A gardener waters the plants with a pipe of dimeter `1mm` . The water comes out at the rate of `10cm^(3)//sec` . The reactionary force exerted on the hand of the gardener is

Options

A. Zero
B. `1.27xx10^(-2)N`
C. `1.27xx10^(-4)N`
D. `0.127N`

Explanation

To find the reactionary force exerted on the hand of the gardener while watering the plants, we can follow these steps:

### Step 1: Convert the diameter of the pipe to meters
The diameter of the pipe is given as 1 mm. We need to convert this to meters for consistency in our calculations.

\[ \text{Diameter} = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \]

### Step 2: Calculate the cross-sectional area of the pipe
The cross-sectional area \( A \) of the pipe can be calculated using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius.

\[ r = \frac{\text{Diameter}}{2} = \frac{1 \times 10^{-3}}{2} = 0.5 \times 10^{-3} \text{ m} \]

\[ A = \pi (0.5 \times 10^{-3})^2 = \pi (0.25 \times 10^{-6}) \approx 7.85 \times 10^{-7} \text{ m}^2 \]

### Step 3: Convert the flow rate from cm³/s to m³/s
The flow rate of water is given as \( 10 \text{ cm}^3/\text{s} \). We need to convert this to cubic meters per second.

\[ \text{Flow rate} = 10 \text{ cm}^3/\text{s} = 10 \times 10^{-6} \text{ m}^3/\text{s} \]

### Step 4: Calculate the mass flow rate
To find the mass flow rate \( \frac{dm}{dt} \), we use the density of water, which is \( 1000 \text{ kg/m}^3 \).

\[ \frac{dm}{dt} = \text{Density} \times \text{Flow rate} = 1000 \text{ kg/m}^3 \times 10 \times 10^{-6} \text{ m}^3/\text{s} = 10^{-2} \text{ kg/s} \]

### Step 5: Calculate the velocity of water
The velocity \( v \) of the water can be calculated using the flow rate and the cross-sectional area.

\[ v = \frac{\text{Flow rate}}{A} = \frac{10 \times 10^{-6} \text{ m}^3/\text{s}}{7.85 \times 10^{-7} \text{ m}^2} \approx 12.74 \text{ m/s} \]

### Step 6: Calculate the reactionary force
According to Newton's second law, the force \( F \) can be calculated using the mass flow rate and the velocity of the water.

\[ F = v \cdot \frac{dm}{dt} = 12.74 \text{ m/s} \times 10^{-2} \text{ kg/s} = 0.1274 \text{ N} \]

### Conclusion
The reactionary force exerted on the hand of the gardener is approximately:

\[ F \approx 0.127 \text{ N} \]