Question

8 mark for review the net force on an object as a function of the position of the object along the x - axis is shown. how many points of stable equilibrium exist for this object along the x - axis? a 1 b 2 c 3

Explanation:

Step1: Recall the condition for stable equilibrium

At stable equilibrium, the force \(F = 0\) (since \(F=-\frac{dU}{dr}\), and at equilibrium \(\frac{dU}{dr} = 0\)), and if the object is displaced slightly, the force acts to restore it to the equilibrium position. Mathematically, this means \(\frac{dF}{dr}>0\) (because \(F =-\frac{dU}{dr}\), so \(\frac{d^{2}U}{dr^{2}}>0\) for stable equilibrium, and using the chain - rule \(\frac{d^{2}U}{dr^{2}}=-\frac{dF}{dr}\)).

Step2: Analyze the graph

We look for the points where \(F = 0\) (the curve crosses the \(r\) - axis). Then, we check the slope of the \(F - r\) curve at those points.

• When \(F = 0\), if the slope of the \(F - r\) curve (\(\frac{dF}{dr}\)) is positive, the equilibrium is stable.
• Count the number of such points.

Answer:

C. 3