As in the preceding example, we haveF<0 whenxis positive,
because the object is being attracted back towardx= 0.
Whenx is negative, the relationship betweenrandxbecomes
r=−x, and the result for the force is the same as before, but with
a minus sign. We can combine the two equations by writing|F|=Gm 1 m 2 /r^2 ,and this is the form traditionally known as Newton’s law of grav-
ity. As in the preceding example, theUandFequations contain
equivalent information, and neither is more fundamental than the
other.
Equilibrium example 42
I previously described the condition for equilibrium as a local max-
imum or minimum ofU. A differentiable function has a zero deriva-
tive at its extrema, and we can now relate this directly to force:
zero force acts on an object when it is at equilibrium.174 Chapter 3 Conservation of Momentum