Question 9.SP.9: Determine the moment of inertia of a slender rod of length L......
Determine the moment of inertia of a slender rod of length L and mass m with respect to an axis that is perpendicular to the rod and passes through one end.
STRATEGY: Approximating the rod as a one-dimensional body enables you to solve the problem by a single integration.
MODELING and ANALYSIS: Choose the differential element of mass shown in Fig. 1 and express it as a mass per unit length.
d m=\frac{m}{L} d x
I_y=\int x^2 d m=\int_0^L x^2 \frac{m}{L} d x=\left[\frac{m}{L} \frac{x^3}{3}\right]_0^L \quad I_y=\frac{1}{3} m L^2
REFLECT and THINK: This problem could also have been solved by starting with the moment of inertia for a slender rod with respect to its centroid, as given in Fig. 9.28, and using the parallel-axis theorem to obtain the moment of inertia with respect to an end of the rod.
