Maximum Extension of Elastic String Attached to Particle on Rough Plane

When a particle is attached to a an elastic string on a rough plane, it will come to rest at a point on the slope where whatever gravitational potential energy the particle started with has been turned into elastic potential energy in the string of been used to overcome friction.

The particle below is attached to an elastic string at a point A.

Suppose we take the gravitational potential energy at A as being zero, then at B the particle has gravitational potential energy equal toIf the particle is released from rest at A, then the initial kinetic energy is zero and the initial elastic potential energy is also zero because the string is not stretched.

The total initial energy is then zero.

If the coefficient of friction isthen the friction forceThe work done against friction in moving from A to B is then

Conservation of energy then gives

Expanding the brackets and collecting like terms gives

The solutions to this areand

We have taken x as positive down the slope so must take the positive square root. The solution is

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