Calculate Pressure on a Dam: Answer 0.33

[Rapier]

Homework Statement

Two dams are both 25 m tall and both hold back lakes with a depth of the 20 m at the dam. The first dam holds back a lake with 100,000 m3 of water in it. The second dam holds back a lake with 300,000 m3 of water in it. What is the ratio of the pressure at the bottom of the first dam to the pressure at the bottom of the second dam?
→3.00
→0.33
→0.8
→1.25
→The pressure is the same at the bottom of both dams.
→You can’t tell without knowing the maximum length of the lake.

Homework Equations

F = aρhg

The Attempt at a Solution

My first thought was that the pressure for both dams are the same since they have the same height. But that doesn't seem to make logical sense. Logically, I believe that if the second dam has 3x the amount of water behind it the pressure should be greater by something along the line of 3x (since my equations don't have a squared or inverse square term it should be a linear relationship...either 3x or 1/3x). I believe that I'm stuck in a second guessing loop.

I believe that the answer is .33. Since the first dam has 1/3 of the water behind it (and identical other statistics to the second dam) the force should be 1/3 that of the second dam.

Can you please confirm/deny and if deny jog me in the right direction? Thanks.

 

[Doc Al]

My first thought was that the pressure for both dams are the same since they have the same height.

That first thought was correct. Static pressure depends only on the depth beneath the surface, not on the amount of water in some body. See: Static Fluid Pressure

Counter to everyday intuition? Perhaps. Counter to 'logic'? No.

 

[Rapier]

That first thought was correct. Static pressure depends only on the depth beneath the surface, not on the amount of water in some body. See: Static Fluid Pressure

Counter to everyday intuition? Perhaps. Counter to 'logic'? No.

Ah hah! Thanks! I am trying to get my intuitive head in line. The water at the back of the dam is supported by the 'land' at the bottom so it doesn't exert any horizontal pressure? I can almost get completely behind this because the pressure is the same at 30' regardless if you are in the ocean or in a deep pool or my gran's lake.

 

[Doc Al]

The water at the back of the dam is supported by the 'land' at the bottom

You can think of it that way.

so it doesn't exert any horizontal pressure?

Sure it does. At any given point in the water, pressure is exerted equally in all directions.

I can almost get completely behind this because the pressure is the same at 30' regardless if you are in the ocean or in a deep pool or my gran's lake.

Exactly!

 

[asteriatic]

Your thinking is correct. The pressure at the bottom of the first dam would be 1/3 of the pressure at the bottom of the second dam. This can be seen by rearranging the equation F = aρhg to solve for pressure (P = F/A = aρhg/A). Since the area (A) and acceleration due to gravity (g) are the same for both dams, the pressure is directly proportional to the density (ρ) and height (h) of the water. Since the height is the same for both dams, the only difference is the density, which is 3 times greater for the second dam. Therefore, the pressure at the bottom of the first dam would be 1/3 of the pressure at the bottom of the second dam.