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exam_2

exam_2 - F = Clnr Cgrzvz F = mMGfrz F = dpmr U =-mMG!r U =...

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Unformatted text preview: F = Clnr + Cgrzvz F = mMGfrz F = dpmr. U = -mMG!r U = mgh U = 10:25: Problem 1 (42 points). . A block of mass m rests on an incline which makes an angle 6' with the horizontal plane (see ﬁgure). There is friction between the block and the surface. The static friction coefficient #3 is larger than the kinetic friction coefficient, to: . The block is attached to a "massless" spring of spring constant k . In the absence of an}.r forces on the spring, its (relaxed) length wouid be )3 . a. (6) We pull on the block and extend the spring till its length is f, + so What is the maximum extension, xm, of the spring for which the block will remain stationary when released? 13. (6} In this position, show a free bod}! diagram for the block. Indicate all forces that act on the block and give their magnitudes. In the following three questions, use the symbol xm c. (10) In this position the block is then gently touched at time t = 0. It starts moving. For what value of x will the block reach its maximum speed? d. {10) As the block moves, the spring will get shorter. At some point in time, 13;, the extension is 1:. How much work was done by (i) gravity, {ii} the spring force, and (iii) by friction between t = 0 and I}. e. (10) As the block moves tip-hill, the spring gets shorter. What is a necessary requirement for the spring to become at least as short as its relaxed length 1’. ‘E’ Problem 2 {32 points). a. (6} I throw an object of mass m up front the ground at an angle of 45° with the vertical. There is a substantlal airdrag on the object. It reaches its highest point after 2 sec. Will it take longer or shorter than 2 sec to fall back to the ground or will it take the same amount of time? Explain your answer clearly. [g = 10 mfsecz] b. (6) A pendulum is hanging from the ceiling of an elevator. Its period (at small angles) is T see when the elevator is at rest. We now accelerate the elevator downwards with 5 mfsecz. What is the period now? Be quantitative. {g = 10 mIsecZ] . c. (6] We release at zero speed an oil drop of radius r in air at 1 atmosphere. The densin of the oil is p. How small should the oil drop be so that the drag force is dominated by the viscous term which is proportional with the speed? C I and C; are the coefficients (for 1 stratosphere air) for the viscous and the pressure term, respectively. A particle moves in one dimension as a function of time: .1: = -0.3 sin (2: + 1&4). a: is in meters. I in sec. d. (6) What is the frequency (in Hz) of this simple harmonic oscillation? e. (3) What are the times (in sec) at which the speed of the particle is maximum? Problem 3 (26 points) A binary star system consists of two stars of mass may and m; orbiting about each outer. The orbits of the stars are circles of radii r; and r2 centered on the center of mass of the system. a. (6) Make a drawing (sketch) of the two orbits. Indicate the positions of the center of mass, and of the stars at; and H12. Mark r; and r; and indicate the direction of motion for each star. 13. (5) What is the magnitude of the gravitational force that My exerts on m2? c. (5) What is the magnitude of the acceleration of H11 and of m2? :1. (10) Derive the orbital period of this binary system Express your answer in terms ofrj, r2, my, m2, and G. ...
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