Angular Displacement, Velocity, and Acceleration

Introduction

Key Concepts

Angular Displacement

Angular displacement refers to the angle through which an object rotates about a fixed axis. It is a vector quantity, possessing both magnitude and direction, and is typically measured in radians (rad) or degrees (°). Unlike linear displacement, which measures movement along a path, angular displacement quantifies the rotational movement about an axis.

Mathematically, angular displacement ($\theta$) can be expressed as: $$ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $$ where:

• $\theta$ is the angular displacement.
• $\omega_0$ is the initial angular velocity.
• $\alpha$ is the angular acceleration.
• $t$ is the time elapsed.

For rotational motion with constant angular velocity, the angular displacement simplifies to: $$ \theta = \omega t $$ where $\omega$ is the constant angular velocity.

Angular Velocity

Angular velocity ($\omega$) measures the rate of change of angular displacement with respect to time. It indicates how quickly an object rotates and is a vector quantity with both magnitude and direction. The direction of angular velocity is determined by the right-hand rule, where curling the fingers of the right hand in the direction of rotation points the thumb in the direction of $\omega$.

The relationship between angular velocity and linear velocity ($v$) is given by: $$ v = r \omega $$ where:

• $r$ is the radius or distance from the axis of rotation to the point of interest.

Angular velocity can also be described in terms of angular displacement over time: $$ \omega = \frac{d\theta}{dt} $$ For constant angular acceleration, angular velocity changes linearly with time: $$ \omega = \omega_0 + \alpha t $$

Angular Acceleration

Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time. It describes how quickly an object is speeding up or slowing down its rotation and is measured in radians per second squared ($\text{rad/s}^2$). Angular acceleration is a vector quantity, with its direction indicating whether the rotational speed is increasing or decreasing.

The formula for angular acceleration is: $$ \alpha = \frac{d\omega}{dt} $$ For rotational motion with constant angular acceleration, angular displacement can be expressed as: $$ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $$ Similarly, angular velocity at any time $t$ is: $$ \omega = \omega_0 + \alpha t $$

Rotational Kinematics Equations

Rotational kinematics involves equations analogous to linear kinematics, adapted for rotational motion. The primary equations are:

1. Angular Displacement: <$$ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $$ For constant angular acceleration.
2. Angular Velocity: $$ \omega = \omega_0 + \alpha t $$
3. Final Angular Velocity: $$ \omega^2 = \omega_0^2 + 2 \alpha \theta $$

Uniform Circular Motion

In uniform circular motion, an object moves in a circle with constant angular velocity. Despite the constant speed, there is a continuous change in the direction of the velocity, resulting in centripetal acceleration directed towards the center of the circle. The key parameters in uniform circular motion include:

• Angular Velocity ($\omega$): Constant rate of rotation.
• Period ($T$): Time taken to complete one full rotation.
• Frequency ($f$): Number of rotations per unit time.

The relationship between period and angular velocity is: $$ \omega = \frac{2\pi}{T} $$ And between frequency and angular velocity: $$ \omega = 2\pi f $$

Non-Uniform Circular Motion

In contrast to uniform circular motion, non-uniform circular motion involves a changing angular velocity, implying the presence of angular acceleration. This type of motion encompasses scenarios where the rotational speed increases or decreases over time, often due to applied torque or other forces.

Applications of Angular Kinematics

Angular kinematics is essential in various real-world applications, including:

• Engineering: Design of rotating machinery, such as turbines and engines.
• Sports: Analysis of rotational movements in activities like gymnastics and figure skating.
• Astronomy: Understanding the rotation of celestial bodies.
• Robotics: Development of joints and limbs with precise rotational control.

Examples and Problem Solving

Consider a wheel that starts from rest and accelerates uniformly. Suppose the wheel has an angular acceleration of $2.0 \, \text{rad/s}^2$. To find the angular displacement after $5$ seconds, use the equation: $$ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $$ Given $\omega_0 = 0$, the equation simplifies to: $$ \theta = \frac{1}{2} \times 2.0 \times (5)^2 = 25 \, \text{rad} $$ Thus, the wheel rotates through an angle of $25$ radians in $5$ seconds.

Comparison Table

Summary and Key Takeaways