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ism_chapter_11

# ism_chapter_11 - Chapter 11 Gravity Conceptual Problems*1(a...

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837 Chapter 11 Gravity Conceptual Problems *1 ( a ) False. Kepler’s law of equal areas is a consequence of the fact that the gravitational force acts along the line joining two bodies but is independent of the manner in which the force varies with distance. ( b ) True. The periods of the planets vary with the three-halves power of their distances from the sun. So the shorter the distance from the sun, the shorter the period of the planet’s motion. 2 Determine the Concept We can apply Newton’s 2 nd law and the law of gravity to the satellite to obtain an expression for its speed as a function of the radius of its orbit. Apply Newton’s 2 nd law to the satellite to obtain: = = r v m r GMm F 2 2 radial where M is the mass of the object the satellite is orbiting and m is the mass of the satellite. Solve for v to obtain: r GM v = Thus the speed of the satellite is independent of its mass and: correct. is ) ( c 3 •• Picture the Problem The acceleration due to gravity varies inversely with the square of the distance from the center of the moon. Express the dependence of the acceleration due to the gravity of the moon on the distance from its center: 2 1 r a' Express the dependence of the acceleration due to the gravity of the moon at its surface on its radius: 2 M 1 R a

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Chapter 11 838 Divide the first of these expressions by the second to obtain: 2 2 M r R a a' = Solve for a : ( ) a a R R a r R a' 16 1 2 M 2 M 2 2 M 4 = = = and correct. is ) ( d 4 Determine the Concept Measurement of G is difficult because masses accessible in the laboratory are very small compared to the mass of the earth. 5 Determine the Concept The escape speed for a planet is given by R Gm v 2 e = . Between v e depends on the square root of M , doubling M increases the escape speed by a factor of 2 and correct. is ) ( a 6 •• Determine the Concept We can take careful measurements of its position in order to determine whether its trajectory is an ellipse, a hyperbola, or a parabola. If the path is an ellipse, it will return; if its path is hyperbolic or parabolic, it will not return. 7 •• Determine the Concept The gravitational field is proportional to the mass within the sphere of radius r and inversely proportional to the square of r , i.e., proportional to . 2 3 r r r = *8 Determine the Concept Let m represent the mass of Mercury, M S the mass of the sun, v the orbital speed of Mercury, and R the mean orbital radius of Mercury. We can use Newton’s 2 nd law of motion to relate the gravitational force acting on the Mercury to its orbital speed. Use Newton’s 2 nd law to relate the gravitational force acting on Mercury to its orbital speed: R v m R m GM F 2 2 S net = = Simplify to obtain: U R m GM R m GM mv 2 1 S 2 1 S 2 1 2 2 1 = = =