5. After the Sun exhausts its nuclear fuel, its ultimate fate will be to collaps

Question

5. After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. a. b. c. Calculate the average density of the white dwarf. Calculate the surface free-fall acceleration. 3.27e+06 m/s2 Calculate the gravitational potential energy associated with a 1.99-kg object at the surface of the white dwarf. 6. A satellite of mass 220 kg is placed into Earth orbit at a height of 250 km above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite's speed? (c) Starting from the satellite on the Earth's surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation.

Explanation / Answer

5)

a)M = 1.989 x 10^30 kg ; R = 6.37 x 10^6 m

we know that,

density = mass/volume

rho = 1.989 x 10^30/(4/3 x 3.14 x (6.37 x 10^6)^3) = 1.84 x 10^9 kg/m^3

Hence, rho = 1.84 x 10^9 kg/m^3

b)we know that

g = G M/r^2

g = 6.67 x 10^-11 x 1.989 x 10^30/(6.37 x 10^6)^2 = 3.27 x 10^6 m/s^2

Hence, g = 3.27 x 10^6 m/s^2

c)We know that

U = - G M m/r

U = -6.67 x 10^-11 x 1.989 x 10^30 x 1.99/(6.37 x 10^6) = -4.14 x 10^-13 J

Hence, U = - 4.14 x 10^13 J

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