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Unformatted text preview: THEORETICAL PROBLEM No. 1 EVOLUTION OF THE EARTHMOON SYSTEM Scientists can determine the distance EarthMoon with great precision. They achieve this by bouncing a laser beam on special mirrors deposited on the Moon´s surface by astronauts in 1969, and measuring the round travel time of the light (see Figure 1). With these observations, they have directly measured that the Moon is slowly receding from the Earth. That is, the EarthMoon distance is increasing with time. This is happening because due to tidal torques the Earth is transferring angular momentum to the Moon, see Figure 2. In this problem you will derive the basic parameters of the phenomenon. Figure 1. A laser beam sent from an observatory is used to measure accurately the distance between the Earth and the Moon. 1. Conservation of Angular Momentum. Let 1 L be the present total angular momentum of the EarthMoon system. Now, make the following assumptions: i) 1 L is the sum of the rotation of the Earth around its axis and the translation of the Moon in its orbit around the Earth only. ii) The Moon’s orbit is circular and the Moon can be taken as a point. iii) The Earth’s axis of rotation and the Moon’s axis of revolution are parallel. iv) To simplify the calculations, we take the motion to be around the center of the Earth and not the center of mass. Throughout the problem, all moments of inertia, torques and angular momenta are defined around the axis of the Earth. v) Ignore the influence of the Sun. 1a Write down the equation for the present total angular momentum of the EarthMoon system. Set this equation in terms of E I , the moment of inertia of the Earth; 1 E ϖ , the present angular frequency of the Earth’s rotation; 1 M I , the present moment of inertia of the Moon with respect to the Earth´s axis; and 1 M ϖ , the present angular frequency of the Moon’s orbit. 0.2 This process of transfer of angular momentum will end when the period of rotation of the Earth and the period of revolution of the Moon around the Earth have the same duration. At this point the tidal bulges produced by the Moon on the Earth will be aligned with the line between the Moon and the Earth and the torque will disappear....
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 Spring '11
 NA
 Angular Momentum, Mass, Moment Of Inertia, Rotation

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