PS3-4 - f = kv , we have dv dt = F kv m = dv F kv = dt m...

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14.4. CHAPTER 14, PROBLEM 4 49 14.4 Chapter 14, Problem 4 Problem: Aconstantforce , F , causes a piston of mass m to advance into a cylinder filled with oil. The piston has several cylindrical tubes that permit the oil to pass through it as the piston moves. The oil exerts a friction force, f = kv , which opposes the motion. The coefficient k is a constant of dimensions mass per unit time and v is the piston’s speed. Solve for the piston’s position, x , as a function of F , t , k , m and v , assuming the piston is at rest with x =0 at t =0 . NOTE: The piston’s speed is not constant — you must solve for it. Solution: The driving force, F , is opposed by the friction force, f , wherefore 3 i F i = F f = ma = m dv dt
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Unformatted text preview: f = kv , we have dv dt = F kv m = dv F kv = dt m Integrating once and using the fact that the piston starts from rest so that v = 0 when t = 0 , we find 1 k f n ( F kv ) + 1 k f nF = t m Regrouping terms yields f n w F kv F W = kt m = F kv = Fe kt/m Solving for v yields v = F k p 1 e kt/m Q Now, to solve for piston position, we use the fact that v = dx/dt so that dx dt = F k p 1 e kt/m Q Integrating once over t , x = F k t F k 8 t e kt/m dt = Ft k + F k m k p e kt/m 1 Q Finally, substituting for v as determined above, we conclude that x = Ft mv k...
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This note was uploaded on 10/12/2009 for the course AME 301 at USC.

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