Consider the equation v = (1/7)zxt2. The dimensions of the variables v, x, and t are [L/T], [L], and [T] respectively. The numerical factor 7 is dimensionless. What must be the dimensions of the variable z, such that both sides of the equation have the same dimensions?
z=L/T*L*T=L^2
To determine the dimensions of the variable z in the equation v = (1/7)zxt^2, we need to equate the dimensions on both sides of the equation.
Let's break down the dimensions of each variable:
- The variable v has the dimension [L/T] (length divided by time), as given in the problem statement.
- The variable x has the dimension [L] (length), as given in the problem statement.
- The variable t has the dimension [T] (time), as given in the problem statement.
Now, let's analyze the dimensions of the equation:
On the left side, we have the variable v, which has the dimension [L/T].
On the right side, we have (1/7)zxt^2.
The numerical factor 1/7 is dimensionless, meaning it does not have any dimensions.
The variables x and t have dimensions [L] and [T], respectively.
To make both sides have the same dimensions, we need to adjust the variable z.
Since v = (1/7)zxt^2, the dimensions of the right side of the equation should also be [L/T].
We can start by multiplying the variables on the right side:
z * [L] * [T]^2 = [L/T]
Simplifying this equation, we get:
[L * T^2] = [L/T]
To have the same dimensions on both sides, the powers of length and time must be equal, so:
1 = -1
Since the dimensions of [L * T^2] do not equal [L/T], there is no possible dimension for z that would make both sides of the equation have the same dimensions. This suggests that there may be an error or inconsistency in the equation or the given dimensions.