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An aluminum cylinder weighing 30 N, 6 cm in diameter and 40 cm long, is falling concentrically through a long vertical sleeve of diameter 6.04 cm. The clearance is filled with SAE 50 oil at 20°C. Estimate the terminal (zero acceleration) fall velocity. Neglect air drag and assume a linear velocity distribution in the oil. Hint: You are given diameters, not radii.

An aluminum cylinder weighing 30 N, 6 cm in diameter and 40 cm long, is falling concentrically through a long vertical sleeve of diameter 6.04 cm. The clearance is filled with SAE 50 oil at 20°C. Estimate the terminal (zero acceleration) fall velocity. Neglect air drag and assume a linear velocity distribution in the oil. Hint: You are given diameters, not radii.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Fluid mechanics problem

**Problem Statement: Terminal Fall Velocity Calculation** An aluminum cylinder weighing 30 N, 6 cm in diameter and 40 cm long, is falling concentrically through a long vertical sleeve of diameter 6.04 cm. The clearance is filled with SAE 50 oil at 20°C. Estimate the terminal (zero acceleration) fall velocity. Neglect air drag and assume a linear velocity distribution in the oil. *Hint: You are given diameters, not radii.* --- To approach this problem, you'll need to consider the dynamics of viscous fluid flow around the cylinder and apply the concept of terminal velocity where the net force acting on the cylinder is zero. The following steps and considerations are key: 1. **Calculate the gap between the cylinder and the sleeve:** - Diameter of the sleeve (D_sleeve): 6.04 cm - Diameter of the cylinder (D_cylinder): 6 cm - Radial gap (clearance): (D_sleeve - D_cylinder) / 2 2. **Oil Properties at 20°C for SAE 50 oil:** - Dynamic viscosity (μ) - Density (ρ) (Note: You need to check standard tables for these properties) 3. **Force Balance:** - Gravitational force (Weight) acting downward: 30 N - Buoyant force (Archimedes' principle) - Viscous drag force in the oil (can be derived from the linear velocity profile assumption) 4. **Velocity Profile Assumption:** - Consider the linear velocity distribution in the oil within the annular gap. 5. **Equations and Solving:** - Use the Stokes' drag equation adapted for the annular flow geometry - Set up an equilibrium equation where the sum of forces equals zero (indicating terminal velocity). By solving these equations with the given parameters and assumptions, you will find the terminal fall velocity of the aluminum cylinder in the oil-filled sleeve.
Transcribed Image Text:**Problem Statement: Terminal Fall Velocity Calculation** An aluminum cylinder weighing 30 N, 6 cm in diameter and 40 cm long, is falling concentrically through a long vertical sleeve of diameter 6.04 cm. The clearance is filled with SAE 50 oil at 20°C. Estimate the terminal (zero acceleration) fall velocity. Neglect air drag and assume a linear velocity distribution in the oil. *Hint: You are given diameters, not radii.* --- To approach this problem, you'll need to consider the dynamics of viscous fluid flow around the cylinder and apply the concept of terminal velocity where the net force acting on the cylinder is zero. The following steps and considerations are key: 1. **Calculate the gap between the cylinder and the sleeve:** - Diameter of the sleeve (D_sleeve): 6.04 cm - Diameter of the cylinder (D_cylinder): 6 cm - Radial gap (clearance): (D_sleeve - D_cylinder) / 2 2. **Oil Properties at 20°C for SAE 50 oil:** - Dynamic viscosity (μ) - Density (ρ) (Note: You need to check standard tables for these properties) 3. **Force Balance:** - Gravitational force (Weight) acting downward: 30 N - Buoyant force (Archimedes' principle) - Viscous drag force in the oil (can be derived from the linear velocity profile assumption) 4. **Velocity Profile Assumption:** - Consider the linear velocity distribution in the oil within the annular gap. 5. **Equations and Solving:** - Use the Stokes' drag equation adapted for the annular flow geometry - Set up an equilibrium equation where the sum of forces equals zero (indicating terminal velocity). By solving these equations with the given parameters and assumptions, you will find the terminal fall velocity of the aluminum cylinder in the oil-filled sleeve.
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