Chapter 22 : Torque and Angular Momentum

Introduction

Torque and angular momentum are fundamental concepts in the study of rotational motion. Torque is the rotational analog of force, responsible for causing changes in angular motion, while angular momentum is the rotational analog of linear momentum, representing the quantity of rotation an object possesses. This chapter explores the definitions, calculations, and the intricate interrelation between torque and angular momentum.

 

Torque: A Rotational Force

 

Definition of Torque

Torque (τ) is the measure of the tendency of a force to rotate an object about an axis. Mathematically, torque is defined as the product of the applied force (F) and the lever arm or perpendicular distance (r) from the axis of rotation to the point where the force is applied:

τ=r⋅F

 

Direction of Torque

The direction of torque is determined by the right-hand rule. If you align your thumb along the direction of the force and your fingers along the direction of the lever arm, then the direction your fingers curl is the direction of the torque.

 

Torque Units

Torque is measured in Newton-meters (Nm) in the International System of Units (SI).

 

Angular Momentum: Measure of Rotation

 

Definition of Angular Momentum

Angular momentum (L) is the rotational analog of linear momentum and is defined as the product of the moment of inertia (II) and the angular velocity (ωω) of an object:

L=Iω

 

Conservation of Angular Momentum

One of the fundamental principles in rotational motion is the conservation of angular momentum. In the absence of external torques, the total angular momentum of a system remains constant. This principle is expressed mathematically as:

L_initial=L_final​

 

Interrelation Between Torque and Angular Momentum

Torque and Angular Acceleration

Newton's second law for rotation relates torque to angular acceleration (α) and moment of inertia (I):

τ=Iα

This equation states that the torque applied to an object is equal to the product of its moment of inertia and angular acceleration.

 

Torque and Change in Angular Momentum

The relationship between torque and change in angular momentum is given by:

τ=ΔLΔt

Here, ΔL is the change in angular momentum, and Δt is the time over which the torque is applied. This equation highlights that torque is the rate of change of angular momentum.

 

Conservation of Angular Momentum

In the absence of external torques, the total angular momentum of a system remains constant. This principle is expressed by the equation:

τ_external=ΔL_totalΔt

When τ_external is zero, the angular momentum of the system is conserved.

 

Practical Applications

Understanding the interrelation between torque and angular momentum is crucial in various real-world scenarios, such as:

• Rotational Motion of Wheels: Torque is responsible for the rotation of wheels, and the conservation of angular momentum ensures the stability of a rotating wheel.
• Gyroscope Stability: Gyroscopes utilize the principles of torque and angular momentum to maintain stability and resist changes in orientation.
• Figure Skating Spins: Figure skaters manipulate their body shape to control torque and angular momentum, showcasing the conservation of angular momentum.

 

Conclusion

Torque and angular momentum are essential concepts in the study of rotational motion. Their interrelation is governed by fundamental principles, including Newton's second law for rotation and the conservation of angular momentum. The understanding of torque and angular momentum provides insights into the behavior of rotating objects, guiding the analysis of mechanical systems and contributing to the design of various devices and technologies.Top of Form

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