# Problem involving Newtonian Gravity and a tunnel through the Earth

• mat5041
In summary: After that, I can give you a step by step explanation. I think it would be much better if you would show your attempt first. After that, I can give you a step by step explanation.
mat5041

## Homework Statement

The problem is: Consider a tunnel that connects any two points A and B on the spherical Earth (assuming constant density). The tunnel is a vacuum, and the train traveling through the tunnel is traveling on a frictionless track with no engine (i.e. it is just falling through the tunnel). Show that the transit time is independent of the positions (e.g. San Fran to LA, or NYC to Sydney). How long is the ride?

## Homework Equations

g=-G*integral(dv*rho(r)/r^2[direction vector])
volume integral
a=dv/dt

## The Attempt at a Solution

The only thing that I can think of is integrating the function that determines the g value to get the velocity and somehow showing that has a constant time not dependent on the distance that the "train" travels. Could anyone give me some insight? Thanks in advance.

The magnitude of g at a radius r is linearly proportional to r since the mass contained is proportional to r^3. Now draw a chord passing through the Earth and use trig to find the component of g parallel to the chord as a function of distance from the center of the chord. You will find that it is a linear function of the distance and the constant of proportionality is the same for all chords. They are all harmonic oscillators with the same spring constant.

Dick said:
The magnitude of g at a radius r is linearly proportional to r since the mass contained is proportional to r^3. Now draw a chord passing through the Earth and use trig to find the component of g parallel to the chord as a function of distance from the center of the chord. You will find that it is a linear function of the distance and the constant of proportionality is the same for all chords. They are all harmonic oscillators with the same spring constant.

Can you please show the calculation. I tried the question, but I'm not getting the component as directly proportional to distance from mean position. Please reply as soon as possible.

particlemania said:
Can you please show the calculation. I tried the question, but I'm not getting the component as directly proportional to distance from mean position. Please reply as soon as possible.

I think it would be much better if you would show your attempt first.

## 1. How does Newtonian gravity affect a tunnel through the Earth?

Newtonian gravity is the force of attraction between two objects with mass. In the case of a tunnel through the Earth, the gravitational force is strongest at the surface of the Earth and decreases as you get closer to the center. This means that as an object falls through the tunnel, it will accelerate until it reaches the center of the Earth and then decelerate as it moves towards the opposite surface.

## 2. What is the significance of a tunnel through the Earth in understanding Newtonian gravity?

A tunnel through the Earth allows us to study the effects of gravity at different depths and distances from the center of the Earth. It also helps us understand the concept of gravitational potential energy, as an object at the surface has the most potential energy and this energy decreases as the object moves towards the center.

## 3. Does the mass of the Earth have an impact on the gravitational force in a tunnel through the Earth?

Yes, the mass of the Earth does have an impact on the gravitational force in a tunnel through the Earth. According to Newton's law of gravitation, the force of gravity is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between them. This means that the more massive the Earth is, the stronger the gravitational force will be at any given point in the tunnel.

## 4. How does the diameter of the tunnel affect the gravitational force in a tunnel through the Earth?

The diameter of the tunnel does not affect the gravitational force in a tunnel through the Earth. As long as an object is located within the tunnel, the gravitational force acting on it will depend solely on its distance from the center of the Earth and the mass of the Earth.

## 5. Can a tunnel through the Earth be used as a means of transportation?

In theory, a tunnel through the Earth could be used as a means of transportation. However, there are many practical and technical challenges that would need to be overcome, such as maintaining a breathable atmosphere and dealing with extreme temperatures and pressures. It is also unlikely that a tunnel could be dug all the way through the Earth due to the Earth's molten core. As of now, this concept remains purely theoretical and has not been successfully implemented.

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