Applications of Dimensional Analysis Checking Dimensional Consistency of equations o
o
A dimensionally correct equation must have same dimensions on both sides of the equation.
A dimensionally correct equation need not be a correct equation but a dimensionally incorrect equation is always wrong. It can test dimensional validity but not find exact relationship between the physical quantities.
Example, x = x0 + v0t +( 1 / 2) at 2 Or Dimensionally, [ L ] = [ L ] + [ LT-1 ][ T ] + [ LT-
2 ][ T 2 ]
x – Distance travelled in time t, x0 – starting position, v0- initial velocity, a – uniform acceleration.
Dimensions on both sides will be [ L ] as [ T ] gets cancelled out. Hence this is dimensionally correct equation.
Deducing relation among physical quantities o
To deduce relation among physical quantities, we should know the dependence of one quantity over others( or independent variables) and consider it as product type of dependence.
o Dimensionless constants cannot be obtained using this method. Example, T = k l x g y m z Or [ L 0 M 0 T 1 ] = [ L 1 ] x [ L 1 T-2 ] y [ M 1 ] z = [ L x + y T-2y M z ] Means, x + y = 0,-2y = 1 and z = 0. So, x = ½, y =-½ and z = 0 So the original equation reduces to T = k √l / g