Question

Neutron stars, such as the ones at the center of the crab nebula, have about the same mass as our sun but a much smaller diameter. if you weigh 785 n on earth, what would you weigh when at the surface of a neutron star that has the same mass as out sun and a diameter of 15 km?

Answer 1

Givens
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F1 = 785 N on earth
F2 = Force on Sun
F3 = Force on Star
r_earth = 6371 km
r_sun = 695 700 km
r_star = 15/2 km = 7.5 km
m_earth = 5.972 * 10^24 kg
m_sun = 2 * 10^30 kg

Formula
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F_sun / F_earth = G *m_test * m_sun/(r_sun)^2 // G * m_test *m_earth / (re)^2
Again, there are cancellations
F_sun/F _ earth = [m_sun/m_earth ] * [(r_sun)^2 / (r_earth)]^2

Substitution and solution
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F_sun = 785 * (2*10^30 / 6*10^24) * (695700)^2 / (6371)^2 Note the radii do not have to be converted since the conversions would cancel anyway.

Solving this mess gives
F_sun = 3.12 * 10^12 Note I have rounded. You can put in the exact number.
This may not seem possible, but I don't think I've made an error. Now you go through the same process for the star. I leave that for you