# exam2s06 - m in 3-D Cartesian space is trapped in a plasma...

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Exam 2 , ME 861, April 12, 2006 Name_____________________________________ Read each problem carefully. There are three pages and three problems, worth various points. Total points: 25. 1. (5 pts) A massless ring with an embedded concentrated mass m is shown. The ring rolls without slip on a horizontal surface. A horizontal force i t F t F ˆ ) ( ) ( = is applied to the mass. The generalized coordinate is θ . The Cartesian position vector, in terms of θ , is given as j R i R R r ˆ cos ˆ ) sin ( θ - - = . Determine the generalized force Q θ for ) ( t F associated with the equation of motion in terms of coordinate θ .

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2. (10 pts) Find the equation of motion in terms of θ for the rolling massless ring with embedded mass m of problem 1.
3. (10 pts) A charged particle of mass
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Unformatted text preview: m in 3-D Cartesian space is trapped in a plasma field which exerts an applied conservative force k z F ˆ β-= , for which the potential energy is 2 2 1 z V = , under the nonholonomic constraint ) ( =-+ z y x x . The generalized coordinates in the 3-D Cartesian space are ( x, y, z ), and the position vector is k z j y i x r ˆ ˆ ˆ + + = . Treat x as the dependent coordinate (i.e. x is the dependent generalized velocity, while y and z are the independent generalized velocities). Find the equation of motion corresponding to z , using the method of your choice. Neglect gravity. (You do not need to eliminate x from the final answer, if it shows up.)...
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exam2s06 - m in 3-D Cartesian space is trapped in a plasma...

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