The dimension of a physical quantity are the powers of the fundamental quantities to which they are to be raised to represent a unit of that physical quantity. The dimensions of fundamental quantities are expressed as (i) that of length by L, (ii) that of mass by M, (iii) that of time by T, (iv) that of current by I, (v) that of temperature by (K). Symbolically, dimension of a physical quantity is written by putting that physical quantity within bracket such as [A] and it is read as dimension of A.

Dimensions

Although we can divide or multiply quantities having different dimensions but we cannot add or subtract the quantities havingdifferent dimensions. Therefore velocity + displacement has no physical meaning.

Principle of Homogeneity

(i) Finding dimensions of derived physical quantities (ii) Conversion of one system of units into another(iii) Checking the accuracy of various formulae.(iv) Derivation of formula

(i) acceleration (ii) angle (iii) density (iv) kinetic energy (v) constant of gravitation (vi) permeability of medium.

Find the dimensions of the following quantities:

Check the accuracy of the relation 3.

Check the accuracy of the relation v =

Uses of Dimensional equations

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, where v is the frequency, l is length, T is tension and m is mass per unit length of the string.

where v = velocity of sound and P = pressure. D = density of medium.

F = stretching force, m = mass per unit length of the string.where n = frequency of vibration l = length of the string ,

The number of particles crossing per unit area perpendicular to X-axis in unit time is

where n1 and n2 are number of particles per unit volume for the value of x1 and x2 respectively. The dimensions of diffusion constant D are

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where n1 and n2 are number of particles per unit volume for the value of x1 and x2 respectively. The dimensions of diffusion constant D are

The equation of state of some gases can be expressed as

Here, P is the pressure, V the volume, T the absolute temperature, and a, b, R are constants. The dimensions of a are

In the relation

P is pressure, Z is distance, k is Boltzmann constant and is the temperature. The dimensional formula of will be

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(a) [MoL2To] (b) [M1L2T1](c) [M1LoT-1] (d) [MoL2T-1]

Converting one unit to another unit1.

Convert 1 N into dyne.

Convert 1 erg to Jules

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Deriving formula using dimensional analysis4.

Time period of a simple pendulum depends on length of string and acceleration due to gravity. Find the expression for time period.

Assuming that the critical velocity of flow of a liquid through a narrow tube depends on the radius of the tube, density of the liquid and viscosity of the liquid, find an expression for critical velocity.

If velocity (V), acceleration (A) and force (F) are taken as fundamental quantities instead of mass (M), length (L) and time (T), the dimension of Young's modulus would be

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(a) FA2V-2 (b) FA2V-3

(c) FA2V-4 (d) FA2V-5

A gas bubble, from an explosion under water, oscillates with a period proportional to PadbEc. Where P is the static pressure, d is the density and E is the total energy of the explosion. Find the values of a, b and c.

E, m, J and G denote energy, mass, angular momentum and gravitational constant respectively. Then the dimensions of EJ2/m5G2 are

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(a) angle (b) length (c) mass (d) time

Rules for dimensional analysisOnly Same dimensions quantities can be added or subtracted1.

Ratio of two same dimension quantities is dimension less2.

Log , trigonometric and exponential functions are dimension less.3.

If x and y are lengths and t is time then find dimensions of a, b,c,d,F, k,

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Limitations of Dimensional Analysis 1. Dimension does not depend on the magnitude of the quantity or dimensionless constants involved. Therefore, a dimensionally correct equation need not be actually correct. e.g.:-dimension of 1/T and 2 are same.

2. Dimensional method cannot be used to derived relations other than those involving products of physical parameters, e.g. :

or y = a cos (t - kx) cannot be derived using this method.

T =

cannot be derived by using dimensions.

3. This method cannot be applied to derive formula if in mechanics a physical quantity depends on more than three physical quantities (mass, length, time). e.g.:-

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