Large stars can explode as they finish burning their nuclear fuel, causing a sup
ID: 2110717 • Letter: L
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 about 1 s 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.
I have found that:
g at surface of the neutron star is 4.6 x 10^11 m/s^2
A) How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?
B) What is the radius of a geosynchronous orbit about the neutron star?
Explanation / Answer
(a)
speed = circumference / period
v = 2Ï€r / T
v = 2Ï€( 8000 m ) / ( 1 sec )
v = 50300 m/s
(b)
a = GM / r ²
a = ( 6.673 × 10^-11 N(m/kg) ² )( 1.9891 × 10^30 kg ) / ( 8000 m ) ²
a = 2.07 × 10^12 m/s ²
(c)
f = ma
f = ( 1.2 kg )( 2.07 × 10^12 m/s ² )
f = 2.49 × 10^12 N
(d)
centripetal acceleration = gravitational acceleration
v ²/r = GM / r ²
v = √( GM / r )
frequency = speed / circumference
= v / 2Ï€r
= √( GM / r ) / 2πr
= √[ ( 6.673 × 10^-11 N(m/kg) ² )( 1.9891 × 10^30 kg ) / ( 9400 m ) ] / 2Ï€( 9400 m )
= 2012 rev/sec
(e)
period = 1 / frequency = 1sec
2πr / √( GM / r ) = 1 sec
2πr = ( 1 sec )√( GM / r )
4Ï€ ²r ² = ( 1 s ² )( GM / r )
r^3 = ( 1 s ² )( GM / 4Ï€ ² )
r^3 = ( 1 s ² )[ ( 6.673 × 10^-11 N(m/kg) ² )( 1.9891 × 10^30 kg ) / 4Ï€ ² ]
r^3 = 3.362 × 10^18 m^3
r = 1.498 × 10^6 m
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