P7Amt1_stahler_f05 - Problem 1(25 points.A particle traces...

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Unformatted text preview: -' Problem 1 (25 points) .A particle traces an orbit that repeats over time. The displacement vector is 77(t) = acos(wt) f: + bsin(wt) 3), Where a, b, and w are positive constants, and where a > b. (a) Find 77(t), the velocity vector as a function of time. (b) Find IF] and '17 I, the magnitudes of F and 17. Write both as functions of time. (c) Find F- 17 as a function of time. ((1) Since both F and 17 change with time, so does the angle 0 between them. Find 0 at t = 7r/(4w). Write your answer in terms of the constants a. and b. \ uv V‘ ‘ ' ‘ " " .Problem 2 (25 points) You are looking out your window, which is 2 m in height. Someone on the floor below throws a ball vertically upward, so that is passes just in front of the window. It takes 0.2 s to cross the window. (a) How fast is the ball moving upward when you first see it? (b) How long after the ball disappears from View will it reappear at the top of the window, moving downward? .rProblem 3 (30 points) A11 astronaut is standing inside a rotating space station, with radius R = 100 m. The space station rotates with period T about its central axis. Rotation makes the astronaut feel he has a weight mgefr, where m is his mass and gaff is an effective gravitatiomal acceleration. The astronaut wishes to determine his period T. Since the space station has no windows, he must use an indirect method. He drops his pencil from rest, at a height of Ah = 2 m above the floor (2. e. ., the space station’s outer rim). The pencil hits the floor 1n a time At: 5 s. (a) What is geff? Here, you may assume that geg is spatially constant, i.e., neglect 2 m compared to 100 m. (b) What is T? uProblem 4 (25 points) At t: 0,'two blocks of the same mass M, but composed of different material, are released from rest down an inclined plane. The blocks are initially separated by a distance L. The coeflicient of kinetic friction between the plane and the upper block is pl, while that between the plane and the lower block is #2- Suppose that #1 < #2. The upper and lower blocks both accelerate after release, with accelerations a1 and a2, respectively. At t = t1 they collide, while still traveling down the plane. (a) Draw a free-body diagram for each block, applicable for any time in the interval 0 < t < t1. (b) Derive expressions for 0,1 and a2 in terms of the given variables: M, g, L, 6, m, and #2. (Of course, you may not need to use all the variables.) (c) What is t1 in terms of these same variables? ...
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