Chapter 11 Gravity

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238 Chapter 11 Substitute numerical values and evaluate v: v = −11 2 2 8 2 ( 8. 0m) π ( 6. 673× 10 N ⋅ m /kg )( 6. 37 × 10 kg/m ) = 1. 0m/s 83 ••• Two identical spherical cavities are made in a lead sphere of radius R. The cavities each have a radius R/2. They touch the outside surface of the sphere and its center as in Figure 11-28. The mass of a solid uniform lead sphere of radius R is M. Find the force of attraction on a point particle of mass m located at a distance d from the center of the lead sphere. Picture the Problem The force of attraction of the small sphere of mass m to the lead sphere of mass M is the sum of the forces due to the solid sphere ( FS ) and the cavities ( ) of negative mass. r F r C Express the force of attraction: S C F F F r r r = + (1) Use the law of gravity to express the force due to the solid sphere: Express the magnitude of the force acting on the small sphere due to one cavity: Relate the negative mass of a cavity to the mass of the sphere before hollowing: Letting θ be the angle between the x axis and the line joining the center of the small sphere to the center of either cavity, use the law of gravity to express the force due to the two cavities: Use the figure to express cosθ : r F S GMm = − iˆ 2 d C 2 2 2 ⎟ GM'm F = ⎛ R ⎞ d + ⎜ ⎝ ⎠ where M′ is the negative mass of a cavity. 3 ⎡ R 4 ⎛ ⎞ ⎤ M' = −ρ V = −ρ ⎢ 3 π ⎜ ⎟ ⎥ ⎢⎣ ⎝ 2 ⎠ ⎥⎦ = − 1 8 4 3 1 ( πρ R ) = − M 3 r GMm F 2 cos iˆ C = θ 2 ⎛ 2 R ⎞ 8 ⎜ ⎜d + 4 ⎟ ⎝ ⎠ because, by symmetry, the y components add to zero. cosθ = d 2 d R + 4 2 8
Substitute for cosθ and simplify to obtain: Substitute in equation (1) and simplify: General Problems r F C GMm = 2 ⎛ 2 R ⎞ 4 ⎜ ⎜d + 4 ⎟ ⎝ ⎠ GMmd = 2 ⎛ 2 R ⎞ 4 ⎜ ⎜d + 4 ⎟ ⎝ ⎠ 3 / 2 d 2 iˆ d R + 4 r GMm F iˆ GMmd = − + 2 d 2 ⎛ 2 R ⎞ 4 ⎜ ⎜d + 4 ⎟ ⎝ ⎠ = ⎡ ⎢ GMm ⎢ − 1 2 ⎢ − d ⎢ ⎧ ⎢ ⎨d ⎣ ⎩ 2 Gravity 2 iˆ 3 / 2 3 d 4 2 R ⎫ + 4 89 •• A neutron star is a highly condensed remnant of a massive star in the last phase of its evolution. It is composed of neutrons (hence the name) because the star’s gravitational force causes electrons and protons to ″coalesce″ into the neutrons. Suppose at the end of its current phase, the Sun collapsed into a neutron star (it can’t in actuality because it does not have enough mass) of radius 12.0 km, without losing any mass in the process. (a) Calculate the ratio of the gravitational acceleration at the surface of the Sun following its collapse compared to its value at the surface of the Sun today. (b) Calculate the ratio of the escape speed from the surface of the neutron-Sun to its value today. Picture the Problem We can apply Newton’s second law and the law of gravity to an object of mass m at the surface of the Sun and the neutron-Sun to find the ratio of the gravitational accelerations at their surfaces. Similarly, we can express the ratio of the corresponding expressions for the escape speeds from the two suns to determine their ratio. (a) Express the gravitational force acting on an object of mass m at the surface of the Sun: Solving for ag yields: GM F g = mag = R Sun 2 Sun Sun 2 Sun m ⎬ ⎭ iˆ 3 / 2 ⎤ ⎥ ⎥ iˆ ⎥ ⎥ ⎥ ⎦ GM a g = (1) R 239
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Page 3 and 4: Squaring both sides of the equation
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Page 7 and 8: Gravity 16.7 d. Show that these dat
Page 9 and 10: Gravity Picture the Problem We can
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Chapter 11 Gravity

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