Dynamics_Part43

# Dynamics_Part43 -

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Unformatted text preview: Problem Set 6: Problem 1. Problem: Sphere A is moving with speed U as shown, and it impacts the inclined face of Wedge B. The sphere's mass is m and the wedge's mass is m, where is a constant. The wedge is initially at rest and is free to move horizontally without friction. After the impact, the sphere moves vertically upward. (a) Determine the wedge angle, , as a function of and e, the sphere's coefficient of restitution. (b) Compute the kinetic energy change, T . NOTE: Your answer should depend only upon e, m and U . Our approach to this problem is to first do a brief kinematics computation. This will facilitate relating quantities in the xy coordinate system to those in the nt coordinate system needed to analyze the impact between the sphere and the wedge. Then, we use the Principle of Momentum and Impulse to determine velocities of the sphere and wedge after the impact. This permits evaluation of the wedge angle, . Finally, we compute the kinetic energy lost as a result of the impact. (a) The line of impact is normal to the wedge's inclined face. Thus, unit vectors n and t in the n and t directions, respectively, are related to unit vectors i and j in the x and y directions, respectively, as follows. n = sin i + cos j i = sin n - cos t and and t t = - cos i + sin j j = cos n + sin t v n . . . j ......... . . . . . . . . . . . . . . . . . . .... . ................. ... . .................... . . . . .. . .. . . .. .... .. ... .... . 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B .............................................................................................................. . ............................................................................ . .................................................. . .................. . . U i v The initial velocities of the sphere and wedge are vA = -U i = -U sin n + U cos t After the impact, the velocities are vA = vA j = vA cos n + vA sin t and vB = -vB i = -vB sin n + vB cos t and vB = 0 Tangential-Velocity Invariance. For this impact, the tangential components of vA and vA are equal, wherefore vA sin = U cos ...
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