 # Moment of Inertia

• Moment of inertia is the inability of a body to change its state of rotation.
• Moment of inertia is the rotational inertia.
• Moment of inertia is a scalar quantity.
• unit of moment of inertia is kg m2.
• Moment of inertia of a particle of mass ‘m’ at a distance ‘s’ is I = mr2.
• Moment of inertia of a group of system of particles \tt I=m_{1}r_1^2+m_{2}r_2^2+...m_{n}\ r_{n}^{2}
• For system of particles \tt I=\Sigma\ mr^{2}.
• For a rigid body I = MK2.
• Radium of gyration \tt K=\sqrt{\frac{r_1^2+r_2^2...r_n^2}{n}}.
• Moment of inertia of a body depends on mass of the body
• Moment of inertia of a body depends on distribution of mass of the body
• Moment of inertia of a body depends on position of axis of rotation.
• Moment of inertia is independent of angular velocity of rotation of the body
• Radius of gyration of a body depends on distribution of mass and position of axis of rotation.
• Radian of generation is independent of mass of rotating body.
• Parallel axes theorem I = Ic + Md2 d = perpendicular distance of parallel axes.
• Parallel axes theorem \tt K=k_1^2+d^{2}
• Perpendicular axes theorem IZ = Ix + Iy
• Perpendicular axes there \tt KZ^{2}=k_x^2+k_y^{2}
• Moment of inertia of circular ring axis passing
• Through centre and normal to its plane I = MR2.
• Work done by torque w = Tθ
• Work energy principle \tt \omega=\frac{1}{2}\ I\ \left(\omega_2^2-\omega_1^2\right)
• Power of a torque \tt P=\frac{d\omega}{dt}=\tau\frac{d\theta}{dt}=\tau\omega.
• Velocity at lower point \tt \overline{V}_{p}=\overline{V}_{pc}+\overline{V}_{c}
• In pure rolling Vc = Vpc = V ⇒ Vp = 0
\tt V_{p}=\sqrt{Vpc^{2}+Vc^{2}+2Vpc\ Vc \cos\theta}
• Kinetic Energy translation of pure rolling =\tt \frac{1}{2}\ Mv^{2}
• Kinetic Energy Rotational of pure rolling = \tt \frac{1}{2}\ I\omega^{2}
• Kinetic Energy Rotational of pure rolling = \tt \frac{1}{2}MK^{2}\ \frac{V^{2}}{R^{2}}=\frac{1}{2}\ MV^{2}\left(\frac{K^{2}}{R^{2}}\right)
• Kinetic Energy total = KEtrans + KErot
• Kinetic Energy trans = \tt \frac{1}{2}MV^{2}+\frac{1}{2}\ MV^{2}\left(\frac{K^{2}}{R^{2}}\right)
• Kinetic Energy trans = \tt \frac{1}{2}MV^{2}\left(1+\frac{K^{2}}{R^{2}}\right).
• Fraction of translational \tt KE=\frac{1}{1+\frac{K^{2}}{R^{2}}}=\frac{R^{2}}{R^{2}+K^{2}}.
• Fraction of rotational \tt KE=\frac{1}{1+\frac{R^{2}}{K^{2}}}=\frac{K^{2}}{R^{2}+K^{2}}.
• In the case of rolling bodies KE translation ≥ KE rotaion.
• In the case of rolling circular ring or rolling hollow cylinder KE trans = KE rotation.
• Rolling body Acceleration down an incline \tt a=\frac{Mg\sin \theta-f}{M}.
• Rolling body Acceleration down an incline \tt a=\frac{g\sin \theta}{1+I/\ MR^{2}}.
• Coefficient of friction \tt \mu\ \geq\frac{\tan\theta}{1+\frac{MR^{2}}{I}}
• Velocity acquired by rolling body on incline \tt V=\sqrt{\frac{2gh}{1+\frac{K^{2}}{r^{2}}}}.
• Time taken by the body rolling down an incline \tt t=\frac{1}{\sin \theta}\ \sqrt{\frac{2h}{g}\left(1+\frac{K^{2}}{r^{2}}\right)}
• Frictional force = \tt \frac{mg \sin\theta}{1+\frac{r^{2}}{k^{2}}}

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