Torque and Angular Momentum - System of Particles and Rotational Motion

Torque

• Definition: Torque is the turning or twisting force that acts on an object around an axis or pivot.

• Expression for Torque: $$ \tau= rF\sin\theta$$ Where:

• $\tau$ is torque (measured in Newton-meters, Nm)

• r is the distance from the axis of rotation to the point where the force is applied (in meters, m)

• F is the force applied (in Newtons, N)

• $\theta$ is the angle between the force vector and the position vector from the axis of rotation (measured in radians)

Angular Momentum

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

• Properties:

• It is a vector quantity, both in direction and magnitude.

• For a spinning object, the direction of angular momentum is given by the right-hand rule.

• The SI unit of angular momentum is kilogram meter squared per second (kg m^2/s)

• Angular Momentum has both Orbital and Spin contribution

• Relation between Torque and Angular Momentum: $$\overrightarrow{\tau}=\frac{d\overrightarrow{L}}{dt}$$ Where:

• $\overrightarrow{\tau}$ is torque (measured in Newton-meter rad)

• $\overrightarrow{L}$ is angular momentum (measured in Kg. m2/s)

• t is time (measured in seconds)

Conservation of Angular Momentum:

• Statement: The total angular momentum of a closed system remains constant, regardless of the internal interactions within the system.

• Conservation of Angular Momentum makes analysis and predictions easier. When external torque is absent:

$$I_i \omega_i =I_f \omega_f$$

• Applications:
• Ice skaters spinning faster when they pull their arms in.
• Conservation of angular momentum of planets orbiting the Sun.
• Action-reaction forces that occur during interactions.

Rotational Motion of a Rigid Body

• Angular Velocity and Angular Acceleration

• Angular Velocity($\omega$): The rate at which an object rotates. It is the change in angle per unit of time. $$\omega = \frac{d\theta}{dt}$$

• Angular Acceleration($\alpha$): The rate at which angular velocity changes is known as angular acceleration. $$\alpha = \frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$$

• Equations of Rotational Motion: $$\omega_f=\omega_i+\alpha t$$ $$\theta_f=\theta_i+\omega_i t+\frac{1}{2} \alpha t^2$$

• Kinetic Energy of Rotation: $$K=\frac{1}{2}I\omega^2$$

• Moment of Inertia(I):

• The measure of an object’s resistance to angular acceleration around a given axis.

• Dependent upon

  • Mass distribution
  • Axis chosen

• Parallel and Perpendicular Axes Theorem:

• The moment of inertia about an axis perpendicular to a surface equals the sum of the moments of inertia of that surface about two mutually perpendicular axes in the plane of the surface, passing through the point at which the perpendicular axis intersects the plane.

• The moment of inertia about an axis parallel to an axis through the center of mass is equal to the moment of inertia about the parallel axis plus the product of the total mass and the square of the distance between the axes.

• Moment of Inertia for Some Shapes: