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Phy 241
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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I'm having problems on a question. It says Derive the expression for v(r), the velocity of a satellite orbiting at distance r from the planet of mass M. Find dv/dr. Given the mass of a certain neutron star is about 6*10^30kg and G=6.67*10^-34Js, what is the velocity of a circular orbit at a distance of 19km from the center of the star? What is dv/dr at this distance? Use these results estimate of the velocity change between the given orbit and an orbit at a distance of 19km+17.5m from the center of the star.
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OK, so here's what I know. (mv^2/r)=(GMm/r^2) and v=sqrt(GM/r). So would I do sqrt ((6.67*10^-34Js)*(6*10^30kg)/19km)=0.0145Js/km?? This might be one part of the question but I don't understand where I go next to answer all these questions??
The value of G given on the test is incorrect. It's 6.67 * 10^-11 N m^2 / kg^2.
In any case you would want to express the radius in meters, as 19 000 m.
You've got v=sqrt(GM/r). To get dv/dr just take the derivative with respect to r. After simplifying you get dv/dr = -1/2 sqrt(G M / r^3). Note that the derivative of 1 / sqrt(r) is -1/2 r^-(3/2).
The question is answered by using the differential. dv = dv/dr * dr. dr is 17.5 meters, and dv/dr is -1/2 sqrt( G M / r^3) as found previously.
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Check my notes. I'll be looking for additional questions.