Torque and Angular Momentum - System of Particles and Rotational Motion

Torque

• Definition: Torque is the measure of the turning effect of a force on an object. It is defined as the cross product of the force vector and the position vector from the axis of rotation to the point of application of the force.

• Calculation: The magnitude of torque is given by τ = Fr sin θ, where F is the magnitude of the force, r is the distance from the axis of rotation to the point of application of the force, and θ is the angle between the force vector and the position vector.

• Torque due to a couple: A couple is a pair of equal and opposite forces separated by a distance. The torque due to a couple is equal to the product of the magnitude of one of the forces and the distance between the forces.

• Equilibrium of torques: An object is in equilibrium when the net torque acting on it is zero. This means that the sum of the torques in one direction must be equal to the sum of the torques in the opposite direction.

• Applications of torque: Torque is used in many everyday applications, such as opening a door, turning a screw, and riding a bicycle.

Angular Momentum

• Definition: Angular momentum is a measure of the rotational motion of an object. It is defined as the product of the moment of inertia of the object and its angular velocity.

• Calculation: The magnitude of angular momentum is given by L = Iω, where I is the moment of inertia and ω is the angular velocity.

• Relationship between angular momentum and torque: The rate of change of angular momentum is equal to the net torque acting on the object. This relationship is known as Newton’s second law for rotational motion.

• Conservation of angular momentum: The total angular momentum of a closed system remains constant, regardless of the internal forces acting on the system.

• Applications of angular momentum: Angular momentum is used in many applications, such as gyroscopes, spinning tops, and figure skating.

Moment of Inertia

• Definition: The moment of inertia of an object is a measure of its resistance to rotational motion. It is defined as the sum of the products of the masses of the particles in the object and the squares of their distances from the axis of rotation.

• Calculation: The moment of inertia of a uniform object can be calculated using the formula I = (1/2)MR², where M is the mass of the object and R is the radius of the object.

• Parallel axis theorem: The moment of inertia of an object about any axis parallel to its center of mass is equal to the moment of inertia of the object about its center of mass plus the product of the mass of the object and the square of the distance between the two axes.

• Perpendicular axis theorem: The moment of inertia of an object about an axis perpendicular to its center of mass is equal to the sum of the moments of inertia of the object about two mutually perpendicular axes that are parallel to the center of mass and intersect it.

• Applications of moment of inertia: Moment of inertia is used in many applications, such as the design of flywheels, pulleys, and gears.

Rotational Dynamics

• Equations of rotational motion: The equations of rotational motion are analogous to the equations of linear motion. They include the following:

  • Angular acceleration: α = Δω / Δt, where α is the angular acceleration, ω is the angular velocity, and t is time.
  • Angular displacement: θ = ω0t + (1/2)αt², where θ is the angular displacement, ω0 is the initial angular velocity, α is the angular acceleration, and t is time.
  • Rotational kinetic energy: K = (1/2)Iω², where K is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.
  • Work-energy theorem: W = ΔK, where W is the work done on the object, ΔK is the change in the object’s kinetic energy, I is the moment of inertia, and ω is the angular velocity.

• Relationship between torque and rotational acceleration: The torque acting on an object is equal to the product of its moment of inertia and its angular acceleration.

Rolling Motion

• Rolling motion: Rolling motion is a combination of rotational and translational motion. It occurs when an object rolls on a surface without slipping.

• Velocity and acceleration of a rolling object: The velocity of a rolling object is equal to the product of its angular velocity and the radius of the object. The acceleration of a rolling object is equal to the product of its angular acceleration and the radius of the object.

• Conservation of energy in rolling motion: The total energy of a rolling object is conserved. This means that the sum of the object’s kinetic energy and potential energy is constant.

Collision in Rotational Motion

• Elastic and inelastic collisions: Collisions in rotational motion can be either elastic or inelastic. In an elastic collision, the total angular momentum of the system is conserved. In an inelastic collision, the total angular momentum of the system is not conserved.

• Coefficient of restitution: The coefficient of restitution is a measure of the elasticity of a collision. It is defined as the ratio of the relative velocity of the objects after the collision to the relative velocity of the objects before the collision.

• Applications of rotational collisions: Rotational collisions are used in many applications, such as sports, automotive engineering, and manufacturing.