Where does the crate permanently come to rest?

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The crate, weighing 5.0N, is released from a height of 2.0m and slides down an inclined plane before crossing a smooth floor. It encounters a rough plane with a spring at a height of 0.5m, where the spring constant is 20N/m and the coefficients of friction are 1/√3 for kinetic and 1/√2 for static friction. The maximum height the crate reaches on the right-hand inclined plane is determined by energy conservation principles. The crate will come to rest after compressing the spring, which must be calculated based on the energy lost to friction and the potential energy stored in the spring. The discussion emphasizes the importance of applying the energy-work principle to solve for the crate's final resting position.
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A small crate of weight 5.0N is released from rest at a height 2.0m up a smoyj inclined plane. The crate slides down the plane and across a smooth 1.50m floor to a rough plane where a spring is located.
The bottom of the spring is located at a height of 0.5m. The spring constant is 20N/m .
The coefficient of kinetic friction between the rough plane and crate is $\frac{1}{\sqrt{3}}$ and the coefficient of static friction is $\frac{1}{\sqrt{2}}$/

(a) What is the max height the crate reaaches on the right hand inclined plane?
(b) Where does the crate permanently come to rest? How many times does the crate travel up to the rough plane?
 

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Hint: Energy-work principle.
 
The first part is OK for me,
but the second part seems very complicated.
 
Well, how much must the spring compress in order to keep the crate going?
 

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