In about 5 billion years the Sun will have exhausted its res

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 -> infinity ).

(d) Calculate the escape velocity for the white dwarf.

Solution

Here,

mass of sun, m = 1.989 *10^30 Kg

radius , r = 6.371 *10^6 m

a)

average density of white dwarf = mass/volume

average density of white dwarf = 1.989 *10^30/(4pi/3 * (6.371 *10^6)^3)

average density of white dwarf = 1.836 *10^9 Kg/m^3

the average density of white dwarf is 1.836 *10^9 Kg/m^3

NO , it is not comparable to anything on earth

b)

acceleration due to gravity = G * m/r^2

acceleration due to gravity = 6.673 *10^-11 * 1.989 *10^30/(6.371 *10^6)^2

acceleration due to gravity = 3.27 *10^6 m/s^2

the acceleration due to gravity is 3.27 *10^6 m/s^2

c)

change in potential energy = - m * g * r

change in potential energy = -1 * 3.27 *10^6 * 6.371 *10^6

change in potential energy = -2.083 *10^13 J

the change in potential energy is -2.083 *10^13 J

d)

let the escape velocity is v

0.5 * m * v^2 = 2.083 *10^13

0.5 * 1 * v^2 = 2.083 *10^13

solving for v

v = 6.45 *10^6 m/s

the escape velocity is 6.45 *10^6 m/s