____________________________________ ____________________________________ MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 In-Class Problems 14-16: Uniform Circular Motion Section ______ Table and Group Number ______________________ Names ____________________________________ Hand in one solution per group. We would like each group to apply the problem solving strategy with the four stages (see below) to answer the following two problems. I. Understand – get a conceptual grasp of the problem II. Devise a Plan - set up a procedure to obtain the desired solution III. Carry our your plan – solve the problem! IV. Look Back – check your solution and method of solution 1In-Class-Problem 14: Whirling Objects, U-Control Model Airplane A U-control airplane of mass M is attached by wires of length L (and negligible mass) to the “pilot” who controls the lift provided by the wing. (The wires control the plane’s elevator.) The plane’s engine keeps it moving at constant speed v. a) Briefly describe how you intend to model the motion of the object? What directions are you choosing for analyzing the components of your forces and state why? b) Find the total tension T in the wires when the plane is flown overhead in a circle so that the wires make an angle θ with the ground. Remember that the wings can provide lift only in the direction perpendicular to their area, i.e. in a direction perpendicular to the wires. Think carefully before selecting the angle of your coordinate system. c) The plane will go out of control and crash if the tension is not maintained. Given a particular speed of the plane, v, is there some angle θcrirt which you would advise the pilot not to exceed? If possible, exhibit a speed vsafe, at which the plane would be safe at any angle? θ L 2In-Class Problem 15: Uniform circular motion and the moon’s period) In this problem assume that the moon is only under the influence of the earth’s r egravitational force given by a magnitude F , =−Gmmm rˆ . Also assume that the em 2 emr , ,em moon is moving in a circular orbit around the earth and that the moon travels with a N m kg 2 2constant speed in that orbit. Let G = 6.67 ×10−11 − . The mass of the earth is m = 5.98×1024 kg . The mass of the moon is m = 7.36 ×1022 kg . The radius of thee morbit is r = 3.8×108 m. ,ema) Briefly describe how you intend to model the motion of the object? What directions are you choosing for analyzing the components of your forces and state why? b) Calculate the period of the moon’s orbit around the earth. c) Is this the same period as the time between full moons as seen from the earth? Explain your reasoning. 3In-Class Problem 16: Consider a planet of mass m1 in orbit around a extremely massive star of mass m2. The period of the orbit is T . Assume that there is a uniform distribution of dust, of density ρ throughout the space surrounding the star and 24π 4extending well beyond the planet with > Gπρ . The gravitational effect of this 2T 3 dust cloud is to add an attractive centripetal force on the planet with magnitude r 4Fdust =− Grm 1rˆπρ 3 in addition to the gravitational attraction between the star and the planet. You may neglect any drag forces due to collisions with the dust particles. a) Briefly describe how you intend to model the motion of the object? What directions are you choosing for analyzing the components of your forces and state why? b) Find an expression for the radius of the orbit of the planet. c) If there were no dust present, would the radius of the circular orbit be greater, equal, or less than your result from part a). Briefly explain your reasoning. Several billion years later, the dust cloud has vanished, but now assume that there is a repulsive force acting on the planet that is given by r kFrepulsive = 3 rˆ, r in addition to the gravitational force between the star and the planet The constant k > 0 2vand satisfies G > k / m .1 m2 d) Show that there are two possible circular orbits for the planet that have the same velocity v . Find the radii of these orbits.
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