College

A large aquarium with a height of 5.00 m is filled with fresh water to a depth of 2.00 m. One wall of the aquarium is made of thick plastic and is 8.00 m wide. By how much does the total force on that wall increase if the aquarium is filled to a depth of 4.00 m?

Answer :

Answer:

total force = 4.704 × [tex]10^5[/tex] N

Explanation:

given data

height = 5 m

depth = 2 m

thick plastic = 8.00 m

depth = 4.00 m

solution

first we get here force in which aquarium fill with depth 2 m

dF = PdA ......1

F = ∫PdA

F = ∫ϼgy(wdy)

F = ϼgw ∫ y dy

F = ϼgw × [tex]\frac{y^2}{2}[/tex] .................2

F = [tex]1000 \times 9.8 \times 8 \times \frac{2^2}{2}[/tex]

F = 1.568 × [tex]10^5[/tex] N

and

now we get here force for depth 4 m

put here value in equation 2

F = [tex]1000 \times 9.8 \times 8 \times \frac{4^2}{2}[/tex]

F = 6.272 × [tex]10^5[/tex] N

so that here

total force will be

total force = 6.272 × [tex]10^5[/tex] N - 1.568 × [tex]10^5[/tex] N

total force = 4.704 × [tex]10^5[/tex] N

Final answer:

The increase in the total force on the aquarium wall when the water depth is increased from 2.00 m to 4.00 m is calculated using the expression ΔF = (1000 kg/m³)(9.81 m/s²)(h₂² - h₁²)(L)/2. Inserting the given values for h₁, h₂, and L, allows us to find the answer in newtons (N).

Explanation:

The question asks about the increase in the total force on the wall of an aquarium when the water depth is changed. According to physics, the force exerted by a static fluid in a container on its wall can be found by using the equation F = ρgh²L/2, where:

  • ρ (rho) is the density of the fluid (water in this case), which is 1000 kg/m³,
  • g is the acceleration due to gravity, which is 9.81 m/s²,
  • h is the depth of the water,
  • L is the width of the wall in contact with the water.

To find the increase in force when the depth of water is increased from 2.00 m to 4.00 m, we use the equation twice for each depth and subtract the smaller force from the larger force:

  1. Calculate the force at 2.00 m depth: F₁ = ρgh₁²L/2
  2. Calculate the force at 4.00 m depth: F₂ = ρgh₂²L/2
  3. Find the difference in force: ΔF = F₂ - F₁

Substituting the given values, the width (L) is stated as 8.00 m:

ΔF = (1000 kg/m³)(9.81 m/s²)(4.00 m)²(8.00 m)/2 - (1000 kg/m³)(9.81 m/s²)(2.00 m)²(8.00 m)/2

Simplifying these expressions will give the answer in newtons (N), which is the increase in total force on the wall.