The discussion centers around the physical significance and practical applications of the integral of the cross product of a force vector and an infinitesimal displacement vector, specifically in the context of torque and its representation in Newtonian mechanics. Participants explore whether this integral has relevance beyond known applications like Biot-Savart's Law.
Participants do not reach a consensus on the physical significance of the integral ##\int \vec F \times d\vec{r}##, with multiple competing views regarding its application and relevance to torque.
Participants highlight the distinction between ##d \vec r## as an infinitesimal change in position and ##\vec r## as the position vector, indicating a potential misunderstanding of the integral's application in calculating torque.
Isn't torque defined as ##\vec r \times \vec F## ?Andrew Mason said:In Newtonian mechanics the integral represents torque or rate of change of angular momentum.
The cross-product as a mathematical tool was invented probably because of its practical application. Although its development as a mathematical tool was post-Newton, it is very useful in mechancs.
AM
arpon said:Isn't torque defined as ##\vec r \times \vec F## ?
That was not my point. ##d \vec r## represents infinitesimal change in position vector, while ##\vec r## represents position vector. Could you please give me a practical example where the net torque is calculated by ##\int \vec F \times d \vec r## ?PeroK said:##\vec r \times \vec F = - \vec F \times \vec r##
arpon said:That was not my point. ##d \vec r## represents infinitesimal change in position vector, while ##\vec r## represents position vector. Could you please give me a practical example where the net torque is calculated by ##\int \vec F \times d \vec r## ?
I see your point. PeroK is quite right that ##\int \vec F \times d \vec r## does not represent torque. I am not sure what it would represent. I also don't see how it applies even to the Biot-Savart Law.arpon said:That was not my point. ##d \vec r## represents infinitesimal change in position vector, while ##\vec r## represents position vector. Could you please give me a practical example where the net torque is calculated by ##\int \vec F \times d \vec r## ?