Define radius of gyration and give the physical significance of moment of inertia.
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The radius of gyration of a body about the axis of rotation of a body is the point at which the weighed mass of the body acts. It is also equal to the square root of moment of inertia of all particles of the body about the axis of rotation divided by the total mass of the body i.e., $\text{k}=\sqrt{\frac{\text{m}_1\text{r}^2_1+\text{m}_2\text{r}^2_3+\text{m}_3\text{r}^2_3+\dots+\text{m}_{\text{n}\text{r}^2_\text{n}}}{\text{m}_1+\text{m}_2+\text{m}_3+\dots\text{m}_{\text{n}}}}$ $=\sqrt{\frac{\Sigma\text{mR}^2}{\text{M}}=\sqrt{\frac{\text{I}}{\text{M}}}}$ Its dimensions are those of length and its is measured in metre in SI units. The moment of a body is a quantity which comes in rotational motion and plays same role in rotational motion as does mass in translational motion. Thus a body continues of rotate or be at rest in the absence of any external torque. This is similar to the law of inertia in translational motion. This aspect is used in over creasing the dead points in the engines and crankshafts. Similarly, the kinetic energy of rotation is dependent on the moment of inertia of the body in the same manner the kinetic energy of translation of motion depends, on the mass of the body. For a given angular velocity $(\omega)$ kinetic energy of rotation $\propto\text{I.}$ If equal torques are applied $I_1$ and $I_2$ the their angular acceleration are inversely proportional to the moments of inertia of the bodies. $\frac{\text{I}_1\alpha_1}{\text{I}_1\alpha_2}=1$ $\text{or }\frac{\alpha_1}{\alpha_2}=\frac{\text{I}_2}{\text{I}_1}$ $\text{or }\alpha\propto\frac{1}{\text{I}}$ Similarly if two bodies have same angular acceleration, then the moments of inertia are directly proportional to the torque applied on then. $\frac{\tau}{\tau}=\frac{\text{I}_1}{\text{I}_2}\text{ or }\alpha_1=\alpha_2$ The linear momentum of a body depends on its mass and velocity. If two bodies have same velocity, then their moments are proportional to their masses i.e., $\frac{\rho_1}{\rho_2}=\frac{\text{m}_1}{\text{m}_2}$ Similarly for angular momentum $\frac{\text{L}_1}{\text{L}_2}=\frac{\text{I}_1}{\text{I}_2}$ Thus, moment of inertia of a body plays same role in the rotational motion as does mass in translational motion. The moment of inertia determines the amount of torque to be applied to produce desired angular acceleration.
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