To solve this problem using dimensional analysis, let’s break down the task and variables provided.

We need to find an expression for the mass $M$ of a stone, which depends on the following quantities:

• $v$: the velocity (of water flow in this case),
• $\rho$: the density of water,
• $g$: the acceleration due to gravity.

Step 1: Assign dimensions to each quantity

Let’s list the dimensions of each quantity in terms of the basic units $\text{M}$ (mass), $\text{L}$ (length), and $\text{T}$ (time):

• $M$: Mass of the stone, so dimension is $[M] = \text{M}$.
• $v$: Velocity, which has the dimension $[v] = \text{LT}^{-1}$.
• $\rho$: Density of water, with dimension $[ \rho ] = \text{ML}^{-3}$.
• $g$: Acceleration due to gravity, with dimension $[g] = \text{LT}^{-2}$.

Step 2: Assume a functional relationship

Since $M$ depends on $v$, $\rho$, and $g$, we can assume a relationship of the form: $M \propto v^a \rho^b g^c$ where $a$, $b$, and $c$ are constants to be determined by dimensional analysis.

Step 3: Write the dimensions for both sides

Substitute the dimensions of each variable into the assumed relationship: $\text{M} = (\text{LT}^{-1})^a (\text{ML}^{-3})^b (\text{LT}^{-2})^c$

Expanding each term: $\text{M} = \text{M}^b \text{L}^{a - 3b + c} \text{T}^{-a - 2c}$

Step 4: Set up equations by matching dimensions

For dimensional consistency, the exponents of $\text{M}$, $\text{L}$, and $\text{T}$ on both sides must be equal.

1. For $\text{M}$: $1 = b$
2. For $\text{L}$: $0 = a - 3b + c$
3. For $\text{T}$: $0 = -a - 2c$

Step 5: Solve the equations

1. From the first equation, $b = 1$.
2. Substitute $b = 1$ into the second equation: $0 = a - 3(1) + c \Rightarrow a + c = 3$
3. Substitute $a$ in terms of $c$ in the third equation: $0 = -a - 2c \Rightarrow a = -2c$ Substitute $a = -2c$ into $a + c = 3$: [ -2c + c = 3