hw4_sol

hw4_sol - Dynamics 457 Problem 4.40 Compressed gas is used...

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Dynamics 457 Problem 4.40 Compressed gas is used in many circumstances to propel things within tubes. For example, you can still fnd pneumatic tubes in use in many banks to send and receive items to drive-through tellersand it is compressed gas that propels a bullet out o± a gun barrel. Let the cross-sectional area o± the tube be given by A , the position o± the cylinder by s , and assume that the compressed gas is an ideal gas at constant temperature so that the pressure P times the volume ˝ is a constant, i.e., D constant. Show that the potential energy o± this compressed gas is given by V D ± P 0 s 0 A ln .s=s 0 / , where P 0 is the initial pressure and s 0 is the initial value o± s . Model the cylinder as a particle and assume that the ±orces resisting the motion o± the cylinder are negligible. Solution We frst realize that with D C . The ±orce exerted on the cylinder can be written as F D A C ˝ : (1) Since ˝ D As , we can simpli±y the above equation to F D C s : (2) The work done by the gas can now be calculated using U 0-s D Z s s 0 F ds D Z s s 0 C s ds ) U 0-s D C ln s ˇ ˇ ˇ s s 0 : (3) Knowing that we can express C as the product o± the initial pressure P 0 and the initial volume ˝ D s 0 A , we can simpli±y the expression ±or the work done by the gas to U 0-s D P 0 s 0 A ln ± s s 0 ² : (4) There±ore we can write the potential energy as U 1-2 D Z L 1-2 E F ± d E r D ± .V 2 ± V 1 / ) V D ± P 0 s 0 A ln ± s s 0 ² .
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Dynamics 465 Problems 4.46 through 4.48 A crate, initially traveling horizontally with a speed of 18 ft = s, is made to slide down a 14 ft chute inclined at 35 ı . The surface of the chute has a coefFcient of kinetic friction ± k and, at its lower end, it smoothly lets the crate onto a horizontal trajectory. The horizontal surface at the end of the chute has a coefFcient of kinetic friction ± k2 . Model the crate as a particle and assume that gravity and the contact forces between the crate and the sliding surface are the only relevant forces. Problem 4.46 If ± k D 0:35 what is the speed with which the crate reaches the bottom of the chute (immediately before the crate’s trajectory becomes horizontal)? Problem 4.47 ±ind ± k such that the crate’s speed at the bottom of the chute (immediately before the crate’s trajectory becomes horizontal) is 15 ft = s. Problem 4.48 Let ± k D 0:5 and suppose that, once the crate reaches the bottom of the chute and after sliding horizontally for 5 ft, the crate runs into a bumper. If the weight of the crate is W D 110 lb, ± D 0:33 , modeling the bumper as a linear spring with constant k
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This note was uploaded on 12/04/2010 for the course ECE 134 taught by Professor York during the Fall '08 term at UCSB.

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hw4_sol - Dynamics 457 Problem 4.40 Compressed gas is used...

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