The viscosity of a fluid is to be measured by a viscometer constructed of two 3-ft-long concentric cylinders. The inner diameter of the outer cylinder is 6 in, and the gap between the two cylinders is 0.05 in. The outer cylinder is rotated at 250 rpm, and the torque is measured to be 1.2 lbf⋅ft. Determine the viscosity of the fluid.

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To determine the viscosity of the fluid, you can use the formula for the shear stress (τ) in a viscometer:

τ = (4 * μ * ω * R^2) / (2 * h)

Where:
τ = Shear stress
μ = Viscosity of the fluid (what we want to find)
ω = Angular velocity (in rad/s)
R = Radius of the inner cylinder (in ft)
h = Gap between the two cylinders (in ft)

In this case, we need to convert the given values to the appropriate units before we can calculate the viscosity.

1. Convert the angular velocity (rotations per minute) to rad/s:
ω = (250 rpm) * (2π rad/1 rotation) / (60 s/1 min)
= (250 * 2π) / 60 rad/s
= 26.18 rad/s (rounded to two decimal places)

2. Convert the radius of the inner cylinder (diameter = 6 in) to feet:
R = 6 in * (1 ft/12 in)
= 0.5 ft

3. Convert the gap between the cylinders to feet:
h = 0.05 in * (1 ft/12 in)
= 0.00417 ft (approximated to five decimal places)

Now we have all the values needed to calculate the viscosity:

τ = (4 * μ * ω * R^2) / (2 * h)

Rearrange the formula to solve for μ:

μ = (τ * h) / (2 * ω * R^2)

Plug in the given values:

μ = (1.2 lbf⋅ft * 0.00417 ft) / (2 * 26.18 rad/s * (0.5 ft)^2)
= 0.0245 lbf⋅s/ft² (rounded to four decimal places)

Therefore, the viscosity of the fluid is approximately 0.0245 lbf⋅s/ft².