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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 find 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 of a gun barrel. Let the cross-sectional area of the tube be given by A , the position of 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., P ˝ D constant . Show that the potential energy of 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 of s . Model the cylinder as a particle and assume that the forces resisting the motion of the cylinder are negligible. Solution We first realize that with P ˝ D C . The force exerted on the cylinder can be written as F D A C ˝ : (1) Since ˝ D As , we can simplify 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 of the initial pressure P 0 and the initial volume ˝ D s 0 A , we can simplify the expression for the work done by the gas to U 0-s D P 0 s 0 A ln s s 0 : (4) Therefore 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 .

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

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