The velocity (v) of sound through a medium may be assumed to depend on the density `(rho)` of the medium and modulus of elasticity (E). If the dimensions for elasticity (ratio of stress to strain) are `[ML^(-1)T^(-2)]` then deduce by the method of dimensions the formula for the velocity of sound is

Options

A. `v=sqrt(Erho)`
B. `v=sqrt((E)/(rho))`
C. `v=sqrt((rho)/(E))`
D. `v=(1)/(2)sqrt((E)/(rho))`

Explanation

To deduce the formula for the velocity of sound (v) in terms of the density (ρ) of the medium and the modulus of elasticity (E), we will use the method of dimensional analysis.

### Step-by-Step Solution:

1. **Identify the variables and their dimensions:**
- The velocity of sound (v) has dimensions of [L T^(-1)].
- The density (ρ) has dimensions of [M L^(-3)] (mass per unit volume).
- The modulus of elasticity (E) has dimensions of [M L^(-1) T^(-2)] (ratio of stress to strain).

2. **Assume a relationship:**
We assume that the velocity of sound (v) depends on the density (ρ) and the modulus of elasticity (E). Thus, we can express this relationship as:
\[ v = k \cdot \rho^a \cdot E^b \]
where \( k \) is a dimensionless constant, and \( a \) and \( b \) are the powers to be determined.

3. **Write down the dimensions:**
The dimensions of both sides of the equation must match. The left-hand side has dimensions:
\[ [v] = [L T^{-1}] \]
The right-hand side can be expressed in terms of dimensions:
\[ [\rho^a] = [M^{a} L^{-3a}], \quad [E^b] = [M^{b} L^{-b} T^{-2b}] \]
Therefore, the dimensions of the right-hand side become:
\[ [\rho^a \cdot E^b] = [M^{a+b} L^{-3a-b} T^{-2b}] \]

4. **Set up the equation for dimensions:**
Equating the dimensions from both sides gives:
\[ [L T^{-1}] = [M^{a+b} L^{-3a-b} T^{-2b}] \]

5. **Compare the dimensions:**
From the equation, we can compare the powers of M, L, and T:
- For mass (M): \( a + b = 0 \) (Equation 1)
- For length (L): \( -3a - b = 1 \) (Equation 2)
- For time (T): \( -2b = -1 \) (Equation 3)

6. **Solve the equations:**
From Equation 3:
\[ -2b = -1 \implies b = \frac{1}{2} \]
Substitute \( b \) into Equation 1:
\[ a + \frac{1}{2} = 0 \implies a = -\frac{1}{2} \]

7. **Substitute back into the formula:**
Now that we have \( a \) and \( b \):
\[ v = k \cdot \rho^{-\frac{1}{2}} \cdot E^{\frac{1}{2}} \]
This can be rewritten as:
\[ v = k \cdot \frac{\sqrt{E}}{\sqrt{\rho}} \]

8. **Final expression:**
Assuming \( k = 1 \) for simplicity, the formula for the velocity of sound is:
\[ v = \sqrt{\frac{E}{\rho}} \]