Chapter 11 Gravity

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234 Chapter 11 (b) Express the gravitational field due to an element of the stick of mass dm: Integrate this expression over the r dg = − = − Gdm 2 ( x − x) ( x − x) 0 GCxdx ( ) i 2 x − x 0 L r g = −GC∫ iˆ = − length of the stick to obtain: ( x − x) = 0 0 ˆ xdx 2 iˆ Gλ dx 0 2 iˆ 2GM ⎡ ⎛ x0 ⎞ ⎛ L ⎞⎤ ln iˆ 2 ⎢ ⎜ ⎟ − ⎜ ⎟ ⎟⎥ L ⎣ ⎝ x0 − L ⎠ ⎝ x0 − L ⎠⎦ The Gravitational Field ( g r ) due to Spherical Objects 71 •• Two widely separated solid spheres, S1 and S2, each have radius R and mass M. Sphere S1 is uniform, whereas the density of sphere S2 is given by ρ(r) = C/r, where r is the distance from its center. If the gravitational field strength at the surface of S1 is g1, what is the gravitational field strength at the surface of S2? Picture the Problem The gravitational field strength at the surface of a sphere is 2 given by g = GM R , where R is the radius of the sphere and M is its mass. Express the gravitational field strength on the surface of S1: Express the gravitational field strength on the surface of S2: GM g 1 = 2 R GM g 2 = 2 R Divide the second of these equations GM by the first and simplify to obtain: g 2 2 = R g GM 1 2 R = 1 g = g ⇒ 1 2 75 •• Suppose you are standing on a spring scale in an elevator that is descending at constant speed in a mine shaft located on the equator. Model Earth as a homogeneous sphere. (a) Show that the force on you due to Earth’s gravity alone is proportional to your distance from the center of the planet. (b) Assume that the mine shaft located on the equator and is vertical. Do not neglect Earth’s rotational motion. Show that the reading on the spring scale is proportional to your distance from the center of the planet.
Gravity Picture the Problem There are two forces acting on you as you descend in the elevator and are at a distance r from the center of Earth; an upward normal force (FN) exerted by the scale, and a downward gravitational force (mg) exerted by Earth. Because you are in equilibrium (you are descending at constant speed) under the influence of these forces, the normal force exerted by the scale is equal in magnitude to the gravitational force acting on you. We can use Newton’s law of gravity to express this gravitational force. (a) Express the force of gravity acting on you when you are a distance r from the center of Earth: Using the definition of density, express the density of Earth between you and the center of Earth and the density of Earth as a whole: The density of Earth is also given by: Equating these two expressions for ρ and solving for M(r) yields: Substitute for M(r) in equation (1) and simplify to obtain: Apply Newton’s law of gravity to yourself at the surface of Earth to obtain: Substitute for g in equation (2) to obtain: GM ( r) m Fg = (1) 2 r where M(r) is the mass of Earth enclosed within the radius r: that is, loser to the center of Earth than your position. ( r) () r M ρ = = V M E ρ = = 4 VE 3 M () r = M E M ( r) 3 4 3 π r M π R ⎛ r ⎜ ⎝ R E 3 ⎞ ⎟ ⎠ 3 ⎛ r ⎞ GM E ⎜ ⎟ m R GM Em r = ⎝ ⎠ = (2) 2 r R R F g 2 3 GM Em GM E mg = ⇒g = 2 2 R R where g is the magnitude of the gravitational field at the surface of Earth. 235 ⎛ mg ⎞ Fg = ⎜ ⎟r ⎝ R ⎠ That is, the force of gravity on you is proportional to your distance from the center of Earth.
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Page 3 and 4: Squaring both sides of the equation
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Page 9 and 10: Gravity Picture the Problem We can
Page 11 and 12: Gravity Picture the Problem Let the
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Page 19 and 20: Substitute for cosθ and simplify t
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Chapter 11 Gravity

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