Question

# Four point masses $$1$$ kg, $$2$$ kg, $$3$$ kg and $$4$$ kg are located at the corners A, B, C and D respectively of a square ABCD of side $$1$$ m. Find the moment of inertia and radius of gyration of the system about AB as the axis of rotation.

Solution

## axis of rotation is $$AB$$It is a discrete mass system. We should Individually calculate Moment of Inertia$$r=$$ Distance of particle from axis of rotationfor $$1\ kg$$ particle $$\Rightarrow I_A=1\times r^2=0$$ ( as $$r$$  for $$1\ kg$$ is $$0$$ as it lies on axis)for $$2\ kg$$ particle $$\Rightarrow I_B=2\times r^2=0$$for $$4\ kg$$ $$\Rightarrow I_D=4\times 1^2=4\ kg\ m^2$$for $$2\ kg$$  $$\Rightarrow I_C=3\times 1^2=3\ kg\ m^2$$So total moment of Inertia $$=0+0+4+3$$                                           $$I=7\ kg\ m^2$$Radius of gyration $$(K)=\sqrt{\dfrac{I}{Total\ mass}}$$                                $$K=\sqrt{\dfrac{7}{10}}$$                                $$K=\sqrt{\dfrac{7}{10}}\ m$$.Physics

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