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# lec05 - Systems of Particles Examples 1 2.003J/1.053J...

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 Systems of Particles Examples 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 2/21/2007 Lecture 5 Systems of Particles: Example 1: Linear Momentum and Conservation of Energy, Example 2: Angular Momentum Systems of Particles Examples Example 1: Block Sliding Down an Inclined Plane (continued) Figure 1: Kinematic diagram of block sliding on ramp. The block rests on a frictionless surface and the ramp also acts on a frictionless surface as well. Figure by MIT OCW. Initially released with s = l . r A = x ˆ ı r B = x ˆ ı + se ˆ s = x ˆ ı + s cos θ ˆ ı + s sin θj ˆ r ˙ A = x ˙ˆ ı r ˙ B = x ˙ˆ ı + ˙ s cos θ ˆ ı + ˙ s sin θj ˆ r ¨ A = x ¨ˆ ı r ¨ B = x ¨ˆ ı + ¨ s cos θ ˆ ı + ¨ s sin θj ˆ What is the velocity of M at the moment m reaches point A?

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Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 2 Systems of Particles Examples Free Body Diagrams (FBDs) System: Figure 2: Free body diagram of block on ramp. The forces present are a reaction force, R , and the forces due to gravity on both bodies, Mg and mg . Figure by MIT OCW. Separately: Because we are not trying to calculate each force, apply linear momentum prin - ciple so that N does not appear. Use system. Linear momentum principle To system: d F = P dt No forces in x-direction. ( F ) x = 0 Linear momentum in the x-direction is conserved. Mx ˙ + m x + ˙ s cos θ ) = constant = 0 (1) constant = 0 because system initially at rest.
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