A block of mass 20 g sits at rest on a plate that is at the top of the fluid on the right side of a U-tube. The U-tube contains two different fluids with densities ρ1 = 900 kg/m^3 and ρ2 = 630 kg/m^3 and has a cross sectional area A = 5.3x10^-4 m^2. The surfaces are offset by an amount h. If the fluid on the right side of the U-tube has the greater density, what is h?

To determine the offset distance h between the surfaces of the fluids in the U-tube, we can use the principles of fluid mechanics and hydrostatic pressure.

First, let's define the variables involved:
- m: mass of the block (20 g or 0.02 kg)
- ρ1: density of the left fluid (900 kg/m^3)
- ρ2: density of the right fluid (630 kg/m^3)
- A: cross-sectional area of the U-tube (5.3x10^-4 m^2)
- h: offset distance between the fluid surfaces

Now, let's break down the problem into two parts - the left side and the right side of the U-tube.

On the left side:
The pressure exerted by the left fluid at the same height is given by P1 = ρ1 * g * h, where g is the acceleration due to gravity (9.8 m/s^2).

On the right side:
The pressure exerted by the right fluid at the same height is given by P2 = ρ2 * g * h.

Since the block is at rest, the net pressure acting on the block must be zero. Therefore, we can equate the pressures on both sides:

P2 - P1 = (ρ2 - ρ1) * g * h = 0

Solving for h, we have:

h = 0 / [(ρ2 - ρ1) * g]
h = 0 / [(630 kg/m^3 - 900 kg/m^3) * 9.8 m/s^2]

We can see that the denominator is negative, indicating that the offset distance h will be negative. This means that the left fluid level is higher than the right fluid level.

Therefore, h = 0.