Graviational potential of 2 cones

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In summary, the question asks what the speed of a particle will be when it falls from infinity to the tip of two hollow fixed cones with a base radius R, slant height L, and surface mass density σ. The equations used are F=GMm/r^2 for force and V=GMm/r for potential, and the particle is assumed to be pulled towards the vertex of both cones. The solution involves calculating the particle's energy at infinity and at the tip of the cones, and equating the two to solve for its speed.
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Homework Statement



Consider two hollow fixed cones (such as ice cream cones without the ice cream). They have a base radius R, slant height L,and a surface mass density σ. The cones are vertical, with their apexs touching (say, at the origin). A particle initially at rest falls in from infinity, along a perpendicular bisector line. What is its speed when it reaches the tip of the cones?

Homework Equations



F = GMm/r^2
V= GMm/r

The Attempt at a Solution



So I am trying to write an equation for the cone, where if I pick any arbitrary height on the cone, I would get the circumference at that point. I would then integrate over the height of the cone. However, I am having trouble coming up with such an equations. I know that the forces in the z and y direction will cancel, so the particle will be "pulled" towards the vertex of both cones.

Would it be possible to consider the cones as a point at their center of mass?
 
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  • #2
Hi khemist! :smile:

I don't really understand what your cones have to do with it... :confused:

At infinity a particle of mass m and speed zero would have energy zero (arbitrary choice).
At distance 6000 km from Earth (which is the radius of the earth), you can calculate the energy E=-GMm/6000km.
This will equal the increase in kinetic energy (1/2)mv^2, from which you can calculate the speed v...

Btw, note that the potential is V=-GM/r.
 

1. What is the gravitational potential of 2 cones?

The gravitational potential of 2 cones refers to the potential energy that two cones have due to their gravitational attraction to each other. This potential energy is dependent on the masses of the cones, their distances from each other, and the gravitational constant.

2. How is the gravitational potential of 2 cones calculated?

The gravitational potential of 2 cones can be calculated using the formula: V = -G(m1m2/r), where V is the potential energy, G is the gravitational constant, m1 and m2 are the masses of the cones, and r is the distance between them.

3. What factors can affect the gravitational potential of 2 cones?

The gravitational potential of 2 cones can be affected by the masses of the cones, their distances from each other, and the gravitational constant. Additionally, the presence of other objects with their own gravitational fields can also affect the potential energy of the cones.

4. How does the gravitational potential of 2 cones relate to their gravitational force?

The gravitational potential of 2 cones and their gravitational force are directly related. As the potential energy increases, the gravitational force between the cones also increases. This means that the closer the cones are to each other, the stronger their gravitational force will be.

5. What are some real-life examples of the gravitational potential of 2 cones?

An example of the gravitational potential of 2 cones can be seen in the orbit of two planets around each other. The potential energy between the two planets determines their distance and the strength of their gravitational force. Another example is the gravitational potential between the Earth and the Moon, which affects the tides on Earth.

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