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Unformatted text preview: Problems 3-2 1. A body of weight 32 lb is dropped from a height of 100 ft in a medium offering an air resistance proportional to the velocity. If the limiting velocity is 400 ft/sec, find the velocity and displacement at any time. Find the time at which the velocity is 200 ft/sec. 2. A freely falling body of weight lb encounters an air resistance equal in pounds to where is the speed. Set up the differential equation for the velocity. Without solving the equation, find the limiting velocity. 3. Solve: DE: IC: and show that the limiting velocity is . 4. Repeat Prob. 1 with air resistance proportional to the square of the velocity. 5. A body of mass m is thrown vertically into the air with a speed . The body encounters an air resistance proportional to the velocity. Set up and solve the differential equation for the velocity of the body while it is rising. At what time does the body reach its maximum height? 6. A particle of mass moves along the axis subject to a resisting force proportional to the cube of the velocity. Find the velocity and the displacement in terms of the time t, the initial velocity, and the initial displacement. Answers and hints: 1. when 2. ft/sec 3. 4. when 5. DE: IC: when 6. max height Problems 3-3 1. A particle of unit mass moves along the axis and is acted on by a force . If the particle starts from the rest at , find its position at any time. Also find the time for the particle to reach the origin. 2. A particle of unit mass moves along the axis and is acted on by a force . If the particle starts at with a velocity , find the position of the particle at any time and the limiting position as time increases. 3. If a pendulum hanging at rest is given an initial velocity , how high will it rise? (See Sec. 1-3, Prob. 3.) 4. A particle starts from rest at and moves along the axis under the force . Show that the time to reach the origin is Evaluate this integral in terms of the gamma function (see Prob. 6, page 31) to obtain . Answers and Hints: 1. when 2. as 3. Problems 3-4 1. One end of a 3-ft chain is held so that the other end just touches the floor. It is then released. Find the force with which the high end hits the ground. 2. A chain is coiled up on the ground. One end is lifted with a constant force. Find the velocity. 3. A raindrop falls through dry air. The raindrop is assumed to lose mass at a constant rate by evaporation. Assume that the evaporating mass has the same velocity as the raindrop. If the air resistance is proportional to the velocity, find the velocity of the raindrop. 4. A spherical raindrop falls from the rest through dry air. The raindrop loses mass at a rate proportional to its surface area. Find the velocity of the raindrop at any time, ignoring air resistance. Assume lost mass has zero velocity. 5. A rocket is moving in a straight line along horizontal rails. Assume the rocket loses mass at a constant rate and the lost mass is thrown backwards at a constant speed relative to the rocket. If the initial velocity is zero, find the velocity at any time if: a. There is no air resistance. b. The air resistance is proportional to the velocity. Answers and hints 1. , assuming the portion of the chain on the floor has zero momentum. ( weight.) is specific 2. where 3. 4. 5. c. Prolems 3-5 1. A body at a temperature of 100o is placed in a room of unknown temperature. The room temperature does not change appreciably. If after 10 min the body has cooled to 90o and after 20 min to 85o, find the temperature of the surroundings. 2. A body of temperature 1000 is placed in water of temperature 50o. After 10 min the temperature of the body is 800 and the temperature of the water is 600. Assuming all the heat lost by the body is absorbed by the water, find the temperature of the body and of the water at any time. Find the equilibrium temperature. 3. Assuming all the heat lost by the cooling body is absorbed by the surroundings, find the differential equation satisfied by the temperature of the surroundings. Answers and hints 1. 80o 2. where is the force, the specific weight of the chain. 3. Problems 3-6 and use Eq. (44). 1. A tank initially contains 100 gal of brine solution containing 10 lb of salt. At fresh water is poured into the tank at 3 gal/min and the uniform dilute solution leaves at the same rate. Find the concentration of the salt in the tank at any time. How long will it take for the dilute solution to reach a concentration equal to one-half the initial concentration? 2. Two tanks initially contain 100 gal of pure water. Brine, with a concentration of 2 lb/gal, is poured into the first tank at a rate of 3 gal/min. The uniform solution in the first tank is pumped into the second tank at 3 gal/min and the solution in the second tank is pumped out at the same rate. Find the concentration in each tank at any time. 3. A leaky tank loses liquid at the rate of 5 gal/min. Brine containing 3 lb of salt per gallon, is pumped in at the rate of 4 gal/min, the solution being constantly stirred. If, initially, the tank contains 100 gal of pure water, find the number of pounds of salt in the tank after t min. 4. A 100 gal tank initially contains 50 gal of brine with a concentration of 1 lb of salt per gallon. Pure water is poured into the tank at a rate of 4 gal/min and the solution, which is kept uniform, is pumped out at the rate of 2 gal/min. What is the concentration of salt in the container at the instant of overflow? 5. Tank A contains 100 gal of fresh water. Tank B contains 100 gal of a brine solution with a salt concentration of 2 lb/gal. Both tanks are stirred to keep their contents uniform. The contents of a tank A are pumped into tank B at 3 gal/min and the contents of B are pumped into A at the same rate. Find the concentration of salt in each tank at any time. Find the equilibrium concentration. 6. Generalize Prob. 2 to tanks. Answers and hints 1. 2. 3. 4. 5. Equilibrium concentration = 1 lb/gal Problems 3-7 1. If the half-life of a radioactive substance is 1,000 years, how much of it is left after 100 years? lb/gal 2. If is the amount of mass left at time t in a first-order reaction, show that the rate constant is given by Where and . 3. Complete the derivation of the Eq. (56) in the text. 4. Solve the differential equation (54) for the case 5. The differential equation for a certain reaction is . Find the half-life. Find . where is the initial concentration and concentration. Answers and hints 1. 4. 5. . Solve this equation and find the limiting Problems 3-8 1. In formula (61) set and show that Show also that if are three equally spaced instants of time and corresponding values of , then the constants and are given by are the 2. Using the data in the text and the result of Prob. 1, obtain the Pearl and Reed formula (62). 3. Derive Eq. (61). Problems 3-9 1. Find a particular solution of assuming . Could this solution have been obtained by the "impedance" approach described in the text? 2. An circuit has no voltage source but has an initial current . Find the current at any time and draw a graph of . 3. An circuit contains a battery and a switch in series as shown in the figure. The switch is closed at and opened at Assume that and that is continuous for even though the voltage is discontinuous at . a. Without solving the differential equation, sketch a graph of . b. Solve analytically for for the two intervals and . 4. In Prob. 3 assume the switch is opened and closed periodically. Without solving the differential equation, sketch the graph of the current. 5. The current in a circuit containing a resistance, a capacitance, and a source of voltage in series satisfies the differential equation where C is the capacitance. a. If , show that b. Use the impedance method to find the steady state current if the voltage is sinusoidal, i.e., or . Find the complete impedance of the circuit. Answers and hints 1. 2. 3. yes 5. b. Problem 3-10 1. For each of the following, describe and sketch the family of curves, find the differential equation of the family, and find the orthogonal trajectories: a. b. c. d. e. Answers and hint 1. a. c. e. Supplementary problems for chapter 3 1. The velocity with which water flows out of an orifice is the same as it would have in falling freely from the water surface to the orifice. The stream of water contracts on leaving the orifice so that the effective area of the orifice is about six-tenths of the actual area. If x is the height of water above the orifice and is the cross-sectional area of the vessel at this height, then the height x satisfies the equation b. d. 2. 3. 4. 5. 6. where is the effective area of the orifice. a. Derive the differential equation. b. Find the time to empty a cylindrical tank 3 ft high and 1 ft in radius through an orifice in the bottom with an effective area of 4 sq in. Show that if vessel filled with a liquid is rotated at a uniform velocity about a vertical axis, the surface assumes the shape of a paraboloid of revolution. Hint: The resultant force on a particle of the liquid at the surface is normal to the surface. A particle of mass is projected with an initial speed of at an elevation angle of . If the particle encounters an air resistance proportional to its velocity, find the parametric equations of the trajectory. Two points and are directly opposite each other on the banks of a river of width . A man starts at and rows across the river always heading directly toward . Assume that the river flows with a uniform spped of and the man's rate of rowing in still water is . Find the path followed by the man: a. If . b. If . Curve of pursuit. Find the path of a dog which runs to overtake its master, both moving with uniform speed and the latter in a straight line. A destroyer spots an enemy submarine 4 miles away. The submarine immediately descends and departs at full speed in a straight course of unknown direction. The speed of the destroyer is three times the speed of the submarine. Discover a course the destroyer should take to be sure to overtake the submarine. Answers and hints 1. b. 3. 4. Assume origin is at , a. axis along the river, b. axis perpendicular to river. 5. Let the man start at and move along the axis with speed and let the dog start at and move with speed . At on the curve of pursuit we have where and is arc length. Solution: 6. One possibility: Take position of submarine as origin of polar coordinates. Let speed of submarine, speed of destroyer. Wait hours until submarine is 4 miles away from origin. At time after this submarine will be at miles from D. Move destroyer in a spiral-like course so that distance of destroyer from origin is also . If path of destroyer is , then or . . If when , solution is ...
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This note was uploaded on 03/27/2012 for the course MAP 3202 taught by Professor Hatim during the Fall '11 term at Valencia.
- Fall '11