How Is Force Calculated on a Plate Stopping Water in a Pipe?

[azure kitsune]

Homework Statement

Water flows at 7 m/s in a pipe of radius 5 cm. A plate having an area equal to the cross-sectional area of the pipe is suddenly inserted to stop the flow. Find the force exerted on the plate. Take the speed of sound in water to be 1.4 km/s. (Hint: When the plate is inserted, a pressure wave propagates through the water at the speed of sound v_s. The mass of water brought to a stop in time Δt is the water in a length of tube equal to v_sΔt.)

Homework Equations

P = F/A

The Attempt at a Solution

I am trying to attempt the problem in the following way: Consider a reference frame where the water is still. Then the insertion of the plate would actually be a plate moving at 7 m/s, causing the still water to start moving. However, I am not sure how to get the force from this. It's probably going to be from P = F/A. Can anyone help me get started on this problem?

 

[aim1732]

You can easily work it out using the hint. The amount of water stopped by the plate is v*dt*A where A=pi*r*r. Now you can find mass and momentum stopped using density and initial velocity of flowing water. Force is thus time rate of change of momentum. Since I can not think of a retarding force on water other than that by the plate, this may be the answer.

 

[kerimek]

It seems like you are on the right track. In order to calculate the force exerted on the plate, we need to consider the change in momentum of the water as it comes to a sudden stop.

First, we can calculate the mass of water that is brought to a stop in time Δt. This can be done using the formula m = ρV, where ρ is the density of water and V is the volume of water. In this case, the volume of water is equal to the cross-sectional area of the pipe (πr^2) times the length of the pipe that the water is flowing through (vsΔt).

Next, we can calculate the change in velocity of the water as it comes to a stop. Since the water is initially flowing at a speed of 7 m/s and is then brought to a stop, the change in velocity is simply -7 m/s.

Using the formula F = Δp/Δt, where Δp is the change in momentum and Δt is the time it takes for the water to come to a stop, we can calculate the force exerted on the water.

Finally, we can use the formula P = F/A to calculate the pressure exerted on the plate by the water. This pressure is equal to the force exerted on the plate divided by the area of the plate.

It's important to note that this calculation assumes that the plate is inserted instantaneously, so the time Δt is very small. In reality, the plate would take some time to fully stop the water flow, so the force and pressure calculated using this method would be an approximation.