [ML2T–2] Nm 24. Frequency f=1T [M°L°T–1] s–1 or Hertz (Hz) 25. Velocity gradient dvdx [M°L°T–1] s–1