lecturenotes-March22

lecturenotes-March22 - QuickReview:Gravita0on For 2...

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Quick Review: Gravita0on F = G m 1 m 2 r 2 For 2 particles the magnitude of the attractive force between them is m 1 and m 2 are masses and r is distance between them and. .. G = 6.67 × 10 11 N m 2 / kg 2 = 6.67 × 10 11 m 3 / kg s 2 F 12 = F 21 Newton’s third law Gravitational Constant ( g, 9.8 m/s 2 )
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Quick Review: Newton’s Law of Gravita0on F = G m 1 m 2 r 2 This is always attractive. G = 6.67 × 10 11 N m 2 kg 2 F = m 1 g = G m 1 m 2 r 2 Where does “g” come from? What is force of gravity at surface of earth? = m 1 G m EARTH r EARTH 2 = m 1 6.67 × 10 11 N m 2 kg 2 5.98 × 10 24 kg 6.38 × 10 6 m ( ) 2 = m 1 9.8 m s 2 = m 1 g
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Quick Review: Gravita0on Notes 1) Three objects -- independent of each other Newton’s 3 rd Law 4) A uniform spherical shell of matter attracts an object on the outside as if all the shell’s mass were concentrated at its center (note: this deFnes the position) height = R E + h 2) Gravitational ±orce is a VECTOR - unit vector notation F 12 = G m 1 m 2 r 12 2 ˆ r 12 ±orce on m 1 due to m 2 ˆ r 12 = r 12 r 12 F 21 = G m 1 m 2 r 21 2 ˆ r 21 ±orce on m 2 due to m 1 ˆ r 21 = r 21 r 21 = ˆ r 12 F 21 = F 12 F 21 = F 12 3) Principle of superposition F 1, net = F 12 + F 13 + F 14 + ... + F 1 n = F 1 i i = 1 n VECTOR ADDITION!!
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m 1 m 2 r F 1 m 2 m 1 Newton proved that the net gravitational force on a particle by a shell depends on the position of the particle with respect to the shell. If the particle is inside the shel Gravitation Inside the Earth 1 2 1 2 l, the net force is zero. If the particle is outside the shell, the force is given by: Consider a mass inside the Earth at a distance from the center of the Earth. If we d . ivide the Earth m r m m F G r = ins 2 ins into a series of concentric shells, only the shells with radius less than exert a force on . The net force on is: . Here is the mass of the part of the Earth inside a sphere of ra GmM r m m F r M = 3 ins ins dius : 4 4 is linear with . 3 3 r r Gm M V F r F r π ρ = = = (13-7)
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Quick Review: Gravita0onal Poten0al Energy From Section 8.3 −Δ U = W done by force Conservative force-path independent Δ U g = W done = m g ( ) dy = mg Δ y y i y f At Earth’s surface, F g ~const. W = G mM r 2 dr r i r f = GmM 1 r 2 dr r i r f Δ U g = W done = F g d x x i x f If we de±ne U = 0 at , then the work done by taking mass m from R to U U ( r ) = W = GmM 0 − − 1 R U ( r ) = GmM r Note: 1) As before, Grav. Pot. Energy decreases as separation decreases (more negative) 2) Path independent 3) MUST HAVE AT LEAST TWO PARTICLES TO POTENTIAL ENERGY (& force) 4) Knowing potential, you can get force….
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lecturenotes-March22 - QuickReview:Gravita0on For 2...

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