The force vs. position diagram acting on a 4.0-kg mass is shown. The object has

Question

The force vs. position diagram acting on a 4.0-kg mass is shown. The object has a
velocity of +3.0 m/s as it passes the origin.

i) Why is this a conservative force?
ii) What is the objects kinetic energy and speed at the x= 5m and x=10m.
iii) If we set the potential energy to be U(x=0) = -10 J at the origin, plot the potential energy as a function of the position.
iv) If instead, and object started at the 10-m mark, what would be the minimum speed (velocity in the negative direction) such that the object could reach the origin? For this case, what would be the object

Explanation / Answer

i) It can be seen that it can be derived as gradient of a scalar function known as potential energy. Indeed, from x =0 to x = 3 m, it comes from U = -k x^2, k>=0, as F = - d U/ dx = k x: it grews linearly. In the second step, the potential energy changes as U = -h x, h>0, so that F = -dU/dx = h = const. The third step is as the first one, but with the constant negative. The fourth step is as the second one, but with the constant negative as well. The fifth step is as the first one, but with the constant positive.

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