Fundamentals of Bioengineering (BIOE252)
Conservation Principles in Bioengineering - Textbook Preview
Chapter 1 - Introduction to Engineering Calculations | Chapter 2 - Foundations of Conservation Principles |
Chapter 3 - Conservation of Mass | Chapter 4 - Conservation of Energy |
Chapter 5 - Conservation of Charge | Chapter 6 - Conservation of Momentum | Chapter 7 - Case Studies



Chapter 6 - Conservation of Momentum

 

OVERVIEW

 

Conservation of linear momentum is the thrust of Chapter 6.  The challenge problem is the kinematics of cycling;  worked example problems include the linear momentum of a bicycle, forces on the ankle and knee while cycling, and forces on a helmet during a crash.  Different types of forces that can act on a system are noted.  The derivation of equations for rigid-body statics and fluid statics from the conservation of linear momentum are shown.  Example problems such as forces on the biceps and hydrostatic pressure differences between the shoulder and ankle are given.  Systems with elastic and inelastic collisions are solved with Newton’s Third Law of Motion.  Steady-state systems with mass flow and applied forces such as the flow through a total artificial heart are a more sophisticated application of the conservation of linear momentum.  Dynamic systems, including reductions known as Newton’s Second Law of Motion and the impulse-momentum theorem, are presented.  Finally, Bernoulli’s equation is presented from the mechanical energy accounting equation.  Bernoulli’s equation is given in this chapter since it is used to describe flowing systems, often in conjunction with the conservation of linear momentum equation.  Friction loss and shaft work are introduced, and applications such as friction losses in circulation and bioremediation pump-and-treat are shown.  Conservation of angular momentum is briefly reviewed in Chapter 6. 

 

 

EXAMPLE PROBLEMS

 

1. The sport of gymnastics requires both impressive physical strength and extensive training for balance.  The iron cross is a moderate skill exercise preformed on two suspended rings, which the gymnast grips with his hands.  The muscular demand for this skill is impressive enough for it to be unadvisable to be performed by younger gymnasts. 

Suppose that a male gymnast wishes to execute an iron cross during a gymnastics session.  The total mass of the gymnast is 125 lbm.  Each ring supports half of the gymnast’s weight.  Assume that the weight of one of his arms is 5% of his total body weight.  The distance from his shoulder joint to where his hands hold the rings is 56 cm.  The distance from his hands to the center of mass of his arm is 38 cm.  The horizontal distance from his shoulder to the center of mass of his body is 22 cm (from the shoulder to the middle of the chest, not the actual center of mass of the body).  If the gymnast is at rest, how much force and torque are at one of his shoulder joints?

 

2. The deep-sea diving vessel, Alvin, can dive up to 12,800 feet under water (National Geographic Society, http://www.wonderclub.com/WorldWonders/VentsHistory.html).  The purpose for the vessel is for exploring depths where the hydrostatic pressure is too great for human divers.  What is the maximum pressure that the submersible can withstand?  List some ideas about how you might design such a vessel. 

 

3. Optical tweezers are a tool that utilizes focused laser light to manipulate microscopic objects.  Because cells have an optical density different than water, light bends as it passes through them.  This results in a momentum change and an applied force.  Unlike other manipulation methods, there is no risk of contamination because the tool merely consists of photons carrying momentum.  Determine the forces exerted by a typical laser beam below with a diameter of 1 μm, power of 500 mW, and a wavelength of 1060 nm.  The beam enters at a 45˚ angle from the horizontal and exits at a 78˚ angle from the horizontal.

a)     Using the equation pphoton = , determine  for a single photon passing through a cell.  (Planck’s constant, h = 6.626x10-34 J-s)

b)     Determine the number of photons passing through the optical tweezers beam every second, N, given the equation: 

 

 

where P is the power of the beam, h is Planck’s constant, and f is frequency.

c)     Calculate the constant force exerted by the laser on the cell.  Hint: The force exerted on the cell is opposite to the force that would be required to hold the cell in place.

 

 

4. The water pumped into Puget Sound from the metropolitan area of Seattle must first be cleansed from impurities accumulated during its time in human service (http://dnr.metrokc.gov/wtd/).  One water treatment plant processes about 133 million gallons of wastewater per day.  The steps of wastewater treatment are described in Example 3.8.

a)     There are two pipes that bring the local wastewater to the water treatment plant.  One has a diameter of 144 in., and the other has a diameter of 88 in.  Assume the water is equally distributed between these two pipes.  What is the Reynolds number for the flow in each pipe?

b)     Assume that the water treatment facility is 2 miles away from the final discharge location of Puget Sound.  To calculate the friction loss per mass flow rate in a smooth pipe, use the following equation:

 

                   

 

      where v is the velocity of the fluid in the pipe, L is the length of the pipe, and r is the radius of the pipe (Bird, Stewart and Lightfoot,1960).  The diameter of the discharge pipe is 144 in. Calculate the friction loss in the pipe.

c)     The pipeline from the wastewater treatment plant to Puget Sound contains a 200 hp pump.  Assume that the water treatment facility is at sea level.  What is the maximum height that the pipeline can rise above sea level?

 

 

Chapter 5 - Conservation of Charge | Chapter 7 - Case Studies