Recall than an object floats on water if it has a lower density than water. A floating or submerged object in a liquid has a gravitational pull (downward force) equal to the buoyancy force (upward force). The force due to gravity (gravitational pull) depends only on the mass of the object. In this case, the object can never be submerged because the density of the solid cannot be changed (unless the phase of the object is changed). To submerge, the object has to be placed in a liquid with LOWER density.

The volume of fluid displaced depends on the volume occupied by the object in the liquid. When an object floats, only part of the object is inside the liquid. This means that only part of the object’s volume is occupied inside the liquid (lower volume of fluid displaced). If the object were to be submerged (in a different liquid), the entire volume of the object would be inside the liquid and there would be an increase in the volume of fluid displaced.

Viscosity is a measure of resistance to flow. A liquid with high viscosity is said to have strong forces between its layers, making it harder to flow whereas a liquid with low viscosity has weak forces between its layers, making it easier to flow. Viscosity is quantified by the coefficient of viscosity (higher the coefficient, higher the viscosity and higher the resistance to flow). The question states that fluid A has more trouble flowing; therefore, fluid A has higher coefficient of viscosity and stronger forces between layers.

Poiseuille’s principle quantifies the flow rate of a fluid through a pipe to numerous variables as follows.

where  is the flow rate,  is change in pressure,  is radius,  is viscosity, and  is length. We can see that viscosity and length terms are in the denominator; therefore, flow rate and these two terms are inversely related. This means that increasing viscosity and/or length of pipe will decrease the flow rate (and vice versa). Note that the radius has the biggest exponential factor; therefore, altering radius of the pipe will contribute to the biggest change in flow rate.

To solve this question, we need to use the continuity equation and Bernoulli’s equation. The continuity equation is as follows:

where  is flow rate,  is velocity, and  is the area. This equation states that that the flow rate of a fluid is the same, regardless of changes in velocity and area. The equation implies that as the area is increased (for example, ) the velocity () decreases to maintain a constant flow rate and vice versa. The question states that End A has the larger radius; therefore, the velocity is lower at End A.

Bernoulli’s equation is as follows

where  is pressure,  is density,  is velocity,  is acceleration due to gravity, and  is the height from ground. This equation implies that as velocity is increased on one side, the pressure decreases to compensate. Since we already determined that End B has the higher velocity, it will also have the lower pressure.