After our Sun exhausts its nuclear fuel, its ultimate fate may be to collapse to

Question

After our Sun exhausts its nuclear fuel, its ultimate fate may be to collapse to a white dwarf state, in which it has approximately the same mass as it has now but a radius equal to roughly the size of the Earth's radius.
(a) Calculate the average density of this white dwarf if the Sun were to collapse to a radius of 6.00 106m.
1. kg/m3

(b) Calculate the free-fall acceleration at its surface.
2. m/s2

(c) Calculate the gravitational potential energy of a 2.00 kg object at its surface. (Take Ug = 0 at infinity.)
3. J

Explanation / Answer

Data: Mass of the sun, M = 2 x 10^30 kg Radius of sun, R = 6 x 10^6 m Solution: (a) Density, = M / V [ where V = Volume ]                 = M / [ (4/3) R^3 ]                 = 2 x 10^30 / [ (4/3) * (6 x 10^6)^3 ]                 = 2.21 x 10^9 kg/m^3 Ans: Density, = 2.21 x 10^9 kg/m^3 (b) Freefall acceleration, g = G M / R^2                                    = 6.67 x 10^-11 * 2 x 10^30 / (6 x 10^6)^2                                    = 3.7 x 10^6 m/s^2 Ans: Freefall acceleration, g = 3.7 x 10^6 m/s^2 (c) Mass of the object, m = 2 kg Potential energy, PE = - G M m / R                                 = - 6.67 x 10^-11 * 2 x 10^30 * 2 / (6 x 10^6)                                 = - 4.45 x 10^13 J Ans: Potential energy, PE = - 4.45 x 10^13 J

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