## System of Particles and Rotational Motion

# 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 m
^{2}. - Moment of inertia of a particle of mass ‘m’ at a distance ‘s’ is I = mr
^{2}. - 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 = MK
^{2}. - 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 + Md
^{2}d = perpendicular distance of parallel axes. - Parallel axes theorem \tt K=k_1^2+d^{2}
- Perpendicular axes theorem I
_{Z}= I_{x}+ I_{y } - 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 = MR
^{2}. - 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 = KE
_{trans}+ KE_{rot } - 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}}}

### View the Topic in this video From 02:38 To 54:19

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