Motion under a central force
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Motion under gravity
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The gravitational force is [Equation goes here - download the original to see it.] It is an example of a force that obeys an inverse square law - that is, the magnitude of the force is inversely proportional to the square of the distance between the two masses. When considering a planetary system the mass of the planet can be considered to be negligible in comparison to the mass of the sun. Consequently, the planet can be regarded as moving under a central force given by [Equation goes here - download the original to see it.] This force can be written in vector form as [Equation goes here - download the original to see it.] The negative sign indicates that the force is directed towards the centre of attraction. We remind you that the general equation for a conic section is [Equation goes here - download the original to see it.] where e is the eccentricity. The magnitude of the eccentricity determines the shape of the section. Thus [Equation goes here - download the original to see it.]circle. [Equation goes here - download the original to see it.]ellipse [Equation goes here - download the original to see it.]parabola [Equation goes here - download the original to see it.]hyperbola Kepler's first law states that the path of each planet is an ellipse with the sun at one focus. We will now derive this law - or rather, a generalisation of it, that the orbit of any object subject to only a central force that obeys the inverse square law and is directed towards the origin is an ellipse Proof [Diagram goes here - download the original to see it.] Let be the origin and be the object in orbit. Let [Equation goes here - download the original to see it.] represent the position of the object at time . The object is subject to a central force that obeys the inverse square law. [Equation goes here - download the original to see it.] Where is a constant and is the mass of the object. Newton's second law is [Equation goes here - download the original to see it.] . Let [Equation goes here - download the original to see it.] represent the radial component of the acceleration. Therefore [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] The radial component is [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] It is difficult to solve this differential equation directly. Experience has shown that substituting [Equation goes here - download the original to see it.] effectively brings it into a form that can be solved. Before we do so , we use the fact that [Equation goes here - download the original to see it.] constant To derive formulae for and in terms of . Since [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] However [Equation goes here - download the original to see it.] can also be expressed by the chain rule as [Equation goes here - download the original to see it.] So [Equation goes here - download the original to see it.] But [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] On differentiating again [Equation goes here - download the original to see it.] Once again [Equation goes here - download the original to see it.] so [Equation goes here - download the original to see it.] We substitute [Equation goes here - download the original to see it.] Into [Equation goes here - download the original to see it.] To obtain [Equation goes here - download the original to see it.] On simplifying and dividing by [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] This is a second order, linear constant coefficient in-homogeneous differential equation. The homogeneous equation is [Equation goes here - download the original to see it.] With auxiliary equation [Equation goes here - download the original to see it.] With roots [Equation goes here - download the original to see it.] and solution [Equation goes here - download the original to see it.] Where is the amplitude and is the phase shift. This supplies the complementary function. To find the particular function we try [Equation goes here - download the original to see it.] Where m is a constant. Then [Equation goes here - download the original to see it.] Hence on substituting into [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] so [Equation goes here - download the original to see it.] So finally the solution is [Equation goes here - download the original to see it.] Where [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] are constants determined by the initial conditions. Substituting [Equation goes here - download the original to see it.] gives [Equation goes here - download the original to see it.] The axes can be chosen so that [Equation goes here - download the original to see it.] ; hence [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] This is the polar equation of a conic. The particular form the path of the object takes {whether circle, ellipse, parabola, or hyperbola} will depend on the value of . Determining that the phase angle , requires that the x-axis acts as the major axis of symmetry of the conic. It also requires that when [Equation goes here - download the original to see it.].This also introduces another beneficial simplification. When [Equation goes here - download the original to see it.] and the velocity is purely transverse, [Equation goes here - download the original to see it.] We showed earlier that the law [Equation goes here - download the original to see it. could be expressed as [Equation goes here - download the original to see it.] ...
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Contents of
Motion under a central force
1 Motion under a central force
2 Transverse Velocity
3 Kepler's Second Law
4 Motion under gravity
5 Energy and central force systems
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