"Proving Central Force is a Conservative Force

AI Thread Summary
A central force is defined as one that acts along a line from an object to a specific point, with its magnitude depending solely on the object's distance from that point. To prove that a central force is conservative, one must show that the work done by the force is path-independent and only depends on the initial and final positions. The discussion highlights confusion around the relationship between work, force, and displacement, emphasizing that the work done cannot be simplified to a basic formula due to the variable nature of the force with displacement. A more rigorous approach using calculus is necessary to express the work done by a central force accurately. Understanding these concepts is crucial for demonstrating that central forces conserve mechanical energy.
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 dependant 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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