**Problem Statement:**

A mass m = 0.5 kg is on a track without friction at the end of a spring with a stiffness K = 2000 N / m. The spring is compressed by a length x = 10 cm.

- If the mass is released, determine its speed when the spring reaches its natural length.
- How high will it ascend along the inclined section of the track?
- If the track has a horizontal section of length d = 7 m with friction (μ = 0.15), determine the maximum height reached by the mass.

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**Solution:**

We will solve the problem using the energy conservation principle. We will take point A of the figure as the initial state and point B when the spring has its natural length (x = 0) as the final state. To calculate the speed of the mass at point B, we will compare its energy between these two states:

In state A all the energy is accumulated in the spring in the form of elastic potential energy, since the spring force is a **conservative force**. In state B, when the spring has its natural length, the mass will have acquired kinetic energy and the spring will no longer have elastic potential energy. On the other hand, the gravitational potential energy does not vary between state A and B since the mass is at the same height in these two states.

As no non-conservative force acts between points A and B, the mechanical energy is conserved:

To calculate the maximum height reached by the mass we will compare the energy at point A with that at point C where the mass stops climbing and reaches its maximum height.

As we said previously, in state A all the energy is accumulated in the spring in the form of elastic potential energy. At point C all the energy has turned into gravitational potential energy.

The mechanical energy is preserved again because there is no non-conservative force acting between points A and C.

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If the track has a horizontal section with friction, the mechanical energy will no longer be conserved between points A and C.

The variation of mechanical energy between both points will be equal to the work of the friction force:

The work of the friction force on a mass moving on a horizontal plane is calculated as we saw in Problem 1:

After doing the substitution of the work in the energy equation we obtain:

In the problem we have used g = 10 m/s^{2}

**Do not forget to convert all givens to the International System of Units (SI) and to include the units in the results of the problems.**