A gardener watering the plants from a pipe of diameter 1mm.The water is flowing from pipe at rate of 10cm3sec.find the reactionary force exerted on the hand of gardener?

To determine the reactionary force exerted on the gardener's hand while watering the plants, we need to consider the principles of fluid dynamics and Newton's third law of motion. The water flowing through the pipe creates a force due to its mass and velocity, which in turn exerts a reaction force on the gardener's hand.

Understanding the Flow of Water

The water is flowing through a pipe with a diameter of 1 mm, which is quite small. The flow rate is given as 10 cm³/sec. First, we need to convert the diameter of the pipe into meters for consistency in our calculations.

• Diameter of the pipe = 1 mm = 0.001 m
• Radius of the pipe = Diameter / 2 = 0.001 m / 2 = 0.0005 m

Calculating the Cross-Sectional Area

The cross-sectional area (A) of the pipe can be calculated using the formula for the area of a circle:

A = πr²

Substituting the radius:

A = π(0.0005 m)² ≈ 7.85 × 10⁻⁷ m²

Finding the Velocity of Water

Next, we need to find the velocity (v) of the water flowing through the pipe. The flow rate (Q) is given by the formula:

Q = A × v

Rearranging this gives us:

v = Q / A

Now, we convert the flow rate from cm³/sec to m³/sec:

• 10 cm³/sec = 10 × 10⁻⁶ m³/sec = 1 × 10⁻⁵ m³/sec

Now we can calculate the velocity:

v = (1 × 10⁻⁵ m³/sec) / (7.85 × 10⁻⁷ m²) ≈ 12.74 m/sec

Calculating the Mass Flow Rate

The mass flow rate (ṁ) can be found using the formula:

ṁ = ρ × Q

Where ρ (rho) is the density of water, approximately 1000 kg/m³. Thus:

ṁ = 1000 kg/m³ × 1 × 10⁻⁵ m³/sec = 0.01 kg/sec

Determining the Reactionary Force

According to Newton's third law, for every action, there is an equal and opposite reaction. The force exerted by the water can be calculated using the change in momentum:

Force (F) = ṁ × v

Substituting the values we have:

F = 0.01 kg/sec × 12.74 m/sec = 0.1274 N

Final Thoughts

The reactionary force exerted on the gardener's hand while watering the plants is approximately 0.1274 Newtons. This force is what the gardener feels as the water flows out of the pipe, pushing against their hand. Understanding these principles not only helps in practical applications like gardening but also illustrates fundamental concepts in physics and fluid dynamics.