Solving Torque & Moment: Deriving V=PQsintheta

[DiamondV]

I'm confused about point 2. It says V=PQsintheta. How is that formula derived?

 

[DrPapper]

Hello DiamondV

If you've taken multi-variable calculus you may recall something known as the Vector Cross Product. The magnitude of V can be thought of as how perpendicular the two vectors that produce it are to each other. So if you had P and Q directly on top of each other the "magnitude" of their "perpendicular-ness" would be zero. But if they are completely perpendicular then $$Sin \theta = 1$$ when $$\theta = \pi / 2$$ so it's the regular multiplication of P and Q. A mathematician could give you a more proper answer I'm sure. Check your Calculus textbook under the section of vector cross product.

Hope that helps.

 

[IgorIGP]

Imho, the phrase:

Consept of the moment of a force about a point is more easily understood through applications of the vector product or cross product

is a fantasy humanities.
Moreover, I believe that the one of the ways to understand the cross product is a moment of force concept. That is the moment of force is a good illustration of the vector product that shows its physical sense from the point of view of one of the branches of mechanics.
The world is so constituted that sometimes the things depend on the result of a pair of vector quantities. This is well illustrated by calculating the area of a parallelogram in the course of geometry. In mechanics, the moment of a force is a good example. The power of rotated force is defined by that. Imagine a solid flat body fixed at the point O to rotate in its plane. Suppose there is a pair of forces F1 and F2 applied at the points M1 and M2, respectively. Let the forces F1 and F2 are directed so that they try to turn the body in opposite directions. Let the forces F1 and F2 are perpendicular OM1 and OM2 respectively. Equality of works of these forces in the infinitely small arcs of movement is a condition of equilibrium of the body. The arc of each force is the product of its radius (OM1 and OM2 respectively) an infinitesimal angle. The angles of both are equal but not equal to zero, so they can be cut. This means that the determining factor in the rotation moving is not the force but it is product of of the force and its radius. And that can be a good illustration to a such abstract concept as a vector product.

 

[Chandra Prayaga]

The "formula" V = PQ sin(theta) is not "derived". It is part of the definition of a cross product.

 

[IgorIGP]

The "formula" V = PQ sin(theta) is not "derived". It is part of the definition of a cross product.

I absolutely agree with you. Who can explain me on how to measure the vector of the force moment that "line of action is perpendicular to plane" were the force and its radius (do not know how is named on english this line segment) act? The analogy where the moment of force derives from the cross product is bad. Softly said.
The concepts of mathematics can derive from the concepts of the real world, but not vice versa. And in this case, as the parent concept it is better to choose something from the electrodynamics like Lorentz force ie (imho).