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P7Amt2_stahler_f05 - Problem 1(20 points x 0 A force 1.7“...

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Unformatted text preview: Problem 1 (20 points) x 0' A force 1.7“ exists throughout the plane. The magnitude of the force at any point is F=W°, 1‘ where W0 is a constant, and where 'r is the radial distance from the origin. The direction of F is tangential and counter-clockwise, as depicted. (a) A- particle of mass m moves in the plane from A to C along the solid line segments shown on the right. It goes by way of point B. Here, A and B are-both at distance 1'1, while C and D are at distance r2, where 1'2 > T1. What is WABC, the work done by the force over the path ABC? (b) Now the particle moves from A to C, but going by way of D. What is WADC, the work done by the force over the path ADC? (c) Is F a conservative force? Justify your answer. If F. is conservative, What is the potential energy U at any point? Problem 2 (20 points) :- «Ax-i: @0090 I 'm A small block of mass m is launched along a frictionless table by means of a spring gun of spring constant k. The spring is compressed a distance A2: from equilibrium and then released. After the mass is released, it encounters a frictionless inclined wedge of mass M and inclination angle 6. The block has rounded corners so that it smoothly climbs the wedge. (a) Assume that the wedge is fixed to the table. What is H, the maximum vertical height the block reaches as it moves up the wedge? (b) The block is launched a second time. The wedge is again initially at rest. This time, however, the wedge is free to move along the table as the block climbs it. What is h, the maximum vertical height that the block now reaches? Problem 3 (20 points) 9W———> II H A uniform plate of mass M , height H, and width W has the shape of a parabola. (a) What is the functional form, y(:z:), of the curved border? Here, the origin of the coordinates is at the bottom point of the curve. The z-coordinate is parallel to the straight, top border, and the y-coordinate increases vertically upward. (b) Find I , the moment of inertia of the plate about the central axis shown. Express your answer in the form I=kMW2. Here, k is a nondimensional constant that you must supply. Problem 4 (20 points) A satellite of mass m orbits the Earth (mass M E) Because of a small frictional force f, the orbital radius gradually diminishes from 'r = To to 'r = r0 -— Ar, where Ar << n. During this process, the orbit is close to circular at any time. (a) Using this circular approximation, what is E, the satellite’s total energy at any radius 1'? Express your answer in terms of G and the given quantities: M E, m, and 'r. (b) What is L, the satellite’s angular momentum at any radius 1‘? ‘Again, express your answer in terms of the given quantities. ' (c) What is AN, the number of orbits completed by the satellite as the radius diminishes by Ar? Express your answer as A AN=g—’, 7'0 where g is a nondimensional factor that you supply. Problem 5 (20 points) A uniform ladder of mass M and length L leans against a wall at an angle 0 to the ground. The wall is frictionless, but the ground has a static friction coefficient no. A person of mass m climbs the ladder. Find Lmax, the maximum distance up the ladder the person can go before the ladder slips. 10 ...
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