A

Step 1: Understand the problem
The piston is moving through a cylinder at a speed of 19 ft/s, with a film of oil that has a viscosity of 0.020 lb.s/ft². We are to find the force required to maintain this motion.

Step 2: Identify the equation for viscous flow
The force required to maintain a constant velocity in a viscous fluid can be calculated using the formula:

F = μ * A * (V / d)

Where:
- F is the force (in lb)
- μ is the dynamic viscosity (in lb.s/ft²)
- A is the area of the piston (in ft²)
- V is the velocity (in ft/s)
- d is the thickness of the film (in ft)

Step 3: Gather the values
From the information provided:
- V = 19 ft/s
- μ = 0.020 lb.s/ft²
- The area A can be calculated from the diameter (if given) of the piston; let’s assume diameter = 3 in = 0.25 ft, hence the area:

A = π(d/2)² = π(0.25/2)² = π(0.125)² = 0.0491 ft²

Step 4: Assume a thickness of the film
Assume d = 0.01 ft (10 mils = 0.01 ft) for the thickness of the oil film.

Step 5: Plug the values into the formula
Now substituting the values into F = μ * A * (V / d)

F = 0.020 * 0.0491 * (19 / 0.01) = 0.020 * 0.0491 * 1900 = 0.765 lb

Therefore, the force required to maintain this motion is approximately:
F ≈ 0.77 lb
Thus, the answer is (A) assuming it corresponds to this range.