The blades of a wind mill sweep out a circle of area A. If wind flows with velocity v perpendicular to blades of wind mill and its density is `p`, then the mechanical power received by wind mill is

Options

A. `pAv^(3)/2`
B. `pAv^(2)/2`
C. `pAv^(2)`
D. `2pAv^(2)`

Answer

To find the mechanical power received by a windmill, we can follow these steps:

### Step 1: Understand the parameters
- The area swept by the windmill blades is \( A \).
- The wind flows with velocity \( v \) perpendicular to the blades.
- The density of the wind is \( \rho \).

### Step 2: Calculate the volume of air passing through the windmill
The volume of air passing through the windmill in time \( t \) can be calculated as:
\[ \text{Volume} = \text{Area} \times \text{Velocity} \times \text{Time} = A \cdot v \cdot t \]

### Step 3: Calculate the mass of the air passing through
Using the density of the air, the mass of the air passing through the windmill in time \( t \) is given by:
\[ \text{Mass} (m) = \text{Density} \times \text{Volume} = \rho \cdot (A \cdot v \cdot t) = \rho A v t \]

### Step 4: Calculate the kinetic energy of the air
The kinetic energy (KE) of the mass of air that passes through the windmill is given by:
\[ \text{Kinetic Energy} = \frac{1}{2} m v^2 = \frac{1}{2} (\rho A v t) v^2 = \frac{1}{2} \rho A v^3 t \]

### Step 5: Calculate the power received by the windmill
Power is defined as the rate of doing work or the rate of energy transfer. Therefore, the power \( P \) received by the windmill is the kinetic energy per unit time:
\[ P = \frac{\text{Kinetic Energy}}{t} = \frac{\frac{1}{2} \rho A v^3 t}{t} = \frac{1}{2} \rho A v^3 \]

### Final Answer
Thus, the mechanical power received by the windmill is:
\[ P = \frac{1}{2} \rho A v^3 \]

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