Need help with keplers 3rd law of periods problem

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The discussion focuses on solving a problem related to Kepler's Third Law of periods, involving a satellite's orbit around a planet with an unknown mass. The satellite has a mass of 20 kg, a period of 2.6 hours, and orbits at a radius of 9.4 million meters, while the planet's surface gravity is 8.6 m/s². The user initially applied the law of periods to calculate the planet's mass and then attempted to find its radius using gravitational acceleration, but encountered an unexpectedly small result. Clarification was provided that the radius in the first equation differs from that in the second equation, which is crucial for accurate calculations. The discussion emphasizes the importance of correctly interpreting the variables in the equations used.
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[SOLVED] need help with keplers 3rd law of periods problem

Homework Statement


A 20 kg satellite has a circular orbit with a period of 2.6 h and a radius of 9.4E6 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 8.6 m/s2, what is the radius of the planet?



Homework Equations


T^2=((4pi^2)/(GM))r^3

a gravity=(GM)/r^2


The Attempt at a Solution



I used the law of periods to find the mass and then used the second equation to find the radius of the planet but I got a really small number
 
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Can you show your calculations in more detail? Note that r in your first equation is not the same as r in the second (in case you forgot that).
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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