8. Centripetal Forces & Gravitation

(III) (a) Use the binomial expansion

(1 ± x)ⁿ = 1 ± nx + (n (n-1)/2) x² ±

to show that the value of g is altered by approximately

∆g ≈ -2g ∆r/r_E

at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E .

(III)The value of g is altered by approximately

∆g ≈ -2g ∆r/r_E

at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E .

(b) What is the meaning of the minus sign in this relation?

(III) The value of g is altered by approximately  

∆g ≈ -2g ∆r/r_E

at a height ∆r above the Earth’s surface, where r_E is the radius of the Earth, as long as ∆r ≪ r_E . 

(c) Use this result to compute the effective value of g at 125 km above the Earth’s surface. Compare to a direct use of Eq. 6–1.

A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle θ due to a massive mountain nearby (Fig. 6–37). .

(c) Estimate the angle θ of the plumb bob if it is 5 km from the center of Mt. Everest. <IMAGE>

A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle θ due to a massive mountain nearby (Fig. 6–37).

(a) Find an approximate formula for θ in terms of the mass of the mountain, m_M, the distance to its center, D_M, and the radius and mass of the Earth.

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(II) Suppose the maximum distance you can throw a particular ball on Earth is 45.0 m. When you travel to another planet, the maximum distance for the same ball is 36.5 m. If the planet is the same size as Earth, what is its mass?

Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr.

(b) What must be the mass m of each star?

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(II) Three meteorites in outer space lie along a straight line. At a certain instant in time, the meteorite on the right is 53 m from the center meteorite and the lefthand meteorite is 21 m from the center meteorite. If the mass of the righthand meteorite is 1300 kg and the net gravitational force on the center meteorite is zero, what is the mass of the lefthand meteorite?