10.11
Teorema del eje paralelo
El teorema del eje paralelo proporciona un método conveniente y rápido para encontrar el momento de inercia de un objeto alrededor de un eje paralelo…
Podría haber varios ejes posibles a lo largo de los cuales un cuerpo rígido puede rotar y, por lo tanto, en consecuencia, podría haber varios momentos de inercia para el mismo cuerpo.
Si se conoce el momento de inercia, ICM , alrededor de un eje que pasa por el centro de masa, entonces el momento de inercia alrededor de cualquier otro eje paralelo se puede obtener usando el teorema del eje paralelo.
El teorema establece que el momento de inercia a lo largo de cualquier eje paralelo al eje que pasa por el centro de masa se da como la suma de ICM y el producto de la masa del cuerpo y el cuadrado de la distancia perpendicular entre los dos ejes.
Considere una puerta de masa M y altura 2L. El ancho de la puerta es la mitad de la altura de la puerta. La puerta gira sobre sus bisagras.
El CM I de la puerta es igual a ML2 por doce. Por lo tanto, el momento de inercia a lo largo del eje de rotación se da como la suma de ICM y ML2 por cuatro.
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Q1: What is the parallel-axis theorem and why is it useful?
The parallel-axis theorem states that the moment of inertia about any axis parallel to an axis through the center of mass equals the moment of inertia through the center of mass plus the product of mass and the square of the perpendicular distance between axes. This theorem provides a convenient method for calculating moments of inertia without lengthy integration, making it especially useful when the center-of-mass moment of inertia is already known.
Q2: How does the parallel-axis theorem apply to a rotating door?
For a door of mass M and height 2L rotating about its hinges, the moment of inertia is calculated using the parallel-axis theorem. Given that the center-of-mass moment of inertia is ML²/12, the total moment of inertia about the hinge axis equals ICM plus ML²/4, which accounts for the perpendicular distance between the hinge axis and the center of mass.
Q3: Why is the parallel-axis theorem more efficient than conventional integration methods?
The conventional method for finding moment of inertia requires linear mass density concepts and integration along an object's length, which is cumbersome and lengthy. The parallel-axis theorem bypasses this process by using a known center-of-mass moment of inertia and applying a simple distance correction, yielding identical results with significantly less calculation.
Q4: How does the perpendicular distance between axes affect moment of inertia calculations?
The perpendicular distance between the rotation axis and the center-of-mass axis is squared and multiplied by the body's mass in the parallel-axis theorem formula. This squared distance term means that even small increases in separation significantly increase the moment of inertia, reflecting how rotating objects farther from their center of mass requires more rotational resistance.
Q5: Can the parallel-axis theorem be applied to a thin rod rotating about its end?
Yes, the parallel-axis theorem simplifies finding the moment of inertia of a thin rod about an axis through its end. If the moment of inertia about the center of mass is known, the end-axis moment of inertia equals the center-of-mass value plus the product of mass and the square of half the rod's length, avoiding lengthy conventional calculations.
Q6: What is the mathematical formula for the parallel-axis theorem?
The parallel-axis theorem formula is I = ICM + Md², where I is the moment of inertia about the parallel axis, ICM is the moment of inertia through the center of mass, M is the body's mass, and d is the perpendicular distance between the two parallel axes. This relationship holds for any rigid body rotating about axes parallel to the center-of-mass axis.
Q7: How does the parallel-axis theorem relate to rotational motion analysis?
The parallel-axis theorem is fundamental to analyzing rotational motion because it allows quick calculation of moment of inertia for any parallel axis once the center-of-mass value is known. Understanding moment of inertia through this theorem is essential for studying moment of inertia and rotational kinetic energy in rigid body dynamics.