The Law of Areas

The Law of Areas - related to its angular momentum L....

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The Law of Areas The trajectory of a planet about the sun is described by an ellipse with the sun in one of its focuses. Figure 14.9 shows the position of the planet at two instances (t and t + [Delta]t). The shaded wedge shows the area swept out in the time [Delta]t. The area, [Delta]A, is approximately one-half of its base, [Delta]w, times its height r. The width of the wedge is related to r and [Delta][theta]: Figure 14.9. Area swept out by planet during a time [Delta]t. [Delta]w = r [Delta][theta] We conclude that the area [Delta]A is given by If the time interval [Delta]t approaches zero, the expression for [Delta]A becomes more exact. The instantaneous rate at which the area is being swept out is The rate at which the area is being swept out depends on the velocity of the planet and is also
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Unformatted text preview: related to its angular momentum L. Figure 14.10 shows how to calculate the angular momentum of the planet. The angular momentum of the planet can be calculated as follows Figure 14.10. Angular momentum of planet. Substituting this in the expression obtained for dA/dt we conclude that Since no external torques are acting on the sun-planet system, the angular momentum of the system is constant. This immediately indicates that dA/dt also remains constant. We conclude that " A line joining the planet to the sun sweeps out equal areas in equal time " This shows that the velocity of the planet will be highest when the distance between the sun and planet is smallest. The slowest velocity of the planet will occur when the distance between the sun and the planet is largest....
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The Law of Areas - related to its angular momentum L....

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