Pressure problem but without any given values. How?

[evilrain14]

Homework Statement

Both dams equally wide and equally high. The first dam holds back a very large lake, and the second dam holds back a narrow river.

Homework Equations

Which dam has to be built more strongly?
a.) the first dam
b.) the second dam
c.) both

The Attempt at a Solution

i know it would be the second dam that holds back a narrow river, but how can i prove it mathematically/algebraically?
how can i show the answer in ratio? -.-

i need your help please.

thanks!

 

[mishek]

Do you know what the hydrostatic pressure is?

 

[evilrain14]

yes. but my problem is, there is no any given values/numbers. i know what to answer but i can't prove it.. :(

 

[mishek]

Hello, do You know what a hydrostatic paradox is?

 

[Nickie]

I would approach this problem by looking at the factors that affect the strength of a dam. These factors include the height and width of the dam, the type of materials used, and the pressure exerted by the water being held back.

Since both dams are equally wide and equally high, we can assume that the pressure exerted on each dam is the same. However, the difference lies in the type of water being held back - a large lake versus a narrow river.

In general, a narrow river will have a higher flow rate and therefore exert more pressure on the dam compared to a large lake. This means that the second dam, holding back the narrow river, will need to be built more strongly to withstand the higher pressure.

To prove this mathematically, we can look at the equation for pressure: P = ρgh, where P is pressure, ρ is the density of the water, g is the acceleration due to gravity, and h is the height of the water column.

Since the height of the water column is the same for both dams, the main difference will be in the density of the water (ρ). The density of a narrow river is typically higher than that of a large lake due to the higher flow rate, which means that the pressure exerted by the river will be higher.

To show this in ratio, we can compare the densities of the two bodies of water. Let's say the density of the water in the lake is ρ1 and the density of the water in the river is ρ2. Then, the pressure exerted by the lake (P1) can be compared to the pressure exerted by the river (P2) using the ratio P1/P2 = ρ1/ρ2. Since ρ1 < ρ2, this ratio will be less than 1, meaning that the pressure exerted by the lake is lower than the pressure exerted by the river.

Therefore, based on the equation for pressure and the comparison of densities, we can conclude that the second dam, holding back the narrow river, will need to be built more strongly to withstand the higher pressure exerted by the river.