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
To determine the fluid level in the left container when equilibrium is reestablished, we need to consider the principles of fluid pressure and equilibrium.
At equilibrium, the pressure in both containers must be the same. Since the two containers are connected at the bottom via a tube of negligible volume, the pressure at the bottom of both containers is equal.
Let's assume the height of the fluid column in the left container (filled with water) is 'h' meters. Since the containers are identical, the height of the fluid column in the right container (filled with mercury) will also be 'h' meters.
The pressure at the bottom of the left container is due to the weight of water in the column. The pressure at the bottom of the right container is due to the weight of mercury in the column.
The pressure at the bottom of each container can be calculated using the formula:
Pressure = density × gravity × height
For water, the density (ρwater) is 1000 kg/m^3, and for mercury, the density (ρmercury) is 13546 kg/m^3. The acceleration due to gravity (g) is approximately 9.8 m/s^2.
So, the pressure at the bottom of the left container is:
Pressureleft = ρwater × g × h
And, the pressure at the bottom of the right container is:
Pressureright = ρmercury × g × h
Since the pressure at the bottom of both containers is equal, we can set these two equations equal to each other:
ρwater × g × h = ρmercury × g × h
Canceling out 'h' from both sides, we get:
ρwater = ρmercury
Substituting the densities, we have:
1000 kg/m^3 = 13546 kg/m^3
This equation is not true; hence, our assumption that the fluid levels in both containers will be at the same height is incorrect. In reality, the mercury will have a significantly greater height than the water column due to the difference in density.
Therefore, the fluid level in the left container when equilibrium is reestablished will not be the same as the initial 1.00 m level. It will be lower, and the exact height can be determined by using the densities and the equation mentioned above, taking into account the difference in densities of water and mercury.