Mass dropped into hole through Earth

In summary, the conversation discusses assumptions about the earth being spherical and having uniform density, known values for Earth's radius (r) and mass (Me), and an unknown mass (m) of an object. The conversation also explores the object's velocity and period of oscillation when passing through the center of the earth, using equations for harmonic motion and gravitational force. The final conclusion is that it takes about 45 minutes for any object to pass through the earth at any angle, and the velocity at the center is approximately 7904 m/s. The conversation also touches on the importance of considering Earth's rotation when making calculations, and the use of conservation of energy to account for the mass of the Earth being passed through.
  • #1
MaximumTaco
45
0
assumptions:
Earth is spherical, and it's density uniform.

Me, G, r(earth) are known, m(object) is not known

How fast is the object traveling when it passes the centre, and what is the period of it's oscillation ??

--------------------------------------------------
x(t) = r cos wt

(r is the magnitude of the harmonic motion, radius of earth)

x(t) at centre = 0, thus (wt) = pi/2

v(t) = -r w sin(wt) by differentiation x(t)

sin(wt) = 1,

v = -rw

w = sqrt(k/m), m is the object dropped

k = GMmr^-2, ie the Grav. force

so we can get wr = sqrt(G * Me)

about 19962000 m/s

Does this look OK so far ?

working out T (similar to above, T = 2 pi/w ) i got about 2s, WTF that can't be right.
 
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  • #2
It takes about 45 minutes for any object to pass entirely through the earth, at abotu any angle, as long as it is not affected by forces other than gravity (and the mass is negligible...but if its not, that cna be added in quite simply).

You should also add that the Earth is not spinning in your setup. If it were spinning and an object were to fall in at a vector perpendicular to the Earth's axis, they would bounce back and forth on the walls.
 
  • #3
OK, ignore rotation, you're right.

OK i re did it, T = 84 minutes which i think is right.

T = 2*pi*sqrt(R^3/ G*M) = 84 minutes, this bit is OK

speed at the middle = rw,

w = sqrt (GM/r^3)

use that and you get 7904 m/s, sound right ? I don't think so



G = 6.67 *10^-11
M Earth = 5.9742 *10^24 kg
R Earth = 6378100 m
 
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  • #4
If you need to be concerned with the velocity at the center of the earth, just use conservation of energy:

[tex]E = .5mv^2 - \frac{GMm}{r}[/tex]

where [tex]M[/tex] is the mass that's under the falling object, in other words, if at some arbitrary point inside the earth, your distance from the center is R, then R is the radius of the sphere you need to worry about. The Earth that you've passed through doesn't matter.
 
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1. What would happen if we dropped a mass into a hole through the Earth?

If we were able to create a hole through the entire Earth and dropped a mass into it, the mass would experience a constant acceleration towards the center of the Earth due to the force of gravity. As it falls, it would gain speed until it reaches the center, and then it would continue to fall towards the other side of the Earth, eventually oscillating back and forth between the two sides.

2. How long would it take for the mass to reach the other side of the Earth?

Assuming the Earth is a perfect sphere with a uniform density, it would take approximately 42 minutes for the mass to travel through the center of the Earth and reach the other side. This is known as the "through and back" time.

3. Would the mass experience any changes in weight during its journey?

Yes, the mass would experience changes in weight as it falls towards the center of the Earth. As it approaches the center, the force of gravity would become stronger, causing the mass to feel heavier. However, as it moves towards the other side, the force of gravity would decrease, making the mass feel lighter. At the center of the Earth, the mass would experience weightlessness.

4. What factors would affect the speed of the mass during its journey?

The speed of the mass would be affected by several factors, including the mass of the Earth, the distance between the two points, and the acceleration due to gravity. The mass of the Earth and the distance between the two points would determine the strength of the gravitational force, while the acceleration due to gravity would determine how quickly the mass would fall towards the center.

5. Would there be any complications or obstacles in creating a hole through the Earth?

Creating a hole through the Earth would be a massive engineering feat and would require advanced technology and resources. Some potential complications include maintaining the structural integrity of the hole, dealing with extreme temperatures and pressures, and accounting for the Earth's rotation and the Coriolis effect. Additionally, any disruptions to the Earth's core could have significant consequences for the planet's magnetic field and overall stability.

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