Physical significance of integral of F cross dr

[arpon]

In the vector calculus course, I calculated integrals like,
##\int \vec F \times \vec{dr} ##
Does this kind of integrals have physical significance or practical application other than Biot-Savart's Law?

 

[Andrew Mason]

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]

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

Isn't torque defined as ##\vec r \times \vec F## ?

 

[PeroK]

Isn't torque defined as ##\vec r \times \vec F## ?

##\vec r \times \vec F = - \vec F \times \vec r##

 

[Andrew Mason]

Yes. That is just a convention. The difference is the sign or direction of the torque vector.

AM

 

[arpon]

##\vec r \times \vec F = - \vec F \times \vec r##

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]

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## ?

The line integral doesn't represent the torque on a body. The total torque on a body would be a volume integral:

##\vec{\tau} = -\int \vec F \times \vec r dV##

I'm not sure when you would use the line integral.

 

[Andrew Mason]

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.

AM