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Chapter 1: Problem 79

A piston having a diameter of 5.48 in. and a length of 9.50 in. slides downward with a velocity \(V\) through a vertical pipe. The downward motion is resisted by an oil film between the piston and the pipe wall. The film thickness is 0.002 in., and the cylinder weighs 0.5 lb. Estimate \(V\) if the oil viscosity is \(0.016 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}\) Assume the velocity distribution in the gap is linear.

Short Answer

Expert verified
The velocity \(V\) can be obtained by rearranging the final equation from step 4 and solving for the value of \(V\). Since all necessary inputs are provided, it's a simple matter of arithmetic to get the final result.

Step by step solution

01

Calculate the area of the oil film

We start with calculating the area of the cylinder in contact with the oil film. This is simply the circumference of the cylinder times its length. The formula is \(Area = \pi \cdot diameter \cdot length = \pi \cdot 5.48 \cdot 9.50\) inches squared.
02

Compute the shear stress

Next, we calculate the shear stress, which is given by the formula \(Shear\: Stress = \frac{\mu \cdot V}{h}\), where \(\mu\) denotes the viscosity of the oil, \(V\) is the velocity we need to calculate and \(h\) is the thickness of the oil film. This equation comes from the linear velocity profile assumption (i.e., the law of viscosity, where shear stress is proportional to velocity gradient. Here, velocity changes linearly with respect to the film thickness).
03

Calculate the shear force

The shear force is the product of the shear stress and the area we calculated in step 1. The formula is \(Shear\: Force = Shear\: Stress \cdot Area\). This force resists the downward motion of the piston.
04

Balance the forces

For steady motion, the downward force (the weight of the piston) must balance the resistive force (shear force). Hence, \(Weight = Shear\: Force\). Substituting the expressions for these forces, and then solving for \(V\) gives us our desired result.

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