Understand Physics Moment: Get Answer to Concept, Formula & More

[UCC]

I am trying to understand how the concept of moment was introduced in physics. For example I don't understand how the formula of the moment of impulse was created. Specifically I don't understand why the multiplication with R (L=PXR, or torque=FXR). Is it demonstrated by experimentation, or the multiplication with R is just a "trick" so the formula is depenent on angular acceleration (in this case for torque ) or other rotational components ?
I guess the best way to find out is by seeing the demonstration of the physicst who created the concept of moment but I can't find anything about it.
Can somebody help me ?

 

[UCC]

Anyone ?

 

[pgardn]

Anyone ?

Well one way to look at it is from a rotational kinetic energy point of view.

Say if a disc is just rotating, the particles of mass that make up the disc have differing kinetic energy depending on how far they are from the axis of rotation. The further away from the axis of rotation, the faster a particle that makes up the disc is moving. Yet every particle has the same angular speed omega.

So 1/2mv^2 could be turned into 1/2m(w^2)(r^2) since v =w*r...

So now if you look at your equation you can rewrite it as 1/2mr^2(w^2). Now notice you have a little bit of an equation mr^2 in there... What exactly is that? mr^2... Apparently physics types like to think of this little part as the resistance of a body to rotation which is one way moment of inertia can be thought of.

The larger the moment of inertia, the more difficult it is to start or stop a body from rotating.

And as I read your question again I realized I might have answered a question you did not ask. Sorry if this is so.

 

[pgardn]

I am trying to understand how the concept of moment was introduced in physics. For example I don't understand how the formula of the moment of impulse was created. Specifically I don't understand why the multiplication with R (L=PXR, or torque=FXR). Is it demonstrated by experimentation, or the multiplication with R is just a "trick" so the formula is depenent on angular acceleration (in this case for torque ) or other rotational components ?
I guess the best way to find out is by seeing the demonstration of the physicst who created the concept of moment but I can't find anything about it.
Can somebody help me ?

I will give this another try. Torque is very well defined experimentally and mathematically. A see-saw or any other device with fulcrum and masses on either side can demonstrate the concept experimentally and mathematically.

 

[mikelepore]

As for torque, if a crowbar, wrench, nutcracker, etc. has a long handle, you can cause one end to apply a larger force over a small distance by giving the end of the handle a smaller force over a greater distance.

 

[makyol]

Actually, moment is the effect of forces to make things move or revolve; therefore we multiply them with R.

 

[kipper_t]

I have actually been pondering this myself.
I do not know the complete answer and I would like the explanation I am about to give to be verified if possible.
I am a very visual person, so I like to see what the physics is rather than just the equation and I think this might be what you are looking for too.

I did a short experiment with a door, I pushed it near the hinge and at the very edge. I tried to imagine what was happening. I imagined that rotating the door through the entire angular distance used up the same energy ( in a slightly idealised situation ) regardless of whether i pushed it from near the hinge or away from the hinge.

So pushing the door near the hinge requires more force per unit linear distance covered (as opposed to angular distance) than away from the hinge which required less force per unit distance.

Try thinking of this as you have a machine that can output 5N per second. There is a bar with a pivot at the end. To move the bar near the hinge requires 10N, but pushing the bar away from the hinge only requires 2.5N. Clearly, using the machine to push the bar near the hinge would result in the bar not moving. Using the bar away from the hinge would result in the bar moving due to the force output exceeding the required force. (Pro's out there, please verify this if possible to make sure I have it right.)

So is torque the energy required to move a bar through a specific angle, or is Torque the energy required to just move the bar.

I suspect that Torque is the minimum energy required to move for example a bar with a pivot at one end and so given the minimum energy required to make the bar move, this information can be used to tell us what this equates to as a force some distance from the pivot.

I have spent a good several hours researching and deriving this from first principals. I would be happy to share this derivation as I can type it up, however I must warn you in it's current state it involves heavy vectorial notation and the einstein summation convention.

I really hope this has been helpful to you. I always struggle to imagine what equations mean, interpreting physics as something physical and realistic is difficult I think, and sometimes the reality is lost in the equations.

Regards
Kipper

 

[tiny-tim]

Welcome to PF!

Hi UCC! Welcome to PF!

… Specifically I don't understand why the multiplication with R (L=PXR, or torque=FXR). Is it demonstrated by experimentation, or the multiplication with R is just a "trick" so the formula is depenent on angular acceleration (in this case for torque ) or other rotational components ?

From the PF Library on angular momentum …

Fundamental equation of motion (rotational):

"Crossing" Newton's second law, $\bold{F}_{net}\ =\ d\bold{p}/dt\ =\ d/dt \int\int\int\rho\,\bold{v}\ dxdydz$,
with the position vector of each infinitesimal part of a rigid body (and cancelling out the torques of the internal reaction forces between the parts, and since $d\bold{r}/dt\times\bold{v}\ =\ \bold{v}\times\bold{v}\ =\ 0$) gives:

$\bold{\tau}_{net}\ =\ d/dt \int\int\int\rho\,\bold{r}\times\bold{v}\ dxdydz\ =\ d\bold{L}/dt$:

net torque = rate of change of angular momentum ​