Chapter 10, Problem 78P

Figure 10.31 shows an object of mass M with one axis through its center of mass and a parallel axis through an arbitrary point A.

Both axes are perpendicular to the page. The figure shows an arbitrary mass element dm and vectors connecting the center of mass, the point A, and dm. (a) Use the law of cosines (Appendix A) to show that $r^2=r_{cm}^2+h^2-2\stackrel{\to }{h}\cdot {\stackrel{\to }{r}}_{cm}$. (b) Use this result in I = ∫r² dm to calculate the object’s rotational inertia about the axis through A. Each term in your expression for r² leads to a separate integral. Identify one as the rotational inertia about the CM, an-other as the quantity Mh², and argue that the third is zero. Your result is a statement of the parallel-axis theorem (Equation 10.17).

FIGURE 10.31 Problem 78