Question 14.10: Moment of inertia measurement. An easy way to measure the mo......
Moment of inertia measurement. An easy way to measure the moment of inertia of an object about any axis is to measure the period of oscillation about that axis. (a) Suppose a nonuniform 1.0-kg stick can be balanced at a point 42 \mathrm{~cm} from one end. If it is pivoted about that end (Fig. 14-17), it oscillates with a period of 1.6 \mathrm{~s}. What is its moment of inertia about this end? (b) What is its moment of inertia about an axis perpendicular to the stick through its center of mass?
APPROACH We put the given values into Eq. 14-14 and solve for I. For (b) we use the parallel-axis theorem (Section 10-7).
(a) Given T=1.6 \mathrm{~s}, and h=0.42 \mathrm{~m}, Eq. 14-14 gives
I=m g h T^{2} / 4 \pi^{2}=0.27 \mathrm{~kg} \cdot \mathrm{m}^{2} .
(b) We use the parallel-axis theorem, Eq. 10-17. The \mathrm{CM} is where the stick balanced, 42 \mathrm{~cm} from the end, so
I_{\mathrm{CM}}=I-m h^{2}=0.27 \mathrm{~kg} \cdot \mathrm{m}^{2}-(1.0 \mathrm{~kg})(0.42 \mathrm{~m})^{2}=0.09 \mathrm{~kg} \cdot \mathrm{m}^{2}
NOTE Since an object does not oscillate about its \mathrm{CM}, we cannot measure I_{\mathrm{CM}} directly, but the parallel-axis theorem provides a convenient method to determine I_{\mathrm{CM}}.