Jet of water striking series of radial flat vanes of the wheel is 200 mm and the velocity of the jet is 100 m/s. The velocity of the vanes is 46 m/s
Calculate the force exerted on the series of vanes by the jet of water.
To calculate the force exerted on the series of vanes by the jet of water, you can use the principle of conservation of momentum. The change in momentum of the water jet will be equal to the change in momentum of the vanes.
The momentum of an object is given by the product of its mass and velocity. In this case, the momentum of the water jet will be the product of the mass flow rate and the velocity of the jet, while the momentum of the vanes will be the product of the mass of the vanes and the velocity of the vanes.
First, let's calculate the mass flow rate of the water jet. The mass flow rate is given by the product of the density of water, the cross-sectional area of the jet, and the velocity of the jet.
The cross-sectional area of the jet can be calculated using the formula for the area of a circle: A = π * (d/2)^2, where d is the diameter of the jet. In this case, the diameter of the jet is 200 mm, which is equivalent to 0.2 m. Therefore, the cross-sectional area is A = π * (0.1)^2 = 0.0314 m^2.
Next, we can calculate the mass flow rate using the formula: Mass flow rate = density * A * V, where density is the density of water, A is the cross-sectional area, and V is the velocity of the jet. The density of water is approximately 1000 kg/m^3. Plugging in the values, we get:
Mass flow rate = 1000 kg/m^3 * 0.0314 m^2 * 100 m/s = 314 kg/s.
Now, let's calculate the momentum of the water jet. The momentum is given by the product of the mass flow rate and the velocity of the jet:
Momentum of water jet = Mass flow rate * Velocity = 314 kg/s * 100 m/s = 31,400 kg·m/s.
Next, let's calculate the momentum of the vanes. The momentum of the vanes is given by the product of the mass of the vanes and the velocity of the vanes. In this case, the mass of the vanes is not mentioned, but we can assume it is negligible compared to the mass flow rate of the water jet. Therefore, we can neglect the mass of the vanes in our calculations.
Now, as per the conservation of momentum, the change in momentum of the water jet will be equal to the change in momentum of the vanes. Therefore, the force exerted on the series of vanes by the jet of water will be equal to the change in momentum of the water jet over time.
Since the problem does not specify a time interval, we cannot calculate the force precisely. We can only calculate the change in momentum. To calculate force, we will need to know the time over which the change in momentum occurs.
In conclusion, the force exerted on the series of vanes by the jet of water cannot be determined without knowing the time over which the change in momentum occurs.