The Bernoulli's equation is given by `P+1/2 rho v^(2)+h rho g=k`. Where P= pressure, `rho`= density, v= speed, h=height of the liquid column, g= acceleration due to gravity and k is constant. The dimensional formula for k is same as that for:

Options

A. Velocity gradient
B. Pressure gradient
C. Modulus of elasticity
D. Thrust

Answer

To find the dimensional formula for \( k \) in Bernoulli's equation, we start by analyzing the equation itself:

\[ P + \frac{1}{2} \rho v^2 + \rho g h = k \]

Where:
- \( P \) = pressure
- \( \rho \) = density
- \( v \) = speed
- \( h \) = height of the liquid column
- \( g \) = acceleration due to gravity
- \( k \) = constant

### Step 1: Determine the dimensions of each term in the equation.

1. **Pressure (\( P \))**:
- Pressure is defined as force per unit area.
- The dimensional formula for force is \( [M L T^{-2}] \) (mass × acceleration).
- The area has the dimensional formula \( [L^2] \).
- Therefore, the dimensional formula for pressure is:
\[ [P] = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

2. **Kinetic Energy Density (\( \frac{1}{2} \rho v^2 \))**:
- The term \( \frac{1}{2} \) is a constant and does not affect dimensions.
- Density (\( \rho \)) has the dimensional formula \( [M L^{-3}] \).
- Speed (\( v \)) has the dimensional formula \( [L T^{-1}] \).
- Therefore, \( v^2 \) has the dimensional formula \( [L^2 T^{-2}] \).
- Now, the dimensional formula for \( \rho v^2 \) is:
\[ [\rho v^2] = [M L^{-3}] \cdot [L^2 T^{-2}] = [M L^{-1} T^{-2}] \]

3. **Potential Energy Density (\( \rho g h \))**:
- The acceleration due to gravity (\( g \)) has the dimensional formula \( [L T^{-2}] \).
- Height (\( h \)) has the dimensional formula \( [L] \).
- Therefore, the dimensional formula for \( \rho g h \) is:
\[ [\rho g h] = [M L^{-3}] \cdot [L T^{-2}] \cdot [L] = [M L^{-1} T^{-2}] \]

### Step 2: Combine the terms.

Since all the terms \( P \), \( \frac{1}{2} \rho v^2 \), and \( \rho g h \) have the same dimensional formula \( [M L^{-1} T^{-2}] \), we can conclude that:

\[ k \text{ must also have the same dimensional formula as these terms.} \]

### Step 3: Conclusion.

Thus, the dimensional formula for \( k \) is:

\[ [k] = [M L^{-1} T^{-2}] \]

### Step 4: Compare with the options.

Now, we need to check which of the given options has the same dimensional formula as \( k \).

1. **Velocity Gradient**: This has dimensions of \( [T^{-1}] \) which is not the same as \( [M L^{-1} T^{-2}] \).
2. **Pressure Gradient**: This has dimensions of \( [M L^{-2} T^{-2}] \) which is not the same as \( [M L^{-1} T^{-2}] \).
3. **Modulus of Elasticity**: This has dimensions of \( [M L^{-1} T^{-2}] \) which is the same as \( k \).
4. **Thrust**: This has dimensions of \( [M L T^{-2}] \) which is not the same as \( [M L^{-1} T^{-2}] \).

### Final Answer:

The correct option is **C: Modulus of Elasticity**.

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