A block of mass m, connected to a spring of spring constant k, rests on a rough incline; the angle of incline is θ . The coefficients of friction are μ s and μ k respectively. The spring is slowly

Question

A block of mass m, connected to a spring of spring constant k, rests on a rough incline; the angle of incline is $θ$ . The coefficients of friction are $μ_{s}$ and $μ_{k}$ respectively. The spring is slowly pulled up the incline until the block starts to move. What is the value of $μ_{k}$ such that the block comes to rest when the spring is neither extended or compressed.

TestHub · Accepted Answer

At impending motion up the incline: k x = m g sin θ + μ s ​ m g cos θ . Once released, the block slides down and stops exactly at the natural length, so the stored spring energy is spent against gravity component and kinetic friction: 2 1 ​ k x 2 = ( m g sin θ + μ k ​ m g cos θ ) x Substituting kx: 2 1 ​ ( m g sin θ + μ s ​ m g cos θ ) = m g sin θ + μ k ​ m g cos θ , giving μ k ​ = 2 1 ​ ( μ s ​ − tan θ ) Ans. (A)

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