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Chapter 8: Problem 6
Could you put a satellite in an orbit that keeps it stationary over the south pole? Explain.
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To what fraction of its current radius would Earth have to shrink (with no change in mass) for the gravitational acceleration at its surface to triple?
What's the approximate value of the gravitational force between a \(67-\mathrm{kg}\) astronaut and a \(73,000-\mathrm{kg}\) spacecraft when they're \(84 \mathrm{m}\) apart?
Given the Moon's orbital radius of \(384,400 \mathrm{km}\) and period of 27.3 days, calculate its acceleration in its circular orbit, and compare with the acceleration of gravity at Earth's surface. Show that the Moon's acceleration is lower by the ratio of the square of Earth's radius to the square of the Moon's orbital radius, thus confirming the inverse-square law for the gravitational force.
In November 2013 . Comet ISON reached its perihelion (closest approach to the Sun) at 1.87 Gm from the Sun's center (only \(1.17 \mathrm{Gm}\) from the solar surface); at that point ISON was moving at \(378 \mathrm{km} / \mathrm{s}\) relative to the Sun. Do a calculation to determine whether its orbit was elliptical or hyperbolic. (Most of ISON's cometary nucleus was destroyed in its close encounter with the Sun.)
Show that the form \(\Delta U=m g \Delta r\) follows from Equation 8.5 when \(r_{1} \simeq r_{2} .\) [Hint: Write \(r_{2}=r_{1}+\Delta r\) and apply the binomial approximation (Appendix A).]
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