Rotational Motion – JEE Mains Physics

• by Zenith LMS
• Updated April 2025

1. Centre of Mass

• Centre of Mass (COM) is the point where the entire mass of a system appears to be concentrated.
• For a two-particle system:

  xcom = (m₁x₁ + m₂x₂) / (m₁ + m₂)

• For a rigid body, COM depends on the shape and mass distribution.

2. Basic Concepts of Rotational Motion

• Rotational motion involves a body rotating about a fixed axis.
• Angular Displacement (θ), Angular Velocity (ω), and Angular Acceleration (α) describe rotational motion.

3. Moment of a Force (Torque)

• Torque (τ) is the rotational analog of force.
• Torque is given by:

  τ = r × F

  where r = position vector and F = force vector.

4. Angular Momentum

• Angular momentum (L) for a particle is given by:

  L = r × p

  where p = linear momentum.
• For a rotating rigid body:

  L = Iω

  where I = moment of inertia and ω = angular velocity.

5. Conservation of Angular Momentum

• If no external torque acts on a system, its angular momentum remains conserved.
• Applications:
  • Ice skater pulling arms inward to spin faster.
  • Neutron stars spinning rapidly after collapsing.

6. Moment of Inertia

• Moment of Inertia (I) is the rotational analog of mass in linear motion.
• Depends on the mass distribution about the axis of rotation:

  I = Σ mᵢrᵢ²

7. Radius of Gyration

• Radius of gyration (k) is defined as:

  k = √(I/M)

  where I = moment of inertia and M = total mass.

8. Values of Moments of Inertia for Simple Objects

• Thin Rod (about center) =

  I = (1/12)ML²

• Ring (about center) =

  I = MR²

• Solid Sphere (about diameter) =

  I = (2/5)MR²

• Solid Cylinder (about axis) =

  I = (1/2)MR²

9. Parallel and Perpendicular Axes Theorems

• Parallel Axis Theorem:

  I = Icm + Md²

  where d = distance between axes.
• Perpendicular Axis Theorem (for planar bodies):

  Iz = Ix + Iy

10. Equilibrium of Rigid Bodies

• For a body to be in mechanical equilibrium:
  • Net external force = 0 (translational equilibrium).
  • Net external torque = 0 (rotational equilibrium).

11. Rigid Body Rotation and Equations of Rotational Motion

• Rotational analogs of linear equations:

  ω = ω₀ + αt

  θ = ω₀t + (1/2)αt²

  ω² = ω₀² + 2αθ

12. Comparison of Linear and Rotational Motions

• Linear Motion ↔ Rotational Motion
• Displacement (x) ↔ Angular displacement (θ)
• Velocity (v) ↔ Angular velocity (ω)
• Acceleration (a) ↔ Angular acceleration (α)
• Mass (m) ↔ Moment of inertia (I)
• Force (F) ↔ Torque (τ)