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HSSLIVE Plus One Physics Chapter 7: Systems of Particles and Rotational Motion Notes

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Systems of Particles and Rotational Motion bridges the gap between single-particle mechanics and the behavior of extended rigid bodies. This chapter expands students’ understanding by introducing center of mass, moment of inertia, and angular momentum concepts. Students learn to analyze both translational and rotational motion, applying torque and angular momentum conservation principles to complex systems from spinning tops to planetary motion.

Plus One Physics Notes – Chapter 7: Systems of Particles and Rotational Motion

SCERT Kerala Board

Centre of Mass

Definition: The centre of mass is the point where the entire mass of a system can be considered to be concentrated.

Formula for centre of mass:

  • For a system of discrete particles:
    • xcm = (m₁x₁ + m₂x₂ + … + mₙxₙ) / (m₁ + m₂ + … + mₙ)
    • ycm = (m₁y₁ + m₂y₂ + … + mₙyₙ) / (m₁ + m₂ + … + mₙ)
    • zcm = (m₁z₁ + m₂z₂ + … + mₙzₙ) / (m₁ + m₂ + … + mₙ)

Properties:

  • The centre of mass moves as if all the external forces act on it
  • For a rigid body with uniform density, the centre of mass coincides with its geometric centre

Motion of Centre of Mass

Newton’s Second Law for a system of particles: F₍ₑₓₜ₎ = M × a₍ₑₘ₎

  • F₍ₑₓₜ₎ = total external force
  • M = total mass of the system
  • a₍ₑₘ₎ = acceleration of the centre of mass

Angular Momentum

Definition: Angular momentum is the rotational equivalent of linear momentum.

Formula: L = r × p = mvr sin θ

  • L = angular momentum
  • r = position vector
  • p = linear momentum
  • θ = angle between r and p

Conservation of Angular Momentum: If no external torque acts on a system, its total angular momentum remains constant.

Moment of Inertia

Definition: Moment of inertia is the rotational equivalent of mass, representing resistance to angular acceleration.

Formula: I = Σmᵢrᵢ²

  • I = moment of inertia
  • mᵢ = mass of particle i
  • rᵢ = perpendicular distance of particle i from the axis of rotation

Moments of inertia for common shapes:

  • Solid sphere: I = (2/5)MR²
  • Hollow sphere: I = (2/3)MR²
  • Solid cylinder (about axis): I = (1/2)MR²
  • Ring (about axis): I = MR²
  • Rod (about centre): I = (1/12)ML²
  • Rod (about end): I = (1/3)ML²

Torque

Definition: Torque is the rotational equivalent of force, causing angular acceleration.

Formula: τ = r × F = rF sin θ

  • τ = torque
  • r = position vector
  • F = force
  • θ = angle between r and F

Direction: Determined by the right-hand rule.

Rotational Kinematics

Angular displacement (θ): Measured in radians Angular velocity (ω): Rate of change of angular displacement (rad/s) Angular acceleration (α): Rate of change of angular velocity (rad/s²)

Equations of rotational motion:

  • ω = ω₀ + αt
  • θ = θ₀ + ω₀t + (1/2)αt²
  • ω² = ω₀² + 2α(θ – θ₀)

Rotational Dynamics

Newton’s Second Law for rotation: τ = Iα

  • τ = net torque
  • I = moment of inertia
  • α = angular acceleration

Rotational Kinetic Energy

Formula: KE₍ᵣₒₜ₎ = (1/2)Iω²

  • I = moment of inertia
  • ω = angular velocity

Rolling Motion

Pure rolling motion: A combination of rotation and translation with no slipping Velocity relationship: v = Rω

  • v = linear velocity of centre of mass
  • R = radius
  • ω = angular velocity

Kinetic energy in rolling motion: KE = (1/2)mv² + (1/2)Iω² = (1/2)mv² + (1/2)Iₑₘω²

Equilibrium of Rigid Bodies

Conditions for equilibrium:

  1. Net force = 0 (translational equilibrium)
  2. Net torque = 0 (rotational equilibrium)

Practice Problems

  1. Find the centre of mass of a system with three particles of masses 2 kg, 3 kg, and 5 kg located at coordinates (1,2), (3,4), and (5,6) respectively.
  2. Calculate the moment of inertia of a uniform rod of mass 2 kg and length 3 m about an axis passing through its centre and perpendicular to its length.
  3. A wheel starting from rest is subject to a constant torque of 5 N·m. If its moment of inertia is 2 kg·m², find its angular velocity after 4 seconds.

These notes cover the essential concepts of Systems of Particles and Rotational Motion for Plus One Physics following the SCERT Kerala Board syllabus, presented in a simplified manner for easier understanding.

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