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Chapter 1: Problem 78

A hydraulic lift in a service station has a 32.50 -cm-diameter ram that slides in a 32.52 -cm-diameter cylinder. The annular space is filled with SAE 10 oil at \(20^{\circ} \mathrm{C}\). The ram is traveling upward at the rate of \(0.10 \mathrm{m} / \mathrm{s}\). Find the frictional force when \(3.0 \mathrm{m}\) of the ram is engaged in the cylinder.

Short Answer

Expert verified
The frictional force is calculated by substituting the given and calculated values into the Reynold's equation. The units should match the SI expression of the equation, so some conversion is needed, such as converting cm to m. After solving the equation, the resulting frictional force can be found.

Step by step solution

01

Calculate Area of Annular Space

We begin by finding the area of the annular space. The annular space is the area between the cylinder and the ram. It can be calculated as the difference of the areas of the larger and the smaller circles (A = \( \pi R^{2} - \pi r^{2} \) ), where R is the radius of the outer cylinder (16.26 cm) and r is the radius of the inner ram (16.25 cm). Make sure the radii are converted from cm to m by dividing by 100.
02

Calculate the Relative Velocity of Fluid

We are given that the ram is moving upwards at a rate of 0.10 m/s. Therefore, the velocity of the oil relative to the cylinder (v) will be - 0.10 m/s.
03

Use the Values in Reynold's Equation

The Reynolds equation for thin film flow is used to calculate the frictional force. The equation is \( F=F_{f}=\frac{ \rho v h L}{\mu} \) where: \rho is the density of oil (880 kg/m3 at \(20^{\circ} \mathrm{C}\) for SAE 10 oil), v is the fluid velocity, h is the oil film thickness (equal to the difference in the radii of the cylinder and the ram), L is the length of the ram immersed in the cylinder (3.0 m), µ is the viscosity of the oil. The viscosity of SAE 10 oil at \(20^{\circ} \mathrm{C}\) is approximately 0.00065 Ns/m2. Substituting the given values, we can calculate the frictional force.
04

Calculate the Frictional Force

Use the calculated and given values in the Reynold's equation to find the frictional force.

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Most popular questions from this chapter

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