Exam1-MW

Exam1-MW - NAME: Exam One: Monday/Wednesday Class September...

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Unformatted text preview: NAME: Exam One: Monday/Wednesday Class September 23, 2009 1. (30 Points) RECTILINEAR MOTION SCORE: AME 301: Fall 2009 D. C. Wilcox 2 The acceleration of sliding block A is a = − vo − v2 /τ , where τ is a constant time scale and vo is a constant velocity to be determined. When the block begins its motion at time t = 0, its position and velocity are given by x = and v = 0. At time t = π τ , the block is located at x = 1 . 3 2 Determine vo . HINT: note that 2 vo dv v = sin−1 2 vo −v ...... ...... . ... ..... . ... •.... ... ..... .... ... ..... ... ... .... ...... ..... ... ..... ...... ..... . .. ... .... ... .... ... ... .. .. ... ... ... ... ... ..... ... ..... ... ... . ... ... . ... ... ... ... ... .. ... .. ... ... ... ... ... ..... ... ..... . ... ... ... ... ... ... ... .... ... ... ... ... . ... ... ... ... ... ... ... .... . ... ... ... ... ... ... ... ... . ... ...................... ... ... ................. ... ... ... .............................................. ... .... ... ........................................... . . . ... ......................................... ... . ..... . ... ... ... .... . .... ................................... .................. ... . . . . .... ..... . . ..... ............................. ................... .... . ....... ........................ . ........... ....... . ................ ...... . .. . .. . .......... ....... . . .. . . . .... ........... ......... . ..... .... . ..•... . ................ •................... ........... ........... .................. ...................... . . . . .. . . . .. .. . .......................... ........................................... . . . ........................... . ........................... ............................................. .... .. . . . . . . ......... .... ............ . .. ........................... .............................................. . ............................................... ........................... . . . . . . . . . ......................... . ............................ . ........................... ............................................... ........................... .............................................. . . . . . . . . .. ... ... ... ... . ... . . . . ... ... . . ... ... . .... . .. . .... . .. .. .. ... . ... . . ... . .. . ... . . . . ............................................................................................................................................................................................................................................................................ . . . . . . . . ....................................................................................................................................................................................... . .. .. .. ..... ... ............................................................................................................................................................................................................................................ . . . ... .. .... .. . . ....... .. .. . .. . .. ... .. . . . . . .. .. . .. . . ........ ..... ..... . ... ..... ..... ................ ...... ..... ...... . ....................................... ............................. ..... ........... ............. .... ... ..... . ....... ... . . ............ ......................................................................................................................................................................................................................................................................................................................................... . .. ... . .. .. ... .. . . . .............................. ... . ........... ....... ... ........ .......... ........ ................... .. . ......................................................................................................................................................................................................................................... . . .... ....... .. . . ........................... ........................................ ................... .............. ....... ..................... . . . ...................................... ....................................... . .. . .. . ..................................... ...................................... . . . A x 2. (30 Points) RELATIVE MOTION OF TWO OBJECTS A projectile is launched with initial speed nV as seen by a stationary observer at an angle α to the horizontal from a height h above the ground. The object is to have the projectile strike the center of a vehicle as shown. The center of the vehicle (whose top is also at height h) is at a horizontal distance L from the launch point at time t = 0 when the projectile is launched and the vehicle’s speed at that time is vx (0) = V . The vehicle accelerates horizontally with a = 1 g , where g is 2 gravitational acceleration. Ignore effects of friction on the ball’s motion. Verify that n satisfies a quadratic equation involving the dimensionless quantity gL/V 2 . HINT: Set the vertical origin of your coordinate system at a distance h above the ground. nV • g vx (t) • vv ................................................................................................................................................................................................................................................................................ .. . . . . . . . . . . . . . .. . .. . . ............................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................... .. .. .. ...................... .... .................. .... ....... ..... .... .... .... .... ... .... .... .... . . . . . . . . . . ......................................................................................................................................................................................................................................... ...... ........................ ............................................................................................................................................................................................................................................................................................ α a = 1g 2 1 3. (40 Points) MOTION WITH CYLINDRICAL SYMMETRY A small sphere of mass m is initially at rest at the top of a frictionless cylindrical surface of radius R. At time t = 0, the sphere receives a tiny nudge to the right and it begins sliding along the surface. At an angle θs where the normal reaction force between the sphere and cylinder, N , is zero, the sphere separates from the surface. (a) State the equations governing the motion up to the separation point in cylindrical coordinates. Let g denote the acceleration of gravity. ˙ (b) Multiply the equation for the circumferential motion by θ and verify that 2g ˙ θ2 = (1 − cos θ) R ˙ Assume that the nudge is sufficiently tiny that θ(0) = 0. ˙ (c) Using the result for θ2 from Part (b), solve for N as a function of m, g and cos θ. (d) Determine the separation angle, θs . Express your answer as the arc cosine of a number. . . . . . . . . . . . . . . . . ... .. .. . .. .. . ... .. . ... .. . . . .. . . . . .................. . .. . . ................................... . . ............................. . . . . ...................... ................ . ............................. ........ . . .................... ................... .......... . .... .. ................. .... .............. . . ............ .... .................. ............ . ................................................................................ . ....................................................................................... . . . ..................................................................................... . . . . . .. . . .. .. . . ........... ....... . . ......................................................................... . ..................................................................................................... . . . . . . ..... . ................................................................................... . . ........................................................................................ . ... . . . . ..................... .. . ... . . . .. . .......................................................................... .... ....... .. ............ .. . . . .. ......................................................................................................... .. ...................................................................................................................... ... . . . . . ....... . . . ... . .. ...... .. . . ................................................................................................................ . . . .................................................... ........................ . .. .. .... . . . . . . ... ... ... ...... . . ..... . ... ... ............................................................................................ . . . ......................................................................................................... . . ..................................................................................................................... . .. . .............................. ...................................... ....... . .. ............. ... .................................................................................................................................................................................................................................................. . . ...................................................................................... ...... .................................................................................................................... . .... .. .. v g Rθ v R ............................................................................................................................................................................................................... ............................................................................................................................................................................................................... ............................................................................................................................................................................................................... v 2 ...
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