"Smooth reducing section" tells us that turbulence is not a concern. "Neglecting frictional effects" means we get to leave that out too. Change in potential energy from a difference in height isn't explicitly stated as left out, but we know we can both because the other things say it's simplified and because if they aren't at the same height, that skews the reading on the manometer.
I am also assuming that what looks like a degree sign in the question as posted is actually supposed to be a 3, because that puts the densities provided into the correct units.
We are explicitly told the density of water and the density of mercury. Water is also incompressible, or at least close enough for most practical textbook-related purposes, which also simplifies things. I am assuming the stated diameter is internal, both because we are not told wall thickness and because the way it's written means this is probably from an engineering textbook, likely one of physics, thermodynamics, and fluid mechanics, not something intended for a tradesman such as a plumber.
The question, as posted, does not specify pipe shape, so for this answer I'm assuming it is circular. (For a real world example of relatively complex shapes with ugly math, look at heat exchanger designs; there's a lot of fiddling with what will work best and cost least).
The area of a circle is π*r^2. That means the area of the 10 cm section is 25π square centimeters, and the area of the 5 cm section is 6.25π square centimeters. The same amount of water, as measured both by mass and by volume, is going through both sections.
Velocity is distance divided by time. We have volume divided by time. To go from volume, measured in cubic centimeters, to distance, measured in centimeters, we divide the volume by the cross sectional area. (This "what do I need and how do I get there", using units to sort it out, is referred to as "dimensional analysis", and is useful to double check that you got the answer you were supposed to be looking for).
One liter is 1,000 cubic centimeters.
4,000/25π = 160π cm/s =velocity in the first pipe section, in centimeters/second.
4,000/6.25π = 640π cm/s =velocity in the second pipe section, in centimeters/second.
I'm now pulling in Bernoulli's equation, in the version for incompressible fluids (like water). Because I'm too lazy.. I mean efficient.. to convert after it's been turned into pressure, first the velocities will be divided by 100 to put them into meters per second instead of centimeters per second.
v1=0.160π m/s
v2=0.640π m/s
I then used Bernoulli's equation. It relates velocity and pressure. What we're actually looking for is the difference in height of the mercury in the manometer, which is the difference in pressure; mmHg is a pressure unit.
After setting the equation for the first set of conditions equal to the equation for the second set of conditions, without plugging the actual numbers in, because each is equal to the same constant, I isolated for the difference in pressures. This gave me
density/2 * (v12-v22) = p2-p1
which, with numbers plugged in, is the pressure difference. I also double-checked units at this point, and confirmed that the final units are in kilogram-meters per second squared, which is correct for pressure.
From here, I have two options: I can use the provided density of mercury, and use Bernoulli's equation with all of the pressure difference in the height component instead of the velocity component. This is probably what your professor expects, which we know because the density of mercury is provided.
This yields
h1 - h2 = ((p2 - p1) ÷(density * acceleration due to gravity). The second pressure minus the first pressure is the same for both fluids; that's why manometers work. It's also something you know because if they weren't equal, the mercury wouldn't be at a stable position.
NOTE: BE CAREFUL ABOUT NEGATIVE SIGNS. It is super easy to drop a negative, especially in this last equation, and nothing hurts quite as much as losing a half point because you made a mistake when you knew better.
The way I would do it if permitted is a simple unit conversion. 1 newton per meter squared is equal to .0075 mmHg. Alternately, plug numbers into both and make sure they match.
