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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualification Exam Day 2 (August 2003) (7) A particle of mass m is acted upon by a constant gravitational field with acceleration g. The particle is also acted upon by a force F v that is directed towards the origin. The magnitude of this force is kr F = where r is the distance from the particle to the origin. There is a cylinder of radius R into which the particle cannot enter. The particle is constrained to slide on the cylinder with no friction. (a) Write the kinetic energy K and the potential energy V of the particle in the cylindrical coordinate system ( ) z , , φ ρ . Note well that I assume here that k is negative. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 2 φ ρ φ ρ ρ ρ & & & & & R z m z m K V V V z R k z k krdr V mgz V c g z c g + = + + = + = + = + = = = ∫ + (b) Write the equations of motion for this system independent of initial conditions....
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