Physics Lecture Notes: Rotational Dynamics

Introduction

• Lecturer: सुशांत, Physics Teacher on PhysicsWallah platform
• Addressing fear about upcoming exams with the introduction of PW's "Eklavya 2.0" batch for Maharashtra HSC 12th board
• Main topics covered: Rotational Dynamics

Topics To Be Covered

1. Kinematics and Dynamics of Circular Motion
   • Angular velocity
   • Angular displacement
   • Centripetal & Centrifugal force
2. Uniform Circular Motion (UCM)
   • Applications of UCM (e.g., Death well, Banking of roads)
3. Concept of Rotational Motion
   • Moment of inertia
   • Angular momentum
   • Torque
   • Conservation of angular momentum
4. Rolling Motion
   • Combined translational and rotational motion
   • Kinetic energy in rolling motion

Key Concepts and Definitions

Kinematics and Dynamics of Circular Motion

1. Revolution vs. Rotation

   • Revolution: Object moves around an axis not passing through it (e.g., Earth around the Sun)
   • Rotation: Object moves around an axis passing through it (e.g., Earth's rotation on its axis)

2. Characteristics of Circular Motion

   • Periodic Motion: Object repeats its path after equal intervals of time (e.g., Earth's orbit around the Sun)
   • Accelerated Motion: Direction of velocity changes at every point (velocity is tangential at all points)

3. Key Terms

   • Radius Vector: Vector from the center of the circular track to the object performing circular motion
   • Angular Displacement (θ): Angle traced by the radius vector at the center
   • Angular Velocity (ω): Rate of change of angular displacement (ω = θ/t)
   • Angular Acceleration (α): Rate of change of angular velocity (α = Δω/Δt)

4. Relation Between Linear and Angular Velocity

   • v = ω * r
   • Linear velocity (v) is perpendicular to radius vector (r)

5. Uniform Circular Motion (UCM)

   • Object performs circular motion with constant speed

Circular Motion in a Horizontal Track and Death Well

1. Centripetal Force

   • Force acting towards the center of the circular path
   • fc = (mv²)/r = mω²r

2. Centrifugal Force

   • Pseudo force acting away from the center
   • fcf = (mv²)/r = mω²r

3. Maximum Speed on Horizontal Curve

   • vmax = sqrt(μrg)

4. Death Well (मौत का कुआँ)

   • Minimum speed requirement to avoid falling: vmin = sqrt(μrg)

Banking of Roads

• Banking Angle (θ): Angle at which the outer edge of the road is elevated above the inner edge
• Safe speed: vmax = sqrt(rg tanθ)
• Minimum speed consideration with friction: vmin = sqrt((rg (μ - tανθ)) / (1 + μ tanθ))
• Upper speed limit: Same as safe speed with friction assisting rather than opposing

Other Rotational Motion Topics

1. Conical Pendulum

   • Bob describes horizontal circular motion, string describes a cone
   • Time period: T = 2π sqrt(l cosθ/g)

2. Vertical Circular Motion

   • Minimum velocity at the highest point: vmin = sqrt(rg)
   • Application of energy conservation for other points

Concepts of Moment of Inertia (I)

• Moment of Inertia: Analogous to mass in rotational motion
• For discrete particles: I = Σ mᵢ rᵢ²
• Radius of Gyration (k): I = Mk², where k is the distance from rotation axis where the entire mass could be concentrated to give the same I

Theorems of Moment of Inertia

1. Parallel Axis Theorem: Iₐ = I_c + Mh²

   • Applicable to any body

2. Perpendicular Axis Theorem: I_z = I_x + I_y

   • Applies to planar lamina only

Angular Momentum (L)

• L = Iω (analogous to linear momentum p = mv)
• Conservation of Angular Momentum: L = constant if no external torque acts

Torque (τ)

• τ = r × F
• τ = Iα (using moment of inertia)

Rolling Motion

• Combination of rotational and translational motion
• Total kinetic energy: KE_total = KE_translational + KE_rotational
• KE_total = 1/2 mv² + 1/2 Iω² Using I = mkr², KE_total = 1/2 mv² * (1 + k²/r²)

Important Tables

References for various objects and moments of inertia (Example: Ring, Hollow Cylinder, Thin Ring, Hollow Sphere, Solid Sphere, Rod)

Conclusion

• Solve questions based on these concepts
• Watch previous year's board marathon for comprehensive revision