Question

What is the dimensions of a/b in the equation p=a-t²/bx, where p is pressure, x is distance and t is time?

Answer 1

Given relation

`p=a-t^2/(bx)`

p has dimension of pressure.

So in rhs both the terms `( a and t^2/(bx))`added should have same dimensions of pressure.

Now dimension of pressure is

`[M][L^-1][T^-2]`

x is the distance so it has dimension of length `[L]`

So dimension of `t^2/(bx)`

`=[M][L^-1][T^-2]=([T^2][L^-1])/"dimension of b"`

So

`"dimension of b "=[T^4][M^-1]`

So the ratio of dimension of `a and b` is

`=([M][L^-1][T^-2])/([T^4][M^-1])`

`=[M^2][L^-1][T^-6]`

Answer 2

In the given relation
`p=a-t^2/(bx)`

`p` is pressure

`:.` on the RHS both the terms also represent pressure.

`=>` dimensions of `t^2/(bx)="pressure"`
`=>b=t^2/(x" pressure")`

Since `t->T and x->L`
`=>b=T^2/(L" pressure")`

Dimensions of `a/b="pressure"/(T^2/(L" pressure"))`
`=>a/b=(L" pressure"^2)/(T^2`

Since dimensions of pressure are `ML^-1T^-2`

We get

Dimensions of `a/b=(L(ML^-1T^-2)^2)/(T^2`

`=>a/b=M^2L^-1T^-6`