How Does Atmospheric Pressure Affect Water Stream Shrinkage?

[dmk90]

Homework Statement

Water flows through a hole at the bottom of a tank that is filled to a height h=3m. The radius of the hole is r1 = 1.5 cm.
1. What is the speed of the water immediately after it leaves the hole?

2. At what distance d below the bottom of the tank is the radius of the stream reduced to r2 = 1 cm?

Homework Equations

The Attempt at a Solution

I managed to solve the speed for 7.7 m/s. However, I'm not sure what formula to use to address the atmospheric pressure that cause the stream to shrink.

 

[Chestermiller]

The stream shrinks because it is accelerating gravitationally. As it speeds up, for the flow rate to remain constant, the cross sectional area must decrease.

Chet

 

[dmk90]

Based on your clarification, I calculated the velocity at the bottom using the conservation of volumetric flow rate (R = Av = constant), v₀ = 17.3. Then g = dv/dt -> dt = dv/g = 0.98 s. Then d = d₀ + v₀t + 0.5at² = 12.3, which seems to be the correct result. What do you think?

 

[Chestermiller]

Based on your clarification, I calculated the velocity at the bottom using the conservation of volumetric flow rate (R = Av = constant), v₀ = 17.3. Then g = dv/dt -> dt = dv/g = 0.98 s. Then d = d₀ + v₀t + 0.5at² = 12.3, which seems to be the correct result. What do you think?

I haven't checked your arithmetic, but you seem to have the right idea.

 

[HalfLostAndSearching]

I would suggest using the Bernoulli's equation to address the atmospheric pressure that causes the stream to shrink. This equation takes into account both the kinetic energy and potential energy of the water flow, as well as the pressure exerted by the surrounding atmosphere. By solving for the pressure at the bottom of the tank and at the exit point of the hole, you can then calculate the pressure difference that causes the stream to shrink. Additionally, you can also consider factors such as the viscosity of the water and the shape of the hole to further understand the water pressure problem.