A fire hose exerts a force on the person holding it. This is because the water accelerates as it goes from the hose through the nozzle. How much force is required to hold a 7.5 cm diameter hose delivering 450 L/min through a 0.80cm diameter nozzle?
To calculate the force exerted by the water coming out of the nozzle, we can use the principles of fluid dynamics and Bernoulli's equation.
Step 1: Convert the given measurements to SI units.
- The diameter of the hose is 7.5 cm, which is 0.075 m.
- The flow rate of the water is 450 L/min, which is 0.0075 m^3/s.
- The diameter of the nozzle is 0.80 cm, which is 0.0080 m.
Step 2: Calculate the velocities of the water in the hose and the nozzle.
- The flow rate of the water can be related to the velocity using the equation: Q = A * V, where Q is the flow rate, A is the cross-sectional area, and V is the velocity.
- The cross-sectional area of the hose can be calculated using the formula for the area of a circle: A_hose = π * (diameter_hose/2)^2.
- Substituting the values, we get: 0.0075 m^3/s = π * (0.075 m/2)^2 * V_hose.
- Solving for V_hose, we find: V_hose = (0.0075 m^3/s) / (π * (0.075 m/2)^2).
- The cross-sectional area of the nozzle can be calculated similarly: A_nozzle = π * (diameter_nozzle/2)^2.
- Substituting the values, we get: A_nozzle = π * (0.0080 m/2)^2.
Step 3: Use Bernoulli's equation to calculate the pressure difference between the hose and the nozzle.
- Bernoulli's equation states that the total energy of a fluid is conserved along a streamline, which means the sum of the pressure energy, kinetic energy, and potential energy remains constant.
- In this case, we can neglect changes in potential energy and assume the height of the hose and nozzle is the same.
- Bernoulli's equation can be written as: P_hose + 1/2 * ρ * V_hose^2 = P_nozzle + 1/2 * ρ * V_nozzle^2, where P is the pressure, ρ is the density of water (approximately 1000 kg/m^3), and V is the velocity.
- Rearranging the equation, we get: P_nozzle - P_hose = 1/2 * ρ * (V_hose^2 - V_nozzle^2).
Step 4: Calculate the force exerted by the water.
- The force exerted by the water is equal to the pressure difference multiplied by the cross-sectional area of the nozzle.
- The pressure difference can be calculated using ΔP = P_nozzle - P_hose from Step 3.
- The force can be calculated as: F = ΔP * A_nozzle.
By following these steps, you should be able to calculate the force required to hold the fire hose.