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