question_answer

The radius of gyration of a rod of length L and mass At about an axis perpendicular to its length and passing through a point at a distance L/3 from one if its ends is

A)  \[\frac{\sqrt{7}}{6}L\]                  

B)  \[\frac{{{L}^{2}}}{9}\]

C)  \[\frac{L}{3}\]                                  

D)  \[\frac{\sqrt{5}}{3}L\]

Correct Answer: C

Solution :

                Moment of inertial of the rod about a perpendicular axis PO passing through centre of mass C. Let N be the point which divides the length of rod AB in ratio \[1:3\]. This point will be at a distance \[\frac{L}{6}\] from C. Thus, the moment of inertia \[I\] about ah axis parallel to PQ and passing through the point N. \[I={{I}_{CM}}=M{{\left( \frac{L}{6} \right)}^{2}}\]                    \[=\frac{M{{L}^{2}}}{12}+\frac{M{{L}^{2}}}{36}=\frac{M{{L}^{2}}}{9}\] If K be the radius of gyration, then \[K=\sqrt{\frac{I}{M}}=\sqrt{\frac{{{L}^{2}}}{9}}=\frac{L}{3}\]