Tunnel from north pole to equator

[Matija]

Hi all! I'm not so good at physics so i decided to turn to you. So please help me.

Homework Statement

I have to calculate how the object (ball) moves when droped into the tunnel (as seen in the picture), and how long the journey takes. We assume that Earth is round and there is no friction and air resistance and Earth density is constant.

http://www2.arnes.si/~vvztrzin/pic.jpg

Homework Equations

The Attempt at a Solution

I tried going throug with lagrangian, but i found it hard, how to calculate gravity energy, because the mass of the Earth is larger on the left than on right.

I appreciate all your help.

 

[Matterwave]

I would do this using Newtonian mechanics (probably because I'm partial to it).

I would get the force due to gravity at all times during the trip.

The force is always towards the center of the Earth, and is due to the mass of the Earth that is within the sphere that has boundaries at the point of the object (assuming the tunnel wasn't large enough to affect Earth's symmetry). Thus, you can get a force vs distance along the tube, and from there get the motion.

 

[kerimek]

Hello there,

It seems like you are trying to calculate the motion of an object dropped into a tunnel that goes from the North Pole to the Equator. This is an interesting problem and involves a few different concepts from physics.

First, let's address the shape of the Earth. You mentioned that we can assume the Earth is round, which is a good approximation for this problem. However, the Earth is actually slightly flattened at the poles and bulging at the equator due to its rotation. This is known as the Earth's oblateness and can have an effect on the motion of objects. But for this problem, we can neglect this effect and assume the Earth is a perfect sphere.

Next, we need to consider the gravitational force acting on the object. The Earth's gravity follows an inverse-square law, meaning that the gravitational force decreases with the square of the distance between two objects. This means that as the object moves closer to the center of the Earth, the force of gravity will increase. However, since the Earth's mass is symmetrically distributed, the gravitational force will be the same on all sides of the object as it falls through the tunnel.

Now, let's talk about the motion of the object. Since we are assuming no air resistance or friction, the only force acting on the object is gravity. This means that the object will accelerate towards the center of the Earth at a constant rate. This type of motion is known as free fall or projectile motion. The equations for this type of motion can be found in most introductory physics textbooks.

To calculate the time it takes for the object to reach the center of the Earth, you can use the equation for the time it takes for an object to travel a certain distance with a constant acceleration. In this case, the distance is the radius of the Earth and the acceleration is the acceleration due to gravity. Once the object reaches the center, it will continue to move towards the other side of the tunnel with the same acceleration, until it reaches the other end.

I hope this helps guide you in the right direction for solving this problem. Remember to always check your equations and units to make sure they make sense and give you a reasonable answer. Good luck!