"Proving Central Force is a Conservative Force

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SUMMARY

The discussion centers on proving that a central force is a conservative force. A central force is defined as one that is directed along a line from the object to a specific point, with its magnitude dependent solely on the distance from that point. The participants highlight that the work done by a central force cannot be simplified to the product of force and distance due to the variable nature of the force with displacement. Instead, a calculus-based approach is necessary to accurately express the work done by such a force.

PREREQUISITES
  • Understanding of central forces in physics
  • Familiarity with the concept of conservative forces
  • Basic knowledge of vector notation and displacement
  • Introduction to calculus, specifically in relation to work done by a force
NEXT STEPS
  • Study the mathematical definition of work done by a force using calculus
  • Explore the properties of conservative forces and their implications in physics
  • Learn about central force motion and its applications in orbital mechanics
  • Investigate examples of central forces, such as gravitational and electrostatic forces
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Students of physics, educators teaching mechanics, and anyone interested in the mathematical foundations of force and energy concepts.

m2287
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Homework Statement



If a force on an object is always directed along a line from the object to a given point, and the magnitude of the force depends only on the distance of the object from the point, the force is said to be a central force. Show that any central force is a conservative force.

Homework Equations





The Attempt at a Solution



i really don't have a clue...

im guessing work done = force * distance

and F = ma might be used but i don't know how!
 
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What is the definition of a conservative force?
 
that if you move an object around and place it back in its original position no energy is lost?
 
m2287 said:
that if you move an object around and place it back in its original position no energy is lost?
Correct. Can you therefore write an expression for the work done by a central force?
 
wd= ma * d ?
 
m2287 said:
wd= ma * d ?
Close but don't forget that the magnitude of the force is dependent on the displacement. Use vector notation for the displacement and write ma as F.
 
does than not just give you wd = f* d

im very confused about how to prove this!
 
m2287 said:
does than not just give you wd = f* d
No. Note that the magnitude of the force is a function of displacement and note that we are looking at displacement which is a vector quantity. Now, because the force is not constant (because is varies with displacement), we cannot use the simple relationship you outlined above. Do you know a general definition of the work done by a force (involving calculus)?
 

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