Newton's Laws with one body inside another

In summary: The weight would continue to accelerate until it reached the center and then decelerate until it got to the far side, and repeat this motion forever.
  • #1
saddlestone-man
78
20
Hello All

What would be the motion of a weight dropped into a hole drilled all the way through an Earth-sized planet?

Would the weight accelerate all the way to the centre and then decelerate until it got to the far side, and repeat this motion forever?
OR
Would it accelerate initially and then start to decelerate as more of the planet was "behind" it and therefore slowing it down, and come to rest gently at the centre?
OR
Something else?

Assume the planet isn't spinning and the hole is in a vacuum.

Best regards. Stef
 
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  • #2
If you truly want answers at A level this is something you should have no trouble doing for yourself - it's a textbook application of Gauss' Theorem. I suspect you want to pick I level.

In idealised frictionless circumstances the weight accelerates towards the core and decelerates once it passes. The acceleration profile depends on your model of the Earth.

The application of Gauss' theorem that I mentioned is to use it to realize that (for a spherically symmetric mass distribution) the mass outside the radius you are at is irrelevant. The gravitational acceleration is therefore ##GM(r)/r^2## where ##M(r)## is the mass of the Earth inside radius ##r##. If you assume constant density this is just the density times the volume of a sphere of radius ##r##. A realistic density profile requires some integration. However, note that ##M(0)=0## for any mass distribution, so the acceleration always falls to zero at the core.
 
  • #3
Many thanks for your answer. Yes I was assuming that the planet had a uniform mass distribution.

Does the weight therefore stop accelerating at the centre and starts to decelerate until it stops at the opposite opening and repeats the process ad infinitum?
 
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  • #4
In idealised frictionless circumstances, yes. The acceleration may increase for some of the fall (depending on the mass distribution, and not in the case of a uniform mass distribution) but will always drop to zero at the core. Spherical symmetry means that the deceleration profile is the reverse of the acceleration profile and the weight must come to rest at the surface again.
 
  • #5
saddlestone-man said:
Yes I was assuming that the planet had a uniform mass distribution.
And that results in a simple harmonic motion because the 'restoring force' towards the centre of the Earth is proportional to the displacement from the centre. Practicalities like enormous temperatures and pressures as you go down (and the Earth's rotation), mean that the experiment is a non-starter BUT: the time period for a simple harmonic oscillator is independent of amplitude so you could repeat the experiment for real if you got a spherical asteroid with a hole through it and a small rock. (the outside bits of the Earth don't count).

Interestingly, the period is (nearly) the same as the period of a satellite in low Earth orbit.

Edit PS there is a vast amount of information about SHM but it nearly all gets Mathematical very quickly. I suggest you do a search and find yourself a hit that suits your level.

Just imagine a future Richard Branson setting up such an extravagant experiment just for fun and to show how rich he is.
 
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  • #6
The period is a constant to cross the planet from side to side, it has been calculated in 42 minutes, both to cross any string and to cross the diameter.
By the way, another idealization that has to be abstracted is that the planet should not rotate, and if it does, it will only reach the other ends without colliding with the walls if the hole coincides with the axis of rotation.
 

Related to Newton's Laws with one body inside another

1. What are Newton's Laws with one body inside another?

Newton's Laws with one body inside another refer to the application of Newton's three laws of motion to a system where one body is contained within another body. This can include situations such as a person riding in a car or a planet orbiting around a star.

2. What is Newton's First Law in this context?

Newton's First Law states that an object at rest will remain at rest and an object in motion will remain in motion at a constant velocity unless acted upon by an external force. In the case of one body inside another, this means that the inner body will continue to move at a constant speed and direction unless an external force is applied to it.

3. How does Newton's Second Law apply to this scenario?

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the context of one body inside another, this means that the acceleration of the inner body will depend on the net force acting on it and its own mass.

4. Can you give an example of Newton's Third Law with one body inside another?

Newton's Third Law states that for every action, there is an equal and opposite reaction. In the case of one body inside another, this means that the outer body will exert a force on the inner body, and the inner body will exert an equal and opposite force on the outer body. For example, when a person jumps off a diving board, the person exerts a downward force on the board, and the board exerts an equal and opposite force upwards on the person.

5. How do Newton's Laws with one body inside another relate to real-world situations?

Newton's Laws with one body inside another are applicable to many real-world situations, such as driving a car, riding a bike, or even the motion of planets in our solar system. These laws help us understand and predict the behavior of objects in motion and are essential in fields such as engineering and physics.

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