Mechanics: Chapter 12 – Conservative Forces

A force is conservative if its curl is a zero vector. In other words, a force F is conservative if

  • \bigtriangledown \times \overrightarrow{F} = \overrightarrow{0}

Two common conservative forces are the gravitational force and the spring force. In vector form, the gravitational force is

  • \overrightarrow{F} = \frac{GMm}{|r|^2}|\hat{r}|
  • \Rightarrow \overrightarrow{F} = \frac{GMm}{x^2 + y^2 + z^2} \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}
  • \Rightarrow \overrightarrow{F} = \frac{GMm}{x^2 + y^2 + z^2} \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}
  • \Rightarrow \overrightarrow{F} = (GMm)\frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2+y^2+z^2)^{\frac{3}{2}}}

Taking the curl of this, one gets

  • \bigtriangledown \times \overrightarrow{F} = (GMm)\frac{2yz\hat{i} -2yz\hat{i}}{(x^2+y^2+z^2)^{\frac{5}{2}}} + (GMm)\frac{2xz\hat{j} -2xz\hat{j}}{(x^2+y^2+z^2)^{\frac{5}{2}}} + (GMm)\frac{2xy\hat{k} -2xy\hat{k}}{(x^2+y^2+z^2)^{\frac{5}{2}}} = \overrightarrow{0}

Similarly, in vector form the spring force can be written as

  • \overrightarrow{F} = -k \overrightarrow{s} = -k (x\hat{i} + y\hat{j} + z\hat{k})
  • \bigtriangledown \times \overrightarrow{F} = -k (0\hat{i} + 0\hat{j} + 0\hat{k}) = \overrightarrow{0}

Another (equivalent) way of thinking about a conservative force is that if a force does zero net work when a particle does a round trip (i.e its final position is the same as its initial position). Forces like friction are not conservative the net work done on a round trio by friction is not zero (and hence its curl is not a zero vector. These conditions are equivalent).

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