Work Done by Spring on a Body Sliding down a rod

[Simon Goster]

1. The Question
Unable to find the work done by spring on the object sliding down the rod as shown below:

Homework Equations

Work Done by Spring Force : -1/2 * KΔx² --(1)
where k = Spring Constant
Δx = Change in Spring Length[/B]

3. Where the problem occurred
While seeing the solution of the problem, it was mentioned that we can simply take
W_Spring = -1/2 * K(X₂²-X₁²) --(2)
But my problem is that if the position of ring on rod is changing, the angle of spring with the rod is changing and thus the force components on rod is changing, then how the usual formula
Work Done by Spring Force : -1/2 * KΔx²
is also valid for the given condition.
PS: I also tried to understand this using concept of conservative force, but couldn't get it.
Thanks!

 

[UncertaintyAjay]

Your formula for work done by a spring is wrong. Its X². Not x.

 

[Simon Goster]

Your formula for work done by a spring is wrong. Its X². Not x.

Sorry for this.
Corrected the Formulas
Thanks!

 

[UncertaintyAjay]

Okay so , ignore the fact that you've seen the solution for a moment. Assume you don't know what the answer is. How would you approach the sum?

 

[Simon Goster]

Okay so , ignore the fact that you've seen the solution for a moment. Assume you don't know what the answer is. How would you approach the sum?

1. Integration =>∫(force of spring along rod)*dx or ∫(force of spring along rod as function of angle)*dθ
2. As Work Done = - Change in Potential Energy
Find Potential Energy due spring for both the positions and use the above relation

Are these meathods right.
Wont consider using integration method
Can we use other concepts like conservative forces or just like that?

 

[UncertaintyAjay]

The second method is easier. When in doubt use conservation of energy. Now try it with that and see what answer you get.