Bihar Board - Class 11 physics - Chapter 7: System of Particles and Rotational Motion Handwritten Notes

The chapter "System of Particles and Rotational Motion" extends the principles of mechanics to systems of particles and rigid bodies. It introduces key concepts such as the center of mass, torque, angular momentum, and rotational dynamics, establishing parallels between linear and rotational motion.

Key Points

1. System of Particles

   • Center of Mass (COM):
     • The point at which the total mass of a system can be considered to be concentrated.
     • For a system of particles: $$\text{COM Position: } \vec{R} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$$
     • For a rigid body, COM depends on its shape and mass distribution.

2. Torque

   • Torque ($\tau$) is the rotational equivalent of force.
   • Formula: $$\tau =\vec{r}\times \vec{F}=rF\sin \theta$$
   • Determines the ability of a force to cause rotational motion.

3. Angular Momentum

   • Rotational analogue of linear momentum.
   • Defined as: $$\vec{L}=\vec{r}\times \vec{p}=I\omega$$ where $I$ = moment of inertia, $\omega$ = angular velocity.

4. Moment of Inertia (MI)

   • The rotational analogue of mass, it measures resistance to changes in rotational motion.
   • Depends on the mass distribution and the axis of rotation: $$I = \sum m_i r_i^2$$
   • Common examples:
     • Rod about the center: $I = \frac{1}{12}ML^2$.
     • Ring about its axis: $I = MR^2$.

5. Rotational Motion Dynamics

   • Newton’s Second Law for rotation: $$\tau = I\alpha$$ where $\alpha$ = angular acceleration.
   • Analogous to $F = ma$ in linear motion.

6. Kinematics of Rotational Motion

   • Rotational equations of motion (for constant angular acceleration): $$\omega ={\omega }_0+\alpha t$$$$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$$$${\omega }^2={\omega }_0^2+2\alpha \theta$$

7. Work, Energy, and Power in Rotation

   • Rotational kinetic energy: $$KE=\frac{1}{2}I{\omega }^2$$
   • Power in rotational motion: $$P=\tau \omega$$

8. Rolling Motion

   • Combination of rotational and translational motion.
   • Total kinetic energy of a rolling body: $$KE_{\text{total}} = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$$

Conclusion

The "System of Particles and Rotational Motion" chapter provides a comprehensive understanding of rotational dynamics and the behavior of rigid bodies. It bridges linear and rotational motion concepts, forming the foundation for advanced topics in mechanics and engineering.

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