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Chapter 2: Problem 48

A 73 -mm-diameter aluminum \((\mathrm{SG}=2.64)\) piston of 100 \(\mathrm{mm}\) length resides in a stationary 75 -mm-inner-diameter steel tube lined with SAE \(10 \mathrm{W}-30\) oil at \(25^{\circ} \mathrm{C}\). A mass \(m=2 \mathrm{kg}\) is suspended from the free end of the piston. The piston is set into motion by cutting a support cord. What is the terminal velocity of mass \(m ?\) Assume a linear velocity profile within the oil.

Short Answer

Expert verified
The terminal velocity can be determined by balancing the forces on the system under equilibrium. These will include the weight of the mass and piston, the buoyancy force and the viscous force. A modified form of Stokes' law will be used for determining the viscous force which can be solved to get the terminal velocity.

Step by step solution

01

Calculate properties of the Piston

The Volume \(V_{piston}\) of the piston is given by \(\pi r_{piston}^2 h_{piston}\), where \(r_{piston}\) is the radius and \(h_{piston}\) is the height. The Density \(\rho_{piston}\) is given by \(SG * \rho_{water}\), where \(\rho_{water}\) is 1000 kg/m^3. The weight \(W_{piston}\) of the piston can be calculated as \(\rho_{piston} * g * V_{piston}\), where g is 9.8 m/s^2.
02

Application of Forces - Balance Equation

The mass m is suspended from the piston and will move downwards when the support cord is cut. The system is in equilibrium when the weight of the mass and piston is equal to the Buoyancy force and the frictional (Viscous) force. So, \(W_{piston}+W_{mass} = B_{piston}+F_{viscous}\). Here the buoyancy force \(B_{piston}\) can be calculated by \(\rho_{oil} * g * V_{piston}\), and the weight of the mass is \(m * g\). The Viscous force \(F_{viscous}\) can be calculated by the Modified Stoke's Law \( \mu_{oil} d_{gap}^2 * v_t / h_{piston}\) where \(\mu_{oil}\) is the dynamic viscosity of oil, \(d_{gap}\) implies the gap between the piston and the tube and \(v_t\) is the terminal velocity.
03

Solve for Terminal Velocity

From Step 2, rearrange the equation to solve for \(v_t\), the terminal velocity. This will be equal to \((W_{piston}+W_{mass} - B_{piston}) h_{piston} / ( \mu_{oil} d_{gap}^2 )\). Substitute the calculated or given values in this step in the equation to find the terminal velocity.

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