Vector Products- Angular Velocity and Angular Acceleration

NCERT Books Reference

• NCERT Class 11: Physics, Part I, Chapters 6 and 7
• NCERT Class 12: Physics, Part I, Chapter 8

1. Vector Products

• Cross product of two vectors:
• The cross product of two vectors A and B, denoted by A × B, is a vector that is perpendicular to both A and B.
• Formula: A × B = |A||B|sinθ n, where |A| and |B| are the magnitudes of vectors A and B, respectively, θ is the angle between A and B, and n is a unit vector perpendicular to both A and B.
• Geometrical interpretation of the cross product:
• The cross product of two vectors A and B represents the area of a parallelogram with sides A and B.
• Properties of the cross product:
• A × B = - B × A
• A × (B + C) = A × B + A × C
• (A × B) × C = (A · C)B - (B · C)A
• Applications of the cross product in physics:
• Calculating torque
• Finding angular momentum
• Determining the direction of magnetic force
• Calculating the Lorentz force

2. Angular Velocity

• Definition of angular velocity:
• Angular velocity ( ω) is the rate at which an object rotates or spins about an axis.
• Formula: ω = dθ / dt, where dθ is the change in the angular displacement and dt is the change in time.
• Units: radians/second (rad/s)
• Relation between angular velocity and linear velocity:
• For an object moving in a circular path, the angular velocity is related to the linear velocity ( v) by the formula: v = ωr, where r is the radius of the circular path.
• Instantaneous angular velocity:
• Instantaneous angular velocity is the angular velocity at a specific instant of time.
• It can be calculated by taking the derivative of the angular displacement with respect to time.
• Average angular velocity:
• Average angular velocity is the angular velocity over a specific time interval.
• It can be calculated by dividing the change in the angular displacement by the change in time.

3. Angular Acceleration

• Definition of angular acceleration:
• Angular acceleration ( α) is the rate at which the angular velocity changes.
• Formula: α = dω / dt, where dω is the change in the angular velocity and dt is the change in time.
• Units: radians/second squared (rad/s²)
• Relation between angular acceleration and angular velocity:
• Angular acceleration is related to angular velocity by the formula: α = a_t/r, where a_t is the tangential acceleration and r is the radius of the circular path.
• Instantaneous angular acceleration:
• Instantaneous angular acceleration is the angular acceleration at a specific instant of time.
• It can be calculated by taking the derivative of the angular velocity with respect to time.
• Average angular acceleration:
• Average angular acceleration is the angular acceleration over a specific time interval.
• It can be calculated by dividing the change in the angular velocity by the change in time.

4. Applications of Angular Velocity and Angular Acceleration

• Rotational Motion:
• Uniform circular motion: An object moving in a circular path with constant angular velocity is said to be in uniform circular motion.
• Non-uniform circular motion: An object moving in a circular path with varying angular velocity is said to be in non-uniform circular motion.
• Rolling Motion:
• Rolling motion without slipping: When an object rolls without slipping, its point of contact with the surface remains stationary.
• Rolling motion with slipping: When an object rolls and slips, its point of contact with the surface moves.
• Torque and Moment of Inertia:
• Torque (τ) is the twisting force that causes an object to rotate about an axis.
• Moment of inertia (I) is the resistance of an object to angular acceleration.
• Parallel and perpendicular axis theorem: The moment of inertia of an object about an axis parallel to its axis of symmetry is equal to the sum of its moments of inertia about two perpendicular axes passing through its center of mass.
• Application of torque and moment of inertia in rotational dynamics: Torque and moment of inertia are used to calculate the angular acceleration of an object and its rotational kinetic energy.

5. Three-Dimensional Motion

• Angular velocity and angular acceleration in three dimensions:
• In three-dimensional motion, angular velocity and angular acceleration have both magnitude and direction.
• Cross product of vectors in three dimensions:
• The cross product of two vectors in three dimensions results in a vector that is perpendicular to both of the original vectors.
• Applications of vector products in three-dimensional kinematics and dynamics:
• Vector products are used in three-dimensional kinematics and dynamics to calculate quantities such as torque, angular momentum, and angular velocity.

6. Precession and Nutations

• Precession of a spinning top:
• Precession is the slow, steady change in the orientation of the axis of rotation of a spinning object.
• In the case of a spinning top, precession is caused by the torque exerted by gravity on the top’s center of mass.
• Nutation of a spinning top:
• Nutation is the small, irregular wobble of the axis of rotation of a spinning object.
• In the case of a spinning top, nutation is caused by the uneven distribution of mass in the top.

By thoroughly understanding these topics and practicing the related problem-solving techniques, you can enhance your preparation for the Vector Products- Angular