According to Bernoullis theoram `p/d + (v^2)/2 + gh` = constant, dimensional formula of the constant.

Options

A. `M^0 L^0 T^0`
B. `M^0 LT^0`
C. `M^0L^2T^(-2)`
D. `M^0 L^2 T^(-4)`

Answer

To find the dimensional formula of the constant in Bernoulli's theorem, we start with the equation given by Bernoulli's theorem:

\[ \frac{p}{d} + \frac{v^2}{2} + gh = \text{constant} \]

Where:
- \( p \) is pressure,
- \( d \) is density,
- \( v \) is velocity,
- \( g \) is acceleration due to gravity,
- \( h \) is height.

### Step 1: Analyze each term in the equation

We need to find the dimensions of each term in the equation to ensure they are all consistent and can be equated to the constant.

### Step 2: Find the dimensions of \( \frac{p}{d} \)

- Pressure \( p \) has the dimensional formula:
\[ [p] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

- Density \( d \) has the dimensional formula:
\[ [d] = \frac{[M]}{[L^3]} = [M][L^{-3}] \]

Now, we can find the dimensions of \( \frac{p}{d} \):
\[ \frac{p}{d} = \frac{[M][L^{-1}][T^{-2}]}{[M][L^{-3}]} = [L^2][T^{-2}] \]

### Step 3: Find the dimensions of \( \frac{v^2}{2} \)

- Velocity \( v \) has the dimensional formula:
\[ [v] = [L][T^{-1}] \]

Thus, the dimensions of \( v^2 \) are:
\[ [v^2] = [L^2][T^{-2}] \]

### Step 4: Find the dimensions of \( gh \)

- Acceleration due to gravity \( g \) has dimensions:
\[ [g] = [L][T^{-2}] \]

- Height \( h \) has dimensions:
\[ [h] = [L] \]

Now, we can find the dimensions of \( gh \):
\[ gh = [L][T^{-2}][L] = [L^2][T^{-2}] \]

### Step 5: Confirm all terms have the same dimensions

From our calculations:
- Dimensions of \( \frac{p}{d} = [L^2][T^{-2}] \)
- Dimensions of \( \frac{v^2}{2} = [L^2][T^{-2}] \)
- Dimensions of \( gh = [L^2][T^{-2}] \)

All terms have the same dimensions, which means they can be equated to the constant.

### Step 6: Determine the dimensional formula of the constant

Since all terms are equal to the constant, the dimensional formula of the constant is:
\[ [L^2][T^{-2}] \]

### Step 7: Write the final answer in dimensional formula notation

The dimensional formula can be expressed as:
\[ [M^0][L^2][T^{-2}] \]

### Final Answer

Thus, the dimensional formula of the constant in Bernoulli's theorem is:
\[ \text{Option 3: } [M^0][L^2][T^{-2}] \]

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