Work to Bring Object up Mineshaft to Earth Surface

• sean80439
In summary, the conversation discusses the calculation of work required to bring an object of mass m from the center of a uniform density Earth to the surface. The first part of the question involves integrating the change in gravity with respect to radius and solving for the work, resulting in a solution of -GM/2R. The second part involves dropping the object from the surface to the center and calculating its speed at the center. The conversation also mentions using the fact that the net force at the center is zero and that the force is linear with distance from the center. The correct expression for work is W = ∫GMmr/R^3 δr and the relationship between force and distance changes from linear inside the sphere to inverse square outside.
sean80439
Here is the first part of the question:
If there is an object of mass m at the center of the earth, how much work does it take bring that object up a shaft to the surface? Edit: Earth is uniform density.

Do I integrate g(r) = -GMr/R^3 with respect to 'r' to get the change in gravity from the center of the Earth to the surface and then solve from R-0. If so I get a solution of -GM/2R, and that doesn't look at all right to me.

The second half of the question relates back to the first part and asks if mass m is dropped from the surface down a mineshaft to the center of the earth, what will its speed be when it reaches the center. Again, since gravity is changing with respect to the radius, how do I get a solid answer?

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What's the force at the center? Half-way to the surface?

Use the fact that the mass of the shell outside the radius from the object to the center does not exert a net force.

Can you get a general force equation from these (for objects within the radius of the Earth)? Now how can you get work knowing the force?

The net force would be zero at the center. Halfway it would be F=-GMm/2R^2 (r = R/2). It would be the F Normal = mg(r) wouldn't it, with W = mg(r)R? Why does it matter what the force is halfway to the surface or am I missing something?

Well the point is to see that the force is linear with distance as it increases from the center. From that you can integrate and find the work done.

Notice that the entire mass is not acting on the object when it is at the halfway point. In fact, the mass that is acting on it is the mass of the Earth enclosed by a sphere of radius equal to the distance of the object from the center. Does that make sense? Your current expression for the second bit is incorrect for this reason.

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So the equation should look like: W = ∫Fr δr or
W = ∫GMmr/R^3 δr and evaluate from 0 to R correct?

Also as I understand it, the relationship is linear inside the sphere and changes to the inverse square when outside correct? When you say second expression are you referring to the original integral in the initial post?

Last edit: Yes I think I know what you mean, the only mass acting on the object is the mass of the Earth at that radius ignoring all mass from every radius of r > r_object.

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1. How do you bring an object up from a mineshaft to the Earth's surface?

To bring an object up from a mineshaft to the Earth's surface, a pulley system is typically used. This involves attaching a rope or cable to the object and using a wheel and axle mechanism to lift it up.

2. What materials are needed for this process?

The materials needed for bringing an object up from a mineshaft to the Earth's surface include a pulley system, a strong rope or cable, a winch or hoist to operate the pulley, and a sturdy platform or surface to support the object once it reaches the surface.

3. How is the weight of the object taken into consideration during this process?

The weight of the object is a crucial factor to consider when bringing it up from a mineshaft to the Earth's surface. The pulley system and rope must be able to support the weight of the object, and the winch or hoist must have enough strength to lift it.

4. Are there any safety precautions to take during this process?

Yes, there are several safety precautions that should be taken when bringing an object up from a mineshaft to the Earth's surface. This includes wearing proper safety gear, ensuring the pulley system is securely attached and in good condition, and having a designated person to operate the winch or hoist.

5. What are some potential challenges that may arise during this process?

Some potential challenges that may arise during this process include the weight of the object being too heavy for the pulley system, the rope or cable breaking, or the winch or hoist malfunctioning. It is important to carefully plan and prepare for these challenges to ensure a safe and successful lift.

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