In about 5 billion years the Sun will have exhausted its resources and will coll

Question

In about 5 billion years the Sun will have exhausted its resources and will collapse into a white dwarf. As a white dwarf it will have roughly the same mass as it does now, but its' radius will be about that of the Earth. (a) Calculate the average density of the white dwarf. Is it comparable to any material we have on Earth? (b) What is the acceleration due to gravity near the surface of the white dwarf? (c) What is the change in gravitational potential energy for a 1.00 kg object as it falls to the surface of white dwarf? Assume the mass begins its fall at a distance very far from the white dwarf (i.e. r rightarrow infinity). (d) Calculate the escape velocity for the white dwarf.

Explanation / Answer

Mass of sun = 1.989 x 10^30 kg

New radius = 6.371 x 10^6 m

a) density = mass / earth

= (1.989 x 10^30) / ( 4 pi (6.371 x10^6)^3 / 3)

= 6.12 x 10^8 kg / m^3

it is way greater density than comparable to any material on earth.

b) a = G M / R^2

= (6.67 x 10^-11 x 1.989 x 10^30) / (6.371 x 10^6)^2

= 3268476.8 m/s^2

c) change in GPE = - G M m / r

= - (6.67 x 10^-11 x 1.989 x 10^30 x 1) / (6.371 x 10^6)

= - 2.0823 x 10^13 J

d) using energy conservation,

- 2.0823 x 10^13 + KE = 0

1 v^2 /2 = 2.0823 x 10^13

v = 6.45 x 10^6 m/s

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