8. Centripetal Forces & Gravitation

Two spheres of mass 300 kg and 500 kg are placed in a line 20 cm apart. If another sphere of mass 200 kg is placed between them, 8 cm from the 300 kg sphere, what is the net gravitational force on the 200 kg sphere?

A 2,000-kg spacecraft is blasting away from the surface of an unknown planet the same size as the Earth. At 1500km above the surface, an instrument onboard reads the gravitational force to be 18000 N. What is the planet's mass?

 Two spheres are separated by 10m. If the lighter 40kg sphere feels a gravitational force of 1.6 × 10^-9 N, what is the mass of the heavier sphere?

(II) You are explaining to friends why an astronaut feels weightless orbiting the Earth in a space station, and they respond that they thought gravity was just a lot weaker up there. Convince them that it isn’t so by calculating how much weaker (in %) gravity is 380 km above the Earth’s surface.

(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?

(II) Two objects attract each other gravitationally with a force of 6.5 x 10⁻⁹ N when they are 0.25 m apart. Their total mass is 6.50 kg. Find their individual masses.