Two identical containers are open at the top and are connected at the bottom via a tube of negligible volume and a valve which is closed. Both containers are filled initially to the same height of 1.00 m, one with water, the other with mercury, as the drawing indicates. The valve is then opened. Water and mercury are immiscible. Determine the fluid level in the left container when equilibrium is reestablished.
distance from the bottom
Hydrostatic pressure P1 at the bottom of containing with mercury only,
P1=ρ h
where h=height of mercury
ρ=specific gravity of mercury = 13.6
S.G. water = 1
Hydrostatic pressure P2 at the bottom of the container with mercury and water,
P2=ρ(1-h)+1*1
Equate P1=P2 and solve for h.
To determine the fluid level in the left container when equilibrium is reestablished, we need to consider the relative densities of water and mercury.
Density of water (ρ_water) = 1000 kg/m^3
Density of mercury (ρ_mercury) = 13600 kg/m^3
Since the two containers are connected at the bottom, the fluids will reach the same level in both containers due to the principle of communicating vessels.
Let's assume the final fluid level in both containers is h meters from the bottom.
In the right container (mercury-filled):
The pressure at the bottom of the container is given by:
P_mercury = ρ_mercury * g * h
In the left container (water-filled):
The pressure at the bottom of the container is given by:
P_water = ρ_water * g * h
Since the two containers are connected at the bottom, the pressure in both containers must be the same when equilibrium is established. Therefore:
P_mercury = P_water
ρ_mercury * g * h = ρ_water * g * h
Canceling out the common variables:
ρ_mercury = ρ_water
Substituting the values:
13600 kg/m^3 = 1000 kg/m^3
Since the densities are not equal, this equation cannot be true. Therefore, there won't be an equilibrium when water and mercury are used in these containers.
To determine the fluid level in the left container when equilibrium is reestablished, we need to understand the principles of fluid pressure and how it is affected by the height and density of the fluid.
In this scenario, we have two containers that are open at the top and connected at the bottom via a tube with a valve. One container is filled with water and the other with mercury. The valve is initially closed, but when it is opened, the fluids will flow to establish equilibrium.
To determine the fluid level in the left container, we need to consider the density and pressure of the fluids. The pressure at any point in a fluid is determined by the depth or height of the fluid above that point and the density of the fluid.
In this case, the height of the fluids in both containers is initially 1.00 m. We know that the density of mercury is greater than the density of water. Consequently, mercury exerts a greater pressure at the same height compared to water.
When the valve is opened, the fluids will try to equalize the pressure between the two containers. As a result, the mercury will flow down to the left container, pushing the water up.
To find the equilibrium fluid level in the left container, the pressure due to the height of the mercury column in the left container needs to be equal to the pressure due to the height of the water column in the right container.
Since the density of mercury is greater, the height of the mercury column in the left container will be shorter than the height of the water column in the right container.
To determine the specific height of the fluid level in the left container, we need to use the principle of hydrostatic pressure:
Pressure = density * gravity * height
Since the pressure is equal, we have:
density of water * gravity * height of water = density of mercury * gravity * height of mercury
Given that the density of mercury is approximately 13.6 times greater than the density of water, we can deduce that the height of the mercury column will be approximately 13.6 times less than the height of the water column.
Therefore, the fluid level in the left container, measured from the bottom, will be approximately 1.00 m / 13.6 ≈ 0.0735 m or 7.35 cm.
Hence, when equilibrium is reestablished, the fluid level in the left container from the bottom will be approximately 7.35 cm.