Rotational Motion About A Fixed Axis, Angular Momentum, System of Particles, and Rotational Motion

Detailed Toppers’ Notes

Rotational Motion About A Fixed Axis:

• Angular displacement: Change in angular position of an object in radians (rad).

  • Relationship with linear displacement: $$s=r\theta$$, where s is linear displacement, r is radius, and θ is angular displacement.

• Angular velocity: Rate of change of angular displacement, measured in radians per second (rad/s).

  • Relationship with linear velocity: $$\omega = v/r$$, where ω is angular velocity, v is linear velocity, and r is the radius.

• Angular acceleration: Rate of change of angular velocity, measured in radians per second squared (rad/s²).

  • Relationship with linear acceleration: $$\alpha = a_{tan}/r$$, where α is angular acceleration, a_tan is tangential linear acceleration, and r is the radius.

• Equations of rotational motion:

  • Position (angular displacement): $$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$$, where θ_0 is the initial angular position, ω_0 is the initial angular velocity, t is the time, and α is the angular acceleration.
  • Velocity (angular velocity): $$\omega = \omega_0 + \alpha t$$
  • Acceleration (angular acceleration): $$\alpha = \frac{d\omega}{dt} = \frac{\Delta \omega}{\Delta t}$$

Torque and Rotational Inertia:

• Torque: Force that causes rotation. $$ \tau = rFsinθ$$ , where τ is torque, F is the force, r is the radius vector from the axis of rotation to the point where the force is applied, and θ is the angle between r and F.
• Moment of inertia: Measures an object’s resistance to angular acceleration, depends on mass distribution. (I=\sum mr^2), where I is moment of inertia, m is mass, and r is the distance from the axis of rotation.
  • Parallel axis theorem: $$I_c = I_{CM} + Md^2$$, where I_c is the moment of inertia about an axis parallel to the CM, I_CM is the moment of inertia about an axis through the center of mass, d is the distance between the two parallel axes, and M is the total mass.
  • Perpendicular axis theorem: $$I_x=I_y + I_z$$, where I_x, I_y, and I_z are the moments of inertia about three mutually perpendicular axes intersecting at a point.

Kinetic Energy of Rotation:

• $$ K_R=\frac{1}{2} I \omega^2 $$

Angular Momentum:

• Angular momentum: Rotational analog of linear momentum, measures an object’s rotational motion. $$L = I \omega$$ where L is angular momentum, I is moment of inertia, and ω is angular velocity.
• Conservation of angular momentum: For a closed system, total angular momentum remains constant. $$L_{initial} = L_{final}$$

System of Particles and Rotational Motion:

• Centre of mass (CM): Point where the total mass of a system is considered concentrated, defined as $$\bar{x} = \frac{\Sigma mx_i}{M}$$ $$\\bar{y} = \frac{\Sigma my_i}{M}$$ where (M=\Sigma m)
• Centre of rotation: Point about which an object or system rotates
• Moment of inertia of a system of particles $$I = \Sigma mr^2$$
• Parallel axis theorem and perpendicular axis theorem also applicable for a system of particles.

Rolling Motion:

• Pure Rolling: Motion in which a body rotates about its center of mass without slipping or skidding.
• Rolling without slipping: No sliding between the point of contact and the surface. Linear velocity at contact point (v) must be zero. $$ v = \omega r$$
• Moment of inertia: $$I_{CM}+MR^2$$

Kinetic energy of a Rolling Object: $$ K = K_{CM} + K_R = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$$

Examples and Applications:

• Rotational motion in everyday life: fans, washing machines, wheels, etc.
• Applications in Engineering and Technology: turbines, gear systems, motors, etc.
• Rotational motion in sports and games: baseball curveball, tennis serve, etc.

Problem-Solving Techniques:

• Free body diagrams and torque equations: Identify forces and torques acting on the system and use equilibrium conditions (∑τ = 0 for rotational equilibrium).
• Energy methods in rotational motion: Use the conservation of energy principle (K_i + U_i = K_f + U_f) to solve problems.

References: NCERT Textbook for Class 11 and Class 12 - Physics, Part 1 and 2