Consider the equation
v = 1/4zxt^2.
The dimensions of the variables v, x, and t are [L]/[T], [L], and [T], respectively. The numerical factor 4 is dimensionless. What must be the dimensions of the variable z, such that both sides of the equation have the same dimensions?
L * T^-1 = ? L T^2
? = T^-3
OR DO YOU MEAN???? v = 1/(4zxt^2) ????
L * T^-1 = x^-1 L^-1 T^-2
x^-1 = L^2 T^1
so
L^-2 T^-1
or
1/(L^2T)
1/[T]3
T^-3
To determine the dimensions of the variable z, we need to equate the dimensions of both sides of the equation.
Starting with the dimensions of the right-hand side (RHS) of the equation, we have:
[v] = [1/4zxt^2]
The dimensions for each term on the RHS are as follows:
[1] = dimensionless (since it is a numerical factor)
[4] = dimensionless (as given in the problem)
[z] = [L]^a [T]^b (where a and b are the powers to be determined)
[x] = [L]
[t^2] = [T]^[2c] (where c is the power to be determined)
Substituting these dimensions back into the equation, we have:
[L]/[T] = (dimensionless) * [L]^a [T]^b * [L] * [T]^[2c]
Simplifying, we can equate the dimensions:
[L]/[T] = [L]^[a+1] [T]^[b+2c]
Equating the dimensions for each side, we get the following equations:
1. Length dimension: 1 = a + 1
2. Time dimension: 1 = b + 2c
From equation 1, we find that a = 0.
Substituting a = 0 into equation 2, we have:
1 = b + 2c
From this equation, we can choose any values for b and c as long as they satisfy this equation. For simplicity, let's assume b = 1 and c = 0.
Therefore, the dimensions of the variable z must be:
[z] = [L]^0 [T]^1
[z] = [T]
Thus, the variable z must have the dimensions [T].