Is Calculating Work Done by Gravity as Simple as It Seems?

[Martin23]

So I was thinking about the Universal Law of Gravitation, and the force of an object depends on the radial distance from Earth for simplicity. As an object travels to the center of the Earth the F_g increases as the radial distance decreases.F=(G*m*m*1)/(r^2). Knowing that the Force is not constant, the work done by gravity would be ∫F*dr from lower limit of r to upper limit of 0. Now, if you put the F in terms of r you get ∫[(G*m*m)/(r^2)]*dr from r to 0. As I tried to solve for Work, I arrived at 1/0 and now I am stuck. Was all this valid or just nonsense? If you think about this in vector form 3 dimensions, it is more understandable to me at least. Here is a picture of my work. Help please.

 

[Legaldose]

Think about what you mean by r being 0. Are you really taking the limit of integration as the center of Earth? I would think you would want to substitute 0 for the radius of Earth. Or even better, just use r_initial and r_final as your limits, it gives you a more general answer. Then just plug-n-chug for your solution.

 

[Martin23]

oh your right haha. 0 radial distance would be like super imposed on Earth hahaa
they would occupy the same space right? haha ty.

 

[jtbell]

As an object travels to the center of the Earth the F_g increases as the radial distance decreases.

No, it doesn't. When the radial distance is less than the Earth's radius, only the mass inside the radial distance "counts" in calculating the gravitational force. Google "shell theorem" for details.

 

[PJC01]

Your thinking about the Universal Law of Gravitation is correct. As an object gets closer to the center of the Earth, the gravitational force increases due to the decrease in radial distance. However, your approach to calculating the work done by gravity is not entirely correct.

The formula for work is W = F * d, where F is the force applied and d is the distance over which the force is applied. In this case, the distance over which the force is applied is not a constant value, as the object is moving closer to the center of the Earth. Therefore, we cannot simply use the formula W = F * d to calculate the work done by gravity.

Instead, we need to use the concept of work as a line integral. This means that we need to integrate the force over the path that the object travels. In this case, as the object moves closer to the center of the Earth, the path it takes is along the radial distance, which is changing. So we need to integrate the force over this changing distance.

The correct formula for calculating the work done by gravity is: W = ∫F(r) * dr, where F(r) is the force at a given radial distance and dr is the change in radial distance.

Therefore, your integral should be: W = ∫[(G * m * m)/(r^2)] * dr from r to 0.

To solve this integral, you need to use techniques such as substitution or integration by parts. It is not valid to simply substitute 0 for r, as this would result in a division by 0.

In vector form, the work done by gravity can be written as W = ∫F(r) * dr, where F(r) is the gravitational force in the radial direction and dr is the change in radial distance. This approach is equivalent to the one described above.

In conclusion, your understanding of the Universal Law of Gravitation is correct, but your approach to calculating the work done by gravity needs to be adjusted. You should use the concept of work as a line integral to correctly calculate the work done by gravity.