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