A The Crab Nebula is a cloud of glowing gas about 10 light-years across, located

Question

A The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light years from the earth (the figure (Figure 1t). It is the remnant of a star that underwent a supernova explosion, seen on earth in 1054 a.d. Energy is released by the Crab Nebula at a rate of about 5 Times 10^31 W. about 10^5 times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every 0.0331 s, and this period is increasing by 4.22 Times 10^-13s for each second of time that elapses. If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. Express your answer using two significant figures. Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. Express your answer using two significant figures. What is the linear speed of a point on the equator of the neutron star? Express your answer using two significant figures. Compare to the speed of light. Express your answer in terms of c. Express your answer using two significant figures. Assume that the neutron star is uniform and calculate its density.

Explanation / Answer

A)
Pn=Power Release by the Nebula=5E31 Watts
I=rotational inertia
w=angular frequency
T=period=.0331 s
T'=dT/dt=4.22E-31
w=2*pi/T

E=rotational energy = I*w^2 = 4*pi^2*I*T^(-2)
Ps=Power radiate by the star=dE/dt=E'=4*pi^2*I*(-2)*T^(-3)*T'=-8...

Ps = -Pn <-------Plug in the expression, solve for I and compute.

B)
I=(2/5)M*r^2, moment of inertia of a solid sphere
solve for R and compute.
I is computed in part A)
M=1.4 Solar mass

C)v=w*r
w=2*pi/T
r from part B

D)Density = mass/volume
mass = 1.4 solar mass
volume=(4/3) pi r^3
r from part B

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