How long does it take to fall to the center of the earth?

AI Thread Summary
The discussion centers on calculating the time it would take for an object to fall to the center of the Earth, assuming only gravitational forces are at play. The initial focus is on deriving acceleration as a function of distance, expressed mathematically. The challenge lies in converting this acceleration into a function of time to determine the fall duration. Participants are encouraged to explore the gravitational force as a function of distance from the center to find a solution. The problem remains unresolved, with ongoing contemplation about the next steps in the calculation.
raul_l
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Let's assume that there are no other forces acting besides the gravitational pull of the earth. If a body started falling from the surface how much time would it take before it reached the center of the earth?
It seems like an easy problem, but so far I have had no luck. So far I have only found acceleration as a function of distance (which is quite easy), which would be a=\frac{4G\pi \rho}{3}(r-s) (rho: Earth's density, r: Earth's radius, s: traveled distance)
But that doesn't help me since I need acceleration (or velocity) as a function of time and I have no idea how to proceed.
 
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Write the gravitational force as a function of distance from the center. Then see if it reminds you of anything.
 
F=\frac{4G\pi \rho m}{3}s

still thinking...
 

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