Rotational Equilibrium and Dynamics

[Greg Bernhardt]

Author: Dr. Donald Luttermoser of East Tennessee State University

 

[arildno]

Lutermoser does a grievous error here, in conflating centre of gravity with centre of mass.

Although he has the correct definition of centre of gravity (i.e, as the point where we might consider the weight concentrated (if such a point exists)), he sets it equivalent to the centre of mass, which is totally differently defined! :grumpy:

The centre of gravity is just where the force of gravity can be considered to ACT.
Where a force acts, is of course, mainly of importance when computing torques, and for a 2-D situation, in which direction vectors and forces are coplanar, net torque $\tau$ and net force $\vec{F}=(F_{x},F_{y})$ we define the centre of action $\vec{r}_{c.a}=(x_{c.a},y_{c.a})$ to be the that point with least magnitude that satisfies:
$$x_{c.a}F_{y}-y_{c.a}F_{x}=\tau$$
(The origin being the point we compute the torque with respect to, say C.M)

This yields:
$$\vec{r}_{c.a}=\frac{\tau}{F_{x}^{2}+F_{y}^{2}}(F_{y},-F_{x})$$

For a uniform gravitational field, C.M and C.G. (that is, c.a.) coincides, but not necessarily with a varying field.


Finally, in the 3-D case, we must assume that the net force&torque are perpendicular vectors in order to be able to define a common centre of action.
If that is the case, we have, as above:
$$\vec{r}_{c.a}=\frac{\vec{F}\times\vec{\tau}}{||\vec{F}||^{2}}$$

I'd like to close with saying that I don't regard concepts like "centre of gravity" to be particularly useful, in that the positin of C.G. depends on such factors as the orientation of the object and which point we happen to compute the torque with respect to.

In engineering, particularly in STATICS, the concept of "centre of pressure" has proven useful, though.

 

[Astronuc]

Hyperphysics has a good discussion on moments of inertia.

Rotational-Linear Parallels
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

http://hyperphysics.phy-astr.gsu.edu/hbase/mi2.html


And these might be useful

Area Moment of inertia
http://em-ntserver.unl.edu/NEGAHBAN/EM223/note18/note18.htm

Mass moment of inertia
http://em-ntserver.unl.edu/NEGAHBAN/EM223/note19/note19.htm

Proofs of moment of inertia equations
http://homepages.which.net/~paul.hills/Spinningdisks/MOI/MOIproofs.html