JEE Physics Notes: System of Particles and Rotational Motion

Introduction

The study of the system of particles and rotational motion is essential in understanding the motion of rigid bodies. Unlike linear motion, rotational motion involves movement about a fixed axis, which introduces new physical quantities such as torque, moment of inertia, and angular momentum. This chapter is crucial for solving advanced mechanics problems in JEE Physics.

1. Center of Mass

• Definition: The point where the entire mass of a system is considered to be concentrated for motion analysis.
• Formula: For a system of particles, the center of mass is given by:
  • Xcm = (m₁x₁ + m₂x₂ + ... + mₙxₙ) / (m₁ + m₂ + ... + mₙ)
  • Ycm = (m₁y₁ + m₂y₂ + ... + mₙyₙ) / (m₁ + m₂ + ... + mₙ)
• Properties:
  • The center of mass moves as if all external forces act on it.
  • For symmetric objects, the center of mass lies at the geometrical center.

2. Motion of Center of Mass

• Newton’s Second Law for a System of Particles: Fext = M acm, where M is the total mass and a_cm is the acceleration of the center of mass.
• Linear Momentum: Given by P = M vcm, where v_cm is the velocity of the center of mass.

3. Torque and Angular Momentum

• Torque (Moment of Force): The rotational equivalent of force, given by τ = r × F.
• Angular Momentum: Defined as L = r × p or L = Iω, where I is the moment of inertia and ω is angular velocity.
• Conservation of Angular Momentum: If no external torque acts on a system, then L = constant.

4. Moment of Inertia

• Definition: The resistance of a body to changes in its rotational motion.
• Formula: I = Σ mᵢrᵢ², where mᵢ is mass and rᵢ is the distance from the axis.
• Parallel Axis Theorem: I = Icm + Md².
• Perpendicular Axis Theorem: Iz = Ix + Iy (for planar bodies).

5. Rotational Kinetic Energy

• Formula: KErot = (1/2) Iω².
• Comparison with Translational Kinetic Energy: KEtrans = (1/2) mv².

6. Rolling Motion

• Rolling motion is a combination of translation and rotation.
• Condition for Pure Rolling: v = Rω.
• Energy in Rolling Motion: KEtotal = (1/2) Iω² + (1/2) mv².

7. Equilibrium of Rigid Bodies

• Types of Equilibrium:
  • Stable Equilibrium
  • Unstable Equilibrium
  • Neutral Equilibrium
• Conditions for Equilibrium:
  • Net force must be zero: ΣF = 0.
  • Net torque must be zero: Στ = 0.

Conclusion

The system of particles and rotational motion provides a fundamental understanding of how rigid bodies move and rotate. Mastering concepts like torque, angular momentum, and moment of inertia is crucial for solving complex problems in JEE Physics.