# Problem: An object of mass, m, is traveling on a horizontal surface. There is a coefficient of kinetic friction, μ, between the object and the surface. The object has speed, v, when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x=0 on its return trip, it stops. Find k, the spring constant. Express k in terms of mu , m, g, and v.

###### FREE Expert Solution

In this problem, we're going to use the equation for the Kinetic energy:

$\overline{){\mathbf{K}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{mv}}}^{{\mathbf{2}}}}$

From the law of conservation of energy:

Initial K = Work done against friction + spring energy.

$\begin{array}{rcl}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}{\mathbf{v}}^{\mathbf{2}}& \mathbf{=}& \mathbf{\mu }\mathbf{m}\mathbf{g}\mathbf{x}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\mathbf{x}}^{\mathbf{2}}\end{array}$........................ (1)

95% (26 ratings) ###### Problem Details

An object of mass, m, is traveling on a horizontal surface. There is a coefficient of kinetic friction, μ, between the object and the surface. The object has speed, v, when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x=0 on its return trip, it stops. Find k, the spring constant. Express k in terms of mu , m, g, and v.