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APC U5 Review1988M2. A 5kilogram object initially slides with speed vo in a hollow frictionless pipe. The end of the pipe contains two springs. one nested inside the other, as shown above. The object makes contact with the inner spring at point A, moves 0.1 meter to make contact with the outer spring at point B, and then moves an additional 0.05 meter before coming to rest at point C. The graph shows the magnitude of the force exerted on the object by the springs as a function of the objects distance from point A.a.Calculate the spring constant for the inner spring.b.Calculate the decrees in kinetic energy of the object as it moves from point A to point B.c.Calculate the additional decrease in kinetic energy of the object as it moves from point B to point C.d.Calculate the initial speed vo of the objecte.Calculate the spring constant of the outer spring1989M3. A 2kilogram block is dropped from a height of 0.45 meter above an uncompressed spring, as shown above. The spring has an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it.a.Determine the speed of the block at the instant it hits the end of the spring.b.Determine the period of the simple harmonic motion that ensues.c.Determine the distance that the spring is compressed at the instant the speed of the block is maximum.d.Determine the maximum compression of the spring. e.Determine the amplitude of the simple harmonic motion.1996M1. A thin, flexible metal plate attached at one end to a platform, as shown above, can be used to measure mass. When the free end of the plate is pulled down and released, it vibrates in simple harmonic motion with a period that depends on the mass attached to the plate. To calibrate the force constant, objects of known mass are attached to the plate and the plate is vibrated, obtaining the data shown below.a. Fill in the blanks in the data table.b. On the graph below, plot T2 versus mass. Draw on the graph the line that is your estimate of the best straightline fit to the data points.c. An object whose mass is not known is vibrated on the plate, and the average time for ten vibrations is measured to be 16.1 s. From your graph, determine the mass of the object. Write your answer with a reasonable number of significant digits.d.Explain how one could determine the force constant of the metal plate.e. Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth? Explain briefly.f. If Columbia is orbiting at 0.3 x 106 m above the Earth's surface, what is the acceleration of Columbia due to the Earth's gravity? (Radius of Earth = 6.4 x 106 m, mass of Earth = 6.0 x 1024 kg)g. Since the answer to part (f) is not zero, briefly explain why objects aboard the orbiting Columbia seem weightless.1995M3. Two stars, A and B. are in circular orbits of radii ra and rb, respectively, about their common center of mass at point P, as shown above. Each star has the same period of revolution T.Determine expressions for the following three quantities in terms of ra, rb, T, and fundamental constants.a. The centripetal acceleration of star Ab. The mass Mb of star Bc. The mass Ma of star ADetermine expressions for the following two quantities in terms of Ma, Mb, ra, rb, T, and fundamental constants.d. The moment of inertia of the twostar system about its center of mass.e. The angular momentum of the system about the center of mass.1999M2 A spherical, nonrotating planet has a radius R and a uniform density throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet.a. Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = Cr. where C = 4/3(Gm).b. On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet.The ball is dropped into the tunnel from rest at point P at the planet's surface.c. Determine the work done by gravity as the ball moves from the surface to the center of the planet.d. Determine the speed of the ball when it reaches the center of the planet.e. Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.f. Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation.2001M2. An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has mass MJ = 1.90 x 1027 kg and radius RJ = 7.14 x 107 m.a. If the radius of the planned orbit is R, use Newton's laws to show each of the following.i. The orbital speed of the planned satellite is given by ii.The period of the orbit is given by b. The explorer wants the satellite's orbit to be synchronized with Jupiter's rotation. This requires an equatorial orbit whose period equals Jupiter's rotation period of 9 hr 51 min = 3.55 x 104 s. Determine the required orbital radius in meters.c. Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs from the desired value in each of the following ways, sketch the resulting orbit on the figure. (J is the center of Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also, describe the resulting orbit qualitatively but specifically.i.When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius.ii.When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly slower than the correct speed for a circular orbit of that radius. ................
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