Concepts to Remember for the “Vector Products- Angular Velocity and Angular Acceleration” for JEE and CBSE Board Exams:

• Definition of Cross Product of Two Vectors:

  • The cross product (also known as the vector product) of two vectors is a new vector perpendicular to both of the original vectors.
  • It is denoted as A x B and has a magnitude equal to the area of the parallelogram formed by the two vectors and a direction that follows the right-hand rule.

• Properties of the Cross Product:

  • The cross product is anti-commutative, i.e., A x B = - B x A.
  • The cross product is distributive over vector addition, i.e., A x (B + C) = A x B + A x C.
  • The magnitude of the cross product of two vectors is equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them, i.e., || A x B || = ||A|| ||B|| sin θ.

• Definition of Angular Velocity Vector:

  • Angular velocity is the rate of change of angular displacement.
  • It is a vector quantity and is defined as ω = dθ/dt, where ω is the angular velocity vector, dθ is the change in angular displacement, and dt is the change in time.

• Definition of Angular Acceleration Vector:

  • Angular acceleration is the rate of change of angular velocity.
  • It is a vector quantity and is defined as α = dω/dt, where α is the angular acceleration vector, dω is the change in angular velocity, and dt is the change in time.

• Relationship Between Angular Velocity Vector and Angular Acceleration Vector:

  • The angular acceleration vector is perpendicular to both the angular velocity vector and the displacement vector.
  • The magnitude of the angular acceleration vector is equal to the rate of increase in the magnitude of the angular velocity vector.

• Applications of Angular Velocity and Angular Acceleration Vectors in Rigid Body Dynamics:

  • Angular velocity and angular acceleration vectors are essential for understanding the rotational motion of rigid bodies.
  • They are used to determine the linear velocity and acceleration of particles in a rotating body and to calculate the forces and torques acting on the body.

• Equation of Motion for a Rotating Object:

  • The equation of motion for a rotating object is: Iα = Σ τ, where I is the moment of inertia, α is the angular acceleration vector, and τ is the net torque acting on the body.

• Torque:

  • Torque is the force that tends to cause rotation.
  • It is defined as the product of the perpendicular distance from the axis of rotation to the point of force application and the magnitude of the force applied.

• Inertia:

  • Inertia is the resistance of an object to changes in its motion.
  • It is a measure of the mass of an object and its distribution of mass.

• Centripetal Force:

  • Centripetal force is the force that causes a particle to move in a circular path.
  • It is directed toward the center of rotation and is equal to the product of the mass of the particle, the square of its tangential velocity, and the radius of the path.

• Tangential Velocity:

  • Tangential velocity is the velocity of a particle tangent to the path it is moving on.
  • For a particle moving in a circular path, the tangential velocity is equal to the product of the angular velocity and the radius of the path.

• Moment of Inertia:

  • Moment of inertia is the sum of the products of the masses of all particles in a body and the squares of their distances from a given rotational axis.
  • It is a measure of the resistance of a body to changes in its angular motion.

• Radius of Gyration:

  • The radius of gyration is the distance from the axis of rotation at which the entire mass of a body could be concentrated without changing its moment of inertia.

• Parallel Axis Theorem:

  • The parallel axis theorem states that the moment of inertia of a body about an axis parallel to its axis of symmetry is equal to the sum of its moment of inertia about the axis of symmetry and the product of its total mass and the square of the distance between the two axes.

• Perpendicular Axis Theorem:

  • The perpendicular axis theorem states that the moment of inertia of a body about an axis perpendicular to two perpendicular axes of symmetry is equal to the sum of its moments of inertia about the two axes of symmetry.

• Angular Momentum:

  • Angular momentum is the product of the moment of inertia of a rotating body and its angular velocity.
  • It is a conserved quantity, which means that it remains constant in the absence of an external torque.

• Conservation of Angular Momentum:

  • The conservation of angular momentum states that the total angular momentum of a system remains constant if no external torque acts on the system.