Rigid bodies and rotational dynamics

Rigid bodies and rotational dynamics

Rigid Bodies

  • A rigid body is an object with a shape that does not change.
  • Rigid bodies don’t deform when forces act on them. The particles within the body do not move relative to each other.
  • An idealised concept, since all real world objects deform to a certain extent under various conditions.

Moment of Inertia

  • Moment of Inertia (I) is a measure of a body’s resistance to rotation and is comparable to mass in linear motion.
  • For a single point mass, Moment of Inertia is calculated as I = m*r^2, where m is the mass and r is the distance of the point from the axis of rotation.
  • For rigid bodies, Moment of Inertia is the sum of moments of inertia of individual point masses.

Angular Momentum

  • In rotational dynamics, Angular Momentum (L) is analogous to linear momentum in linear motion.
  • Angular Momentum of a particle is calculated as L = Iω, where, I is the Moment of Inertia and ω is the angular velocity.
  • Just like linear momentum, angular momentum is conserved if no external torque acts on the system.

Torque

  • Torque (τ) is the turning effect of a force. In rotational motion, it plays a role analogous to force in linear motion.
  • Torque is calculated as τ = r × F, where r is the distance from the axis of rotation to the point of application of the force and F is the force.
  • The magnitude of torque is given by τ = rFsinθ, where θ is the angle between r and F.
  • The net torque on a system is equal to the rate of change of its angular momentum.

Rotational Kinetic Energy

  • Rotational Kinetic Energy (KE) is the kinetic energy due to rotation of the body.
  • Given as KE = ½Iω^2, where, I is the Moment of Inertia and ω is the angular velocity.

Rotational Analogs

  • Newton’s second law in rotational motion is τ = Iα, where, α is angular acceleration.
  • In rotational motion, Work done (W) = τθ, where θ is the angle of rotation.
  • Power (P) in rotational motion is given by P = τω, where ω is the angular velocity.
  • Rotational version of the Impulse-Momentum theorem is given by τdt = dL, where dt is the small change in time and dL is small change in angular momentum.