Where Will the Ball Land After Passing Through Earth's Core?

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SUMMARY

When a ball is dropped into a hole drilled through the center of the Earth, it will initially fall to the center due to gravity, then continue to oscillate back and forth, akin to a pendulum, until it eventually comes to rest at the center. The time taken for this journey is approximately 42 minutes, calculated using the formula T = π * √(r/g), where r is Earth's radius (6,378,100 m) and g is the acceleration due to gravity (9.81 m/s²). This scenario assumes no friction or other forces acting on the ball, which would otherwise prevent it from reaching the opposite side of the Earth.

PREREQUISITES
  • Understanding of gravitational forces and motion
  • Familiarity with simple harmonic motion (S.H.M.) concepts
  • Basic knowledge of physics formulas, particularly those involving time, radius, and gravity
  • Awareness of Earth's physical properties, including density variations
NEXT STEPS
  • Research the derivation of the formula T = π * √(r/g) in the context of gravitational physics
  • Explore the implications of varying Earth's density on gravitational calculations
  • Investigate the concept of antipodes and the likelihood of drilling through land versus water
  • Learn about the effects of friction and other forces on oscillatory motion in physics
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Physics students, educators, and anyone interested in gravitational dynamics and the theoretical implications of drilling through the Earth.

jontyjashan
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Hey
i m a newbie
suppose i drill a hole that passes through the centre of Earth and reaches the
other point on earth
now i drop a ball in this hole
where wud it land??
 
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jontyjashan said:
Hey
i m a newbie
suppose i drill a hole that passes through the centre of Earth and reaches the
other point on earth
now i drop a ball in this hole
where wud it land??

The ball would first go all the way to the center, and because of its momentum, it would keep moving through. It would get to a certain point on the other side (still in the hole) but its momentum would be counteracted by gravity, where then upon it would fall again to the center. It's sort of like a pendulum. This would continue to happen unitl the ball ended up at the center of the earth, where it would remain stationary.
 
I agree with benk99nenm312, but I'd like to add some points. First if there were no friction or other forces other than gravity working on the ball then it would reach exactly the other side of the Earth, then fall back through. It would never stop going back and forth. Any friction though and it would never make it to the other side, it would stop short, and eventually would settle in the center.

Secondly I'd like to mention that no matter where the two exits for the hole are it would always take about 42 minutes for it to fall through. The formula for figuring out how long it will take is:
T = \pi * \sqrt{r / g}

Where T is time taken, r is radius, and g is acceration due to gravity. For Earth r = 6,378,100 m, and g = 9.81 m/s.
 
Correct, about 42 minutes. Unless the hole was from pole to pole, the ride would be bumpy.
 
If you drilled at a random place, how likely is it that you would hit land on the other side, as opposed to ocean?
 
maze said:
If you drilled at a random place, how likely is it that you would hit land on the other side, as opposed to ocean?

75% of the Earth is water... so 25%? :-p
 
protonchain said:
75% of the Earth is water... so 25%? :-p

Ahh.. but the probability also depends on where you first start drilling. It's easy to hit China from where I am, but its hard to hit New Zealand from Greenland. :biggrin:
 
Benk,
You need to pull up Goggle Earth, China is in the Northern hemisphere,think about it.
 
Integral said:
Benk,
You need to pull up Goggle Earth, China is in the Northern hemisphere,think about it.

I wasn't necessarily saying it with the intention of having great accuracy, I was just jokingly stating a point. (China is popularly referred to being America's opposite position on the globe.)
 
  • #10
benk99nenm312 said:
I wasn't necessarily saying it with the intention of having great accuracy, I was just jokingly stating a point. (China is popularly referred to being America's opposite position on the globe.)

Hence the title of the film "the China syndrome"
 
  • #11
Chronos said:
Correct, about 42 minutes. Unless the hole was from pole to pole, the ride would be bumpy.
This is a fairly important point. It highlites the fact that the ball is essentially entering into a very elliptical orbit. In a vacuum it will continue to go back and forth through the center o fthe Earth. But, if the hole were not at the poles, then it is on a part of the Earth's surface that is moving (Eastward). As the ball falls down the hole, it is desending to a lower orbit, which means a faster orbit, which means ti bumps into the Eastern wall of the hole.
 
  • #12
DaleSwanson said:
Secondly I'd like to mention that no matter where the two exits for the hole are it would always take about 42 minutes for it to fall through. The formula for figuring out how long it will take is:
T = \pi * \sqrt{r / g}

Where T is time taken, r is radius, and g is acceration due to gravity. For Earth r = 6,378,100 m, and g = 9.81 m/s.
I'd like to know how did you derive this formula.
 
  • #13
how to derive this formula
 
  • #14
fluidistic said:
I'd like to know how did you derive this formula.

If it is assumed that the Earth has uniform density the object will move with S.H.M.
Maximum acceleration=g
g=-w^2.r (w= angular velocity)
T=2pi/w (T= time period i.e. time to go there and back again)
It is the same time period as for a satellite in close orbit.
 
  • #15
Hey the orbit would be the least of your problems, the second your drill reaches molten rock, and metal under its enormous presure you'd have global warming the likes of Venus.
 
  • #16
protonchain said:
75% of the Earth is water... so 25%? :-p

I don't think so. For example, you could imagine a world where one hemisphere is all land and the other is all water, in which case the answer would be zero.
 
  • #17
Chronos said:
42 minutes

Seriously? 42 minutes? Awesome. I'm going to stop here before the numerologists creep in.
 
  • #18
Dragonfall said:
Seriously? 42 minutes? Awesome. I'm going to stop here before the numerologists creep in.

Yes, but as Dadface pointed out this is only true if we suppose the density of Earth as a constant, which strongly differs from reality.
 
  • #19
fluidistic said:
Yes, but as Dadface pointed out this is only true if we suppose the density of Earth as a constant, which strongly differs from reality.

yah, I was wondering about that. The only way I see to really calculate the time it would take is to split the Earth up into sections according to density. Then, you would have to do some calculations for each section and add them up... maybe?
 
  • #20
benk99nenm312 said:
yah, I was wondering about that. The only way I see to really calculate the time it would take is to split the Earth up into sections according to density. Then, you would have to do some calculations for each section and add them up... maybe?

Welcome to the Wonderful World of Integration.
 
  • #21
protonchain said:
75% of the Earth is water... so 25%? :-p
If you start at land it is pretty hard to hit land on the opposite side:
http://www.antipodemap.com/

This is from http://en.wikipedia.org/wiki/Antipodes:
600px-Antipodes_LAEA.png
 
Last edited by a moderator:
  • #22
A few years ago, a very funny fellow named Ze Frank proposed making the Earth into a sandwich by placing two pieces of bread directly opposite each other. People all over the world got in on it and had some successes. In the process, Ze made a tool that might help you visualize the chance of hitting land when you drill through the earth. I was surprised to find that I'd end up in the Indian Ocean near Australia.

http://www.zefrank.com/sandwich/tool.html
 

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