10.11
平行轴定理
平行轴定理提供了一种方便快捷的方法来计算物体绕与通过其质心的轴平行的轴的惯性矩。 以细棒为例。 使用传统方法求出一根细杆绕着穿过其中部（质心所在）的轴的惯性矩和绕着穿过其末端的轴的惯性矩的过程有着惊人的相似之处。 在传统方法中，利用线性质量密度和沿杆长度积分的概念。 假设要确定该细杆绕一端旋转的惯性…
刚体可能绕多个不同的轴旋转，因此同一个物体可能对应多个不同的转动惯量。
如果转动惯量， 我CM , 若已知物体绕通过其质心的轴的转动惯量，则可利用平行轴定理求得物体绕任何其他平行轴的转动惯量。
该定理指出，对于任意一条与通过质心的轴平行的轴，其转动惯量等于质心处的转动惯量ICM加上物体质量与两轴之间垂直距离平方的乘积。
考虑一扇质量为 M 、高度为 2L 的门。该门的宽度为其高度的一半。门绕其铰链旋转。
门的质心转动惯量 ICM 等于 ML2 除以 12。因此，绕转动轴的转动惯量等于 ICM 与 ML2 除以 4 之和。
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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.