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You push a block of mass m against a horizontal spring, compressing the spring a distance x. Then you release the block, and the spring sends it sliding across a tabletop. It stops a distance d from where you release it. Let k be the spring constant. What is the coefficient of kinetic friction between the block and the table? (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)

μk =

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  1. Use conservation of energy E = K + U
    Ui (initial potential from spring) = Kf (final kinetic energy

    1/2 kx^2 = 1/2 mv^2
    v = x*sqrt(k/m)

    From kinematics:
    d = (Vf^2-Vi^2) / (2a)
    d = (x*sqrt(k/m))^2 / (2a) (substituting our v from above)

    d = (kx^2) / (2*ma)

    We know from forces that friction = µ*N and N = mg
    the net force on the object after leaving the spring will be that from friction, therefore F = µmg = ma

    µmg = (kx^2) / (2d)

    µ = (kx^2) / (2dmg)

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