Express the potential energy U in terms of M and v

In summary, the conversation discusses deriving the total mass and potential energy of a finite isothermal sphere using the radius and circular velocity. The potential energy is expressed as U = -GM^2/R and the virial theorem is used to find the kinetic energy and total energy in terms of M and v. The answers for K and E are -v^2M/2 and v^2M/2, respectively.
  • #1
leonne
191
0

Homework Statement


For a finite isothermal sphere with a radius R and circular velocity v (remember,
the rotation curve is constant), we derived the total mass and potential energy as
M =v2R/G

U = -GM2/R
Express the potential energy U in terms of M and v. Use the virial theorem to
fi nd the kinetic energy K and total energy E (again in terms of M and v).



Homework Equations





The Attempt at a Solution


Kind of confused about the first part to find potential in terms of m and v.
Would I just sub total mass equation into potential? I get U=-v4R/G
Is this right? Hmm seems like what i did is wrong. Is it saying that the potential energy can only have M and V?
 
Physics news on Phys.org
  • #2
It said m and v so I would think you would want to have
U=-v^2 M
 
  • #3
O yea seems that would work you would get the same units when u get the answer thxs
 
  • #4
For this part
Use the virial theorem to
fi nd the kinetic energy K and total energy E (again in terms of M and v).
Would the answers be k=-v^2M/2 E=v^2m/2
I think the virial theorem is E=-(K)=(U/2)
 
  • #5


Yes, you are correct in thinking that you can substitute the total mass equation into the potential energy equation. However, the equation you wrote is not quite correct. The correct equation for potential energy in terms of M and v would be:

U = -GMv^2/2

To find the kinetic energy K, we can use the virial theorem which states that the total kinetic energy is equal to half the total potential energy:

K = U/2 = -GMv^2/4

And finally, the total energy E can be found by adding the potential and kinetic energies together:

E = U + K = -GMv^2/4 - GMv^2/2 = -3GMv^2/4

So in summary, the potential energy U can be expressed in terms of M and v as -GMv^2/2, the kinetic energy K is equal to -GMv^2/4, and the total energy E is equal to -3GMv^2/4.
 

1. What is potential energy?

Potential energy is the energy that an object possesses due to its position or configuration. It is stored energy that has the potential to do work.

2. What is the relationship between potential energy and mass (M)?

The potential energy of an object is directly proportional to its mass. This means that as the mass of an object increases, its potential energy also increases.

3. How does velocity (v) affect potential energy?

The potential energy of an object is directly proportional to the square of its velocity. This means that as the velocity of an object increases, its potential energy increases exponentially.

4. How do you express potential energy in terms of M and v?

The potential energy (U) of an object can be expressed as U = 1/2 * M * v^2, where M is the mass of the object and v is its velocity.

5. What is the unit of measurement for potential energy?

Potential energy is measured in joules (J), which is the same unit used for energy and work.

Similar threads

  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
1K
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
2K
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
3K
  • Advanced Physics Homework Help
Replies
2
Views
2K
  • Other Physics Topics
Replies
14
Views
2K
Back
Top