Dropping a mass in a hole through the Center of Earth

In summary, the conversation discusses a problem involving a spherical mass dropped into the deepest hole drilled into the Earth's surface. The weight of the mass at the bottom of the hole is to be calculated using the formula F=GMm/r^2, where G is the universal gravitational constant, M is the mass of the Earth, m is the mass of the dropped object, and r is the distance between the centers of the Earth and the object. The depth of the hole (40,230 ft) should be subtracted from the radius of the Earth before squaring it to get an accurate value for r.
  • #1
Pruddy
64
0

Homework Statement


Assume that the Earth is a uniform density spherical mass. (This assumption is not correct but we will use it for simplicity in working the problem.) The deepest hole drilled into the Earth's surface went to a depth of 40,230 ft (Wikipedia.org). Imagine that this hole was straight toward the center of the Earth and that a small 50 kg spherical mass was dropped into the hole and ended up at the very bottom of it. What would be the weight of the spherical mass at that location?


Homework Equations



w=mg


The Attempt at a Solution


This is a two shell theorem problem. Since we all know that w=mg, F=GMm/r2. I think we have to find the gravitational force of the object that is at bottom of the Earth and then multiply it by its mass. This will give us the weight of the object.
 
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  • #2
Welcome to PF!

Hi Pruddy! Welcome to PF! :smile:
Pruddy said:
I think we have to find the gravitational force of the object that is at bottom of the Earth and then multiply it by its mass. This will give us the weight of the object.

Yes, that's correct. :smile:

What is worrying you about that? :confused:​

(btw, don't worry about the duplicated posts, that sometimes happens :wink:)
 
  • #3
Thanks for your quick reply. I do not know the right equation to use to get the gravitational force of of the mass at the bottom of the earth:confused:.
 
  • #4
it's the formula you've already given … F=GMm/r2

(and of course GM/R2 = g where R is the radius of the Earth)

(btw, you mean the bottom of the hole, not the bottom of the Earth :wink:)
 
  • #5
Thanks again for your quick reply. Yeah you are right, I meant the bottom of the hole. But what happends to the depth of the hole which is 40,230ft. Is it irrelevant to the question? and if it is not, will is be right to add it to the radius of the Earth before squaring it(radius)...
 
  • #6
Pruddy said:
… will is be right to add it to the radius of the Earth before squaring it(radius)...

uhh? :confused:

it's a hole …

subtract it! :smile:​
 
  • #7
Thanks Tim, That was very helpful...
 

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

The time it takes for a mass to reach the other side of Earth would depend on the mass of the object, the density of the Earth's core, and the distance between the two points. However, using the equation for gravitational acceleration, it can be calculated to take approximately 38 minutes and 11 seconds for a mass to travel through the center of Earth.

2. Would the mass reach terminal velocity while falling?

No, the concept of terminal velocity only applies to objects falling through an atmosphere. Since there is no atmosphere in the center of Earth, the mass would continue to accelerate due to the force of gravity until it reaches the other side.

3. What would happen to the mass as it reaches the center of Earth?

As the mass approaches the center of Earth, the force of gravity would decrease. This is because at the center, the gravitational forces from all sides would cancel out, resulting in a net force of zero. Therefore, the mass would continue to move through the center without slowing down.

4. What would be the impact of air resistance on the mass?

Since there is no air in the center of Earth, there would be no air resistance acting on the mass. This means that the mass would not experience any additional force slowing it down, and it would continue to accelerate towards the other side.

5. Would the mass experience any changes in weight as it travels through the center of Earth?

Yes, the mass would experience a change in weight as it travels through the center of Earth. This is because the force of gravity is directly proportional to the mass of an object. As the mass approaches the center, the gravitational force would decrease, resulting in a decrease in weight. However, once the mass passes the center and begins to move towards the other side, the gravitational force would increase again, resulting in an increase in weight.

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