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Levi Tate
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xirow said:id be very impressed if anyone could build a tunnel through the earth, but if it was perfectly through the center of the earth id imagine one would get stuck in the middle
Levi Tate said:Homework Statement
A tunnel is drilled straight through the Earth from Detroit to Zxxiuw. At the middle of the tunnel it is at 1/2 Earth radius from the center. Assuming that a mass m placed in the tunnel will slide without friction, show that it experiences a force that is directly proportional to its distance from the center of the tunnel. Give an expression for this force in terms of the distance and the weight mg at the surface. Assume the Earth is not rotating, a bad approximation, but hey, you're not going to drill this tunnel anyhow.
Levi Tate said:Homework Statement
A tunnel is drilled straight through the Earth from Detroit to Zxxiuw. At the middle of the tunnel it is at 1/2 Earth radius from the center. Assuming that a mass m placed in the tunnel will slide without friction, show that it experiences a force that is directly proportional to its distance from the center of the tunnel. Give an expression for this force in terms of the distance and the weight mg at the surface. Assume the Earth is not rotating, a bad approximation, but hey, you're not going to drill this tunnel anyhow. I am thinking I need to use energy and then take the derivative to get the force, but I am having trouble finding out the right assumptions to make such that I can do that. That there is the problem, I am on an iPad so in the attachment is a picture of the problem as well as what I've been able to work out.
Thank you in advance
xirow said:its acceleration towards the opposite direction its in is equal to its distance from the center of the Earth divided by Earth's radius times 9.8m/s/s
sorry should have explained this more, before i was explaining the end result. The reason for the acceleration is because the force of gravity is based on the amount of mass on the other side of the object. From the surface of the Earth there is an acceleration of 9.8 m/s/s as it moves closer to the center of the Earth it decreases. then becomes zero at the center of the tunnel. then as it is at the other end of the tunnel it has an acceleration of 9.8 m/s/s in the other directionDick said:Nothing you've said proves that, now does it?
Levi Tate said:Right xirow that's the general idea of what's going on but the hard part is solving the problem.
Levi Tate said:I have that expression and tried to integrate using the second law, but I got a pretty ridiculous solution to the differential equation
Levi Tate said:Cos I have to write v=dv/dt in one dimension don't i?
A "tunnel through the earth" is a hypothetical tunnel that would go straight through the center of the Earth, connecting two points on the surface. It is often used as a thought experiment to explore concepts such as gravity, physics, and geology.
Yes, a "tunnel through the earth" is theoretically possible, but not currently feasible with our current technology. It would require extreme heat and pressure-resistant materials and advanced drilling technology.
If a tunnel were dug through the earth, the person or object inside the tunnel would experience a gradual increase in temperature and pressure as they traveled towards the center. Once they reached the center, they would begin to feel a decrease in both temperature and pressure until they reached the opposite side of the tunnel.
The time it would take to travel through a "tunnel through the earth" would depend on various factors, such as the depth of the tunnel, the speed of the object or person, and the air resistance inside the tunnel. However, it would take approximately 42 minutes and 12 seconds to travel from one side of the earth to the other, assuming a constant acceleration of gravity and no air resistance.
If a tunnel were dug through the earth, it would not have a significant impact on the earth's rotation or orbit. However, it could potentially alter the distribution of mass within the earth, which could cause small changes in the earth's rotation and orbit over a long period of time.