Question

Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newton’s third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of ?1s and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars.

A) Consider a neutron star with a mass equal to the sun, a radius of 13 km , and a rotation period of 1.0 s. What is the speed of a point on the equator of the star?

B) What is g at the surface of this neutron star?

C) A stationary 1.0 kg mass has a weight on earth of 9.8 N. What would be its weight on the star?

D) How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?

E) What is the radius of a geosynchronous orbit about the neutron star?

Answer 1

a)

v = distance / time

v = 2 * pi * r / t

v = 2 * pi * 13000 / 1

v = 81681.4 m/s

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b)

g = GM / r²

where M is mass of star

g = 6.67e-11 * 1.989e30 / 13000²

g = 7.85e11 m/s²

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c) mass on earth = 1 kg

mass on star = 1 kg

so, weight on star = mg = 1 * 7.85e11

W = 7.85e11 N

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d)

v = sqrt ( GM / r)

here

r = 13000 + 1000 = 14000 m

so,

v = sqrt ( 6.67e-11 * 1.989e30 / 14000)

v = 9.7345e7 m/s

so,

time taken

T = 2 * pi * 14000 / 9.7345e7

T = 9.036e-4 sec

so,

f = 1/T

f = 1106 rev/sec

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(e)

geosynchronous means period of satellite should be 1 second

r³ = T * GM / 4²

r³ = 3.36e18

so,

r = 1.497e6 m