A Hole through the Centre of the Earth - Part 4

Friday, February 15, 2008

A Hole through the Centre of the Earth - Part 4

Continued from here...
See Part 1, Part 2

Ok, I'm back...

Let's analyze the process... now.

We got the hole dug from Shanghai to Entre Rios. We got a rock to put into the hole (like the rock volunteered or something !!! ).

Let's do it...

Rock is moving from the surface of the earth. The acceleration due to gravity (g) is 9.8 m/s2. Wait, how come it is 9.8 ??? Let me check...

I went here...

The gravitational acceleration towards an object with mass m is given by:

Contant g


where:
r - distance from center of the object to the location we are considering,
G - is the gravitational constant of the universe.

I went here...

Constant G





Then g = 5.9736 x 1024 kg x 6.67428 x 10-11 m3 kg-1 s-2 / (6371 x 103 m)2

Big calculation, let me use MS Excel :)

g = 3.99 x 1014 / 4.06 x 1013

= 9.82 !!! (OMG, I just found out g :)

Let's seperate the radius component from the calculation...

So, g = 398694790 / r2

Ok, fine.

When we go 1 km deep into the earth, what happens to g. OMG, that is a gud question.

When we go to poles, g becomes 10 as the poles are flattened, radius is less.

But, does the fact that one body is inside the other changes anything. I don't know. Let's put it away for now. :) May be come to it later.

When we reach half way to the centre, g will be 398694790 / (3185 x 3185), i.e., 39.3. That's a hell of an acceleration.

So, the process.

1. Stone is released. Initial state : Velocity = 0; Acceleration = 9.82;

2. Stone on the way to the centre. The acceleration goes on increasing, velocity goes on increasing.

3. Stone reaches the centre. Maximum g (inifinity !!!). Maximum speed.

4. Stone goes away from the centre. Pull in the opposite direction. Deceleration maximum. Velocity decreasing. Deceleration decreasing.

5. Stone reaches the other end of the hole : Velocity = 0; Deceleration = 9.82.

The process repeats in the opposite direction. This can continue for infinity.

BUT... there is FRICTION, the great player in everything.

So, the speed always goes on decreasing than that is happening in the ideal case. So, the stone stops before reaching the other end, farther and farther away time. This happens for a long long time until the stone comes to rest at the centre.

Wow, now what? How will we get the stone back? What happens when another stone hits the first stone at the centre? What if two stone are put from opposite sides at the same time? Lots and lots and lots of questions... ???

We will talk about them also... next time. Bye.

To be continued...


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