Consider a relation connecting three physical quantities A, Band C given by ABC. The dimensions of A, B and Care [LT], [LT] and [LT] respectively. Therefore, the exponents n and mhave values as

To solve this problem, we need to determine the exponents \(n\) and \(m\) for a relation involving three physical quantities \(A\), \(B\), and \(C\), which all have dimensions \([LT]\). Let's denote the relation as:

\[
A^n B^m C^p = \text{dimensionless}
\]

The quantities \(A\), \(B\), and \(C\) have dimensions \([A] = [LT]\), \([B] = [LT]\), and \([C] = [LT]\). Therefore, their respective dimensions can be written as:

\[
[A] = L^1 T^1, \quad [B] = L^1 T^1, \quad [C] = L^1 T^1
\]

Now, substituting these dimensions into the equation, we get:

\[
[A^n] [B^m] [C^p] = (L^1 T^1)^n (L^1 T^1)^m (L^1 T^1)^p
\]

Simplifying further:

\[
L^{n+m+p} T^{n+m+p}
\]

For the expression to be dimensionless, the exponents of both \(L\) and \(T\) must be zero:

\[
n + m + p = 0 \quad \text{(for L)}
\]
\[
n + m + p = 0 \quad \text{(for T)}
\]

Since both conditions are the same, we only need one equation:

\[
n + m + p = 0
\]

To solve for \(n\) and \(m\), we would need an additional relationship involving these variables. However, if we are considering that \(A\), \(B\), and \(C\) must combine to form a dimensionless quantity, then we can conclude that the only requirement is:

\[
n + m + p = 0.
\]

If no further constraints are provided, this equation alone does not determine unique values for \(n\), \(m\), and \(p\); it simply indicates their sum is zero. 

### Conclusion

The exponents \(n\), \(m\), and \(p\) must satisfy the condition:

\[
n + m + p = 0.
\]

Without additional information, we cannot uniquely determine \(n\) and \(m\). However, possible sets of \((n, m, p)\) could be:

- \(n = 1, m = 1, p = -2\)
- \(n = 2, m = -1, p = -1\)
- \(n = 0, m = 0, p = 0\), etc.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How can dimensional analysis be used to derive equations relating physical quantities?
2. What are the steps to perform dimensional consistency checks on equations?
3. How do different dimensions of physical quantities affect their representation in equations?
4. Can you provide examples of other dimensionless quantities in physics?
5. How do dimensional exponents relate to physical laws, like in Newton's equations or Maxwell's equations?

**Tip:** Always ensure the dimensions on both sides of any physical equation match to maintain consistency.

Explore dimensional analysis in physics by solving for exponents in a relation involving physical quantities with dimensions [LT]. Learn how to apply the fundamental equation and derive possible values for the exponents. Suitable for advanced high school students and physics enthusiasts.

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