Consider the equation 𝜐 = 1/3 z xt 2. The dimensions of the variables υ, x, and t are [L]/[T], [L], and [T], respectively. The numerical factor 3 is di- dimensionless. What must be the dimensions of the variable z, such that both sides of the equation have the same dimensions? Show how you determined your answer.

Let's consider the equation given:

\[
\upsilon = \frac{1}{3} z x t^2
\]

where:
- The dimension of \(\upsilon\) is \(\frac{[L]}{[T]}\).
- The dimension of \(x\) is \([L]\).
- The dimension of \(t\) is \([T]\).

We are asked to determine the dimensions of the variable \(z\) such that both sides of the equation have the same dimensions. 

### Step 1: Determine the dimensions on the right-hand side of the equation

The right-hand side involves the terms \(z\), \(x\), and \(t^2\). Let's express the dimensions of each:

- The dimension of \(x\) is \([L]\).
- The dimension of \(t^2\) is \([T]^2\).
- Let the dimension of \(z\) be \([Z]\).

So, the dimensions of the right-hand side are:

\[
\frac{1}{3} \cdot [Z] \cdot [L] \cdot [T]^2
\]

Since the factor \(\frac{1}{3}\) is dimensionless, it doesn't affect the dimensional analysis. Therefore, we can ignore it:

\[
\text{Dimensions of RHS} = [Z] \cdot [L] \cdot [T]^2
\]

### Step 2: Equate the dimensions of both sides of the equation

The left-hand side of the equation has the dimension:

\[
\text{Dimensions of } \upsilon = \frac{[L]}{[T]}
\]

So, we equate the dimensions on both sides:

\[
\frac{[L]}{[T]} = [Z] \cdot [L] \cdot [T]^2
\]

### Step 3: Solve for the dimensions of \(z\)

To isolate the dimension of \(z\), divide both sides by \([L]\) and \([T]^2\):

\[
[Z] = \frac{\frac{[L]}{[T]}}{[L] \cdot [T]^2}
\]

Simplify:

\[
[Z] = \frac{1}{[T]^3}
\]

So, the dimensions of \(z\) must be:

\[
[Z] = [T]^{-3}
\]

### Conclusion:

The dimensions of the variable \(z\) must be \([T]^{-3}\) to ensure that both sides of the equation have the same dimensions.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the dimensional analysis change if \(t\) were replaced with \(t^3\) in the equation?
2. Can you determine the dimensions of \(z\) if \(\upsilon\) had different dimensions, such as \([L]^2/[T]\)?
3. How does dimensional analysis ensure the physical validity of an equation?
4. What happens to the dimensions of \(z\) if the equation involved a square root of \(t\) instead of \(t^2\)?
5. Could the factor \(\frac{1}{3}\) ever affect the dimensional analysis if it were not dimensionless?

**Tip:** Always check the dimensions on both sides of an equation to ensure they match; this is crucial in physics and engineering to verify the correctness of your equations.

Understand how to determine the dimensions of variable z in equations involving units and dimensions. Solve the problem where υ = (1/3) z xt^2 and dimensions of υ, x, and t are [L]/[T], [L], and [T] respectively. Learn the step-by-step dimensional analysis to find the required dimensions of z.

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