In my lecture today my professor briefly mentioned that force is the derivative of energy but I did not really get what he meant by that. I tried to express it mathematically:
$$\frac{d}{dt}K_E=\frac{d}{dt}\left(\frac{1}{2}mv^2\right)=mv\frac{dv}{dt}$$
This looks really close to Newton's second law $F=ma$ but there is an extra "$v$" in there. Am I missing something here?
Answer
It is important to understand to which derivative you are referring to, i.e. derivative with respect to what?.
For conservative systems, it is true that the force can be expressed as minus the gradient of the potential energy: $$ \tag{1} \textbf F(\textbf x) = -\nabla V( \textbf x),$$ which can be though of as the defining property of a conservative system.
The gradient $\nabla$ reduces for one-dimensional systems to the derivative with respect to the space coordinate, i.e. you have in this simple case $$ \tag{2} F = -\frac{dV}{dx}.$$
Taking as an example the case of a mass $m$ in the gravitational field of the earth, you have the potential energy $$ \tag{3} V(z) = mgz, $$ where $z$ is the distance from the ground. The force in the $z$ direction is then given by $$ F_g = - \frac{dV(z)}{dz} = -mg,$$ which is what you would expect.
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