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.
1. find k, the spring constant.
2. express k in terms of µ, m, g, and v.

Explanation:

The mass of object is m

The coefficient of friction is μ

the speed of object at x= 0 is v

when the object compress the springs stops so,

The kinetic energy of the spring is equal to and friction work is done

...........

after that it returns from the rest and finally comes to rest after coming to x=0

position.

So, now the potential energy of the spring is equal to the work done by the friction force.

substitute the x value in equation 1

mv^2 = kx^2+2\mu (mg)x

solving this we get

therefore the force constant

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