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sample mechanics problems

Page history last edited by Julia Gouvea 12 years, 7 months ago

Sample biological context problems University of Minnesota Introductory Physics for Biology Kenneth Heller

 

  1. Your task is to design an artificial joint to replace arthritic elbow joints in patients. After healing, the patient should be able to hold at least a gallon of milk (3.76 liters) while the lower arm is horizontal. The biceps muscle is attached to the bone at a distance of 1/6 of the bone length from the elbow and makes an angle of 80o with the horizontal arm bone which weighs 3.2 N. How strong does the artificial joint need to be?

 

  1. You have been asked to evaluate a safety device proposed for a centrifuge. A small 0.1 g block is attached by a spring so that it can slide on a horizontal disk that spins around its center. One end of the spring is attached to a point 6.0 cm from the center of the disk and the other end is attached to the block. When the centrifuge is not spinning the spring is fully compressed. If the centrifuge spin reaches 1000 revolutions per minute (rpm), the length of the spring is such that the block intercepts the light from a light emitting diode (LED) and turns down the motor. To calculate where to put the LED, you must first determine how much the spring stretches at the maximum allowable spin rate if the spring constant is 6.25 N/m. Next describe qualitatively what happens if the safety system does not shut down the motor and the spin increases reaching a value of 2387 rpm.

 

  1. You are investigating the energetics of single cell organisms. A single cell organism such as the bacteria E. Coli, is able to swim in the direction of increasing food concentration using its flagella. E. Coli has a constant power output from converting (for the purpose of swimming) ATP at the rate of about 160 molecules per second. Each ATP molecule provides energy 13kBT of energy at room temperature. At its maximum speed, the bacteria can go as fast as 30m/s, or about 30 times its own diameter in a second. 1 m means one micron and is 10-6 m. You need to calculate the speed of the bacteria as a function of time when starting from rest.

 

  1. You are studying the breathing process in physiology class and wonder about the energy necessary for the process under stressful situations. When you breathe, your diaphragm causes the volume of your chest cavity to increase slightly. The lungs are at the other end of the chest cavity from the diaphragm and their expansion or contraction is caused by the change of pressure in the chest cavity. Assuming that the chest cavity contains a constant amount of an ideal gas, you decide to begin your investigation by calculating the gas energy transfer as heat when the chest cavity expands reasonably rapidly as in a gasp. Because of contact with numerous blood vessels, you believe that the gas in the chest cavity remains at a constant temperature during this expansion. You also assume that during this part of the process the lungs do not move. You will calculate the heat exchanged by the gas in the chest cavity for this part of the breathing process as a function of the ratio of the final volume to the initial volume of the chest cavity, the temperature and amount of the gas in the chest cavity, and any relevant constants so that you can use it in a computer simulation.

 

  1. You have been asked to evaluate the development of a machine that has the potential to eliminate toxins from a water supply. The prototype is supposed to take a solution of a toxic substance in water at the temperature 22oC. The concentration of the toxin in five liters of this solution is 80 millimoles/liter (mmole/liter). The specification of the machine requires it to get one liter of pure water leaving behind 4 liters of contaminated water using no more than 200 J of energy. Is this possible?

 

  1. You are the medical advisor to a TV show about "death defying" stunts. Your task is to design a stunt in which a 5 ft 6 in, 120 pound actor jumps off a 100-foot tall tower with an elastic cord tied to one ankle with the other end of the cord tied to the top of the tower. This 75 ft cord is very light but very strong and needs to stretch so that it can stop the actor without pulling a leg off. Such a cord exerts a force with the same mathematical form as a spring. To minimize the force that the cord exerts on the leg, you want it to stretch as far as possible. You must determine the elastic force constant that characterizes the cord so that you can purchase it. For maximum dramatic effect, the jump will be off a diving board at the top of the tower. From tests you have made, the maximum speed of a person coming off the diving board is 10 ft/sec.

 

  1. You know that rocks from Mars have been found on Earth and may show evidence of fossilized microbes. One theory is that the rocks were shot off Mars by the large volcanoes. You are skeptical so you decide to calculate how fast Mars volcanoes eject rocks. You know the gravitational force on Mars is only 40% that on the Earth. You look up the height of Martian volcanoes and find some evidence of the distance rocks from the volcano hit the ground from pictures of the Martian surface. If you assume the rocks farthest from a volcano were ejected at an angle of 45 degrees, how fast was the furthest rock ejected as a function of its distance from the volcano and the height of the volcano?

 

  1. You have been asked to evaluate a new device designed to hold arteries open. The design has three small spheres connected in an equilateral triangle by three identical small springs. When inserted into an artery, the springs are compressed. To determine their safety, you have been asked to calculate the force exerted on the artery wall by one of the spheres as a function of the spring constant, the compression distance of each spring, and the angle between the sides of the triangle.. For the first calculation you decide to ignore the blood flowing through the artery, the weight of the spheres and springs, and assume that the cross-section of the artery is a circle.

 

  1. You have a job in a research laboratory investigating cardio-vascular disease. In particular, your group is investigating how the pumping capacity of the heart is affected by constrictions in an artery. Your group is working on a computer program which simulates these conditions. Your task is to determine the rate of blood flow through an artery that is wide when it leaves the heart and becomes narrow as it goes into the leg. For each case, numbers will be input to the program that give the diameter of the narrow part of the artery, the diameter of the wide part of the artery, the blood pressure in the narrow part of the artery, the blood pressure in the wide part of the artery, the density of blood, and the difference in height between the narrow part of the artery and the heart. You assume that the patient is standing in this simulation.

 

  1. You are in a research group investigating the mechanisms by which a virus attaches itself to a healthy cell and injects its genetic material into the cell. In the virus you are studying, the head of the virus is attached to one end of a thin tubule. When the virus collides with a cell, the free end of the tubule attaches to the cell and the tubule acts as a spring. Just after the collision, the head of the cell oscillates along direction of the tubule. You need to determine the maximum speed of the head as it oscillates because you think that this helps the virus inject its genetic material into the cell. From a micro-video of the process, you know the maximum distance of the oscillation and the period of the oscillation. You do not know anything else about the virus.

 

  1. You have been asked to evaluate a procedure for reattaching severed fingers at the scene of an accident. In this procedure the finger has been found and preserved by keeping it at 0oC. To successfully reattach it, it must be brought closer to body temperature, approximately 30oC. The authors of this procedure have rejected warming the finger using an active device such as an oven because something could go wrong and the finger either heated too fast or overheated. Instead they propose the following: first add 500 grams of water at room temperature to an insulated aluminum container with a mass of 200 grams also at room temperature of 23oC; then add the finger; finally add a piece of stainless steel that has been in a pot of boiling water, making sure that the stainless steel does not touch the finger. To see if this procedure is practical, you decide to calculate the mass of stainless steel needed. You estimate that the typical mass of a finger is 100 grams and that its specific heat is 3000 J/(kg oC). You know that the specific heat of aluminum is 900 J/(kg oC), water 4200 J/(kg oC), and stainless steel 500 J/(kg oC).

 

  1. You are working with an ecology group investigating the habits of eagles. During this research, you observe an eagle circling in the air at a height that you estimate to be 300 feet. You suspect that the eagle is circling around its prey on the ground below. You wonder how well the eagle can see so you decide to calculate how far the eagle is from its prey. Looking up at the eagle, you determine that its wings are banked to make an angle of 15o from the horizontal and that it takes 18 seconds for it to fly around in a circle. Your colleague tells you that the lift provided by the wings is always perpendicular to the surface of the wing.

 

  1. You have a job in a research laboratory investigating the mechanisms of aging. One study is concerned with the process of expelling damaging toxin from human cells. You estimate that the volume available to put the toxin outside of the cell is three times that of the cell. You know that the cell’s energy source is ATP molecules which can supply 30.5 x 103 Joules per mole. You have been asked to calculate the number of ATP molecules required to expel each molecule of the toxin from inside a cell of a person running a fever of 103oF (39oC or 312 K). Initially the toxin is distributed uniformly inside and outside the cell.

 

  1. You are the technical consultant for a science fiction movie about a society in the near future where cloning of humans is common. In the story, a human clone is enclosed in a thin membrane and is suspended in a fluid that fills a transparent tank. The clone’s body is horizontal and supported by two tubes, one for nutrients and one for waste products. One tube is attached to the membrane at the head of the clone and the other at the foot. Each tube is attached to the clone at a different angle to the vertical which is determined by the external machinery. In addition to these two angles, you know the weight of the clone, its length, its volume, and the volume and density of the fluid in the tank. You need to determine the horizontal position of the clone’s center of mass if this arrangement is to work.

 

  1. You are a member of a research team studying spinal damage caused by routine activities. In particular you need to calculate the force on the spine at the fifth lumbar vertebra caused by people improperly lifting a heavy box from the floor as a function of the weight being lifted. To simplify the situation, you decide to model the back as a uniform horizontal rod with a known weight and length. One end of the rod represents the shoulders supporting the weight of the object to be lifted straight up. The other end of the rod represents where the spine is attached to its pivot at the fifth lumbar vertebra. The principle muscle supporting the back is the erector spinalis attached two thirds of the way up the spine at a known angle.

  1. You are working at a pediatrics clinic in the winter and notice an increase in the number of shoulder injuries to small children. All of these injuries seem to occur in connection to ice skating outings with their parents. One day you are walking around a frozen lake when you see an adult take two small children, who appear to be identical twins, skating across the lake. Actually the adult skates while the twins glide as they are pulled by the adult. One twin holds the adult’s hand by reaching up at an angle to the horizontal. The other twin holds on to the first twin’s hand. Since the twins are the same size, they hold hands keeping those arms horizontal. Thinking of your clinic experience, you determine how the force on each twin’s arm depends on the twin’s mass and acceleration as well as the angle of the first twin’s arm with the horizontal and the coefficient of kinetic friction between the ice and the skates.

 

  1. You are a marine biologist who is studying the impact that the dumping of food scraps into the ocean by ships has on the health and eating habits of marine life living in the top 100m of water. Because some of the scraps sink, they are only available for consumption in this topmost layer for a short time. In order to write a computer program to simulate the food dispersal through the water you need to calculate a relation between the distance fallen and the time spent falling. For each food scrap the program will assign a known volume and mass. From measurements that are also input to the program, the maximum speed of each food scrap is also known as is the density of the water. Assume that the scraps are discarded from the ship so that they are not moving when they enter the water.

 

  1. Parents concerned about the safety of their children playing at a nearby lake have asked you to investigate. A tall tree with many large branches stands on the shore. The children have attached a rope to a branch 20 ft above the lake. The branch reaches out over the lake so that, when the rope hangs straight down, it just touches the lake. Children use this rope to swing from lower branches in the tree to the lake where they let go of the rope at its lowest point and go into the water. You watch as a child climbs 12 feet up the tree, grabs the taut rope 4.0 feet above its end, pushes off the tree, and swings over the water. You wonder if a child pushing off from the tree too fast could break the rope. First you test the rope and determine that it can safely hold 400 lbs. Next you determine that the heaviest child using the lake weighs 100 lbs. With that information you calculate the maximum safe distance from the bottom of the tree that the swinging child could enter the water.

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