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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 rotation
for $$1\ kg$$ 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 $$(K)=\sqrt{\dfrac{I}{Total\ mass}}$$
                                $$K=\sqrt{\dfrac{7}{10}}$$
                                $$K=\sqrt{\dfrac{7}{10}}\ m$$.

1432198_1026097_ans_2ed2fb33adb24e0c8f5eb8ff96a85832.png

Physics

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