Learning Outcomes
i. Master the fundamental equations of angular motion: angular displacement (θ), angular velocity (ω), and angular acceleration (α).
ii. Recognize the applications of angular motion equations in solving problems involving rotational motion.
iii. Effectively apply the equation θ = ωt + ½αt² to calculate angular displacement for objects with constant or non-zero initial angular velocities.
iv. Utilize the equation ω = ω₀ + αt to determine the angular velocity of an object after a certain time, considering its initial angular velocity and angular acceleration.
v. Combine angular motion equations to solve complex problems involving rotational motion, such as determining the angular displacement and angular velocity of an object at different time intervals.
Introduction
In our previous lessons, we delved into the concepts of angular displacement, angular velocity, and angular acceleration, providing a foundation for understanding rotational motion. In this lesson, we embark on a practical journey, equipping ourselves with the tools to solve problems involving rotational motion. We will explore the powerful equations of angular motion and their applications in various scenarios.
i. Angular Displacement: A Measure of Rotational Turn
Angular displacement (θ) represents the angle through which an object rotates about a fixed axis. It is measured in radians, a unit that describes the fraction of a circle's circumference that an object has rotated through.
ii. Angular Velocity: The Speed of Rotation
Angular velocity (ω) represents the rate of change of angular displacement, indicating the speed of rotation. It is measured in radians per second (rad/s) and conveys how quickly an object is rotating.
iii. Angular Acceleration: The Change in Rotational Speed
Angular acceleration (α) represents the rate of change of angular velocity, measuring how quickly the rotational speed of an object is changing. It is measured in radians per second squared (rad/s²) and can be positive (increasing rotational speed) or negative (decreasing rotational speed).
iv. Equations of Angular Motion: Guiding Rotational Motion Analysis
The fundamental equations of angular motion provide a framework for solving problems involving rotational motion:
Angular displacement: θ = ω₀t + ½αt²
This equation relates angular displacement (θ) to initial angular velocity (ω₀), angular acceleration (α), and time (t). It applies to objects with constant or non-zero initial angular velocities.
Angular velocity: ω = ω₀ + αt
This equation relates angular velocity (ω) to initial angular velocity (ω₀), angular acceleration (α), and time (t). It allows us to determine the angular velocity of an object after a certain time.
Applying Angular Motion Equations: Solving Rotational Motion Problems
Let's consider two examples to illustrate the application of angular motion equations:
Determining Angular Displacement:
A fan rotates at a constant angular velocity of 100 radians per second. After 5 seconds, what is the angular displacement of the fan?
Using the angular displacement equation, we can calculate:
θ = ω₀t + ½αt² = 100 rad/s * 5 s + ½(0 rad/s²) * 5² s² ≈ 500 radians
Calculating Angular Velocity: A wheel accelerates uniformly from rest to an angular velocity of 40 radians per second in 5 seconds. What is the angular acceleration of the wheel?
Applying the angular velocity equation, we can determine:
ω = ω₀ + αt = 0 rad/s + α * 5 s = 40 rad/s
Solving for α, we get:
α = 40 rad/s / 5 s ≈ 8 rad/s²
The equations of angular motion provide a powerful toolkit for solving problems involving rotational motion. By mastering these equations and understanding their applications, we can analyze the motion of rotating objects, determine their angular displacement and angular velocity, and gain insights into various physical phenomena involving rotation.
