|Aug17-12, 04:41 PM||#18|
Is angular momentum conserved here?
And more: Why is it only instantaneous? There is a torque around that point all the time? Do you mean that because the stick moves all the time the rotation around it is only instantaneous? But then is the rotation around the center of mass not also instantaneous? Surely that moves too - what distingiushes those two coordinate frames?
Or said in another way:
I don't understand why it is much easier to work with the center of mass. I don't understand why this point is so special. I understand it translates as though only acted on by external forces but it seems that all points on the stick do this in this situation.
Sorry I ask so much, but I actually feel that I get a little closer to a complete understanding every time. I will consider your answer to this reply final for now and try to go back and get a better understanding by thinking everything over. Thanks so much for your help for now.
|Aug17-12, 05:03 PM||#19|
if the body is rotating, then the only point that isn't accelerating is the centre of mass
(and the only point that isn't moving is the centre of rotation*)
if the body is moving freely, there is no torque about any point
* this is in 2D … for a 3D body, there will be an instantaneous axis with zero velocity
|Aug18-12, 06:39 AM||#20|
Okay ill try to explain my problem more pictorially. Why is the stick moving as in scenario 2 and not as in scenario 1 on the attached picture?
|Aug18-12, 05:22 PM||#21|
Graduate mechanics courses present the mathematics of rigid bodies in several representations. Although the physics is “classical”, some of the mathematics seems anti-intuitive at first sight because the descriptions aren’t unique.
One learns in introductory physics to choose the leverage point that is most “convenient” for the calculations. However, “convenient” isn’t the same as “unique”. Very often the center of mass is used as a leverage point because it is most “convenient”. Again, convenient isn’t unique.
Both the leverage point and the axes of rotation are arbitrary. It is possible that your left cerebral hemisphere has chosen one axis and your right cerebral hemisphere has chosen another axis of rotation. Both hemispheres are correct. However, solving a physics involves working with one axis consistently.
“Screw theory refers to the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies.
The conceptual framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics).
The value of screw theory derives from the central role that the geometry of lines plays in three dimensional mechanics, where lines form the screw axes of spatial movement and the lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and addition of vectors. A remarkable result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle.
A remarkable result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle.
Screw theory notes that all rigid-body motion can be represented as rotation about an axis along with translation along the same axis; this axis need not be coincident with the object or particle undergoing displacement. In this framework, screw theory expresses displacements, velocities, forces, and torques in three dimensional space.”
Sometimes the best way to learn a topic is from a graduate student’s thesis. This thesis covers the moment of inertial in rigid body theory.
“Rigid-Body Inertia and Screw Geometry
This paper reviews the geometric properties of the inertia of rigid bodies in the light
of screw theory. The seventh chapter of Ball’s treatise  defines principal screws of inertia for a general rigid body based on Ball’s co-reciprocal basis of screws. However, the application of that work to the important cases of planar- and spherical-motion is not satisfactory. The following paper proposes a new formulation of the screws of inertia which is more easily applicable, and compares it with common mathematical devices for treating rigid body inertia such as the inertia tensor [6, 9]. This brings to light a geometric perspective of inertia that does not often accompany this topic.”
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