# Calculating force of water coming out

1. May 19, 2015

### Raghav Gupta

1. The problem statement, all variables and given/known data
The side wall of a wide vertical cylindrical vessel of height , h = 75 cm has a narrow vertical slit running all the way to the bottom of the vessel. The length of slit is l = 50 cm and width b = 0.1 cm. With the slit closed, the vessel is filled with water. What is the resultant force of reaction of the water flowing out of the vessel immediately after the slit is opened?

A. 4.9 N
B. 0.49 N
C. 49 N
D. 490 N
2. Relevant equations
P = P0 + ρgh
F = PA
g = 9.8 m/s2
ρ = 103kg/m3
b = 10-3m
h = 3/4 m
l = 1/2 m

3. The attempt at a solution

The P0 will be cancelled as outside pressure is also P0
$$F = \int_0^l bρg(h-l)dl$$

$F = bρg [hl - \frac{l^2}{2} ]$

⇒ F= 9.8 /4 = 2.45 N

Where is the flaw?
Why I am getting a wrong answer?

2. May 19, 2015

### haruspex

I get the same answer. Do you know what it is supposed to be?

3. May 19, 2015

### Raghav Gupta

Yes, it is supposed to be 4.9 N.
They have given solution showing Toricelli law.
Is this approach wrong?

4. May 19, 2015

### haruspex

Can you post the whole supposed solution?

5. May 19, 2015

### haruspex

But... it clearly says "immediately after the slit is opened". It will take time to reach a steady velocity, so I do not see how Torricelli's law can be applied here.

6. May 19, 2015

### Raghav Gupta

So if they would not have written "immediately" would then my answer have been wrong and their's correct?

7. May 19, 2015

### Staff: Mentor

They added the word immediately to let you know that you are to use the starting level of fluid in the tank to calculate the velocity out of the slot. As far as the time required to establish a steady velocity distribution is concerned, they intended you to assume that this happens instantaneously, and can be neglected. The time required would be on the order of the depth divided by the speed of sound. The level in the tank would not change significantly during that time interval.

What they are doing here is a horizontal momentum balance on the fluid. They are finding the force that the tank has to exert on the fluid to accelerate it out of the slot at the indicated velocity. This is equal and opposite to the horizontal force that the fluid exerts on the tank.

Their analysis is correct.

Chet

8. May 19, 2015

### haruspex

Chet, can you explain what is wrong with the analysis adopted by both Raghav and me? Prior to removal of the strip, the water exerted a force on the strip of 2.45N. The instant the strip is removed, that force is no longer exerted, but all the other forces on the container remain. So the reaction force should be 2.45N.
To make the Torricelli solution correct, it should have asked for the force when the flow is established but the level has barely fallen.

9. May 19, 2015

### Staff: Mentor

I agree. The problem is poorly posed.

Chet

10. May 19, 2015

### Raghav Gupta

Thanks guys.
I posted this problem since I was amazed that why my attempt was giving a solution not in options and I tried and tried but still same 2.45 N, but I see the solution which they have given is interpreting something else.