Learning Outcomes
i. Comprehend the relationship between angular arc length (S), radius (r), and angular displacement (θ), expressed by the equation S = rθ.
ii. Apply the equation S = rθ to solve problems involving circular motion and calculate the angular arc length traveled by an object.
iii. Understand the relationship between linear velocity (v), angular velocity (ω), and radius (r), represented by the equation v = rω.
iv. Utilize the equation v = rω to solve problems involving circular motion and determine the linear velocity of an object moving in a circular path.
v. Effectively combine the equations S = rθ and v = rω to analyze and solve problems involving both angular and linear motion in circular paths.
Introduction
In our previous lesson, we delved into the fundamental concepts of angular displacement, angular velocity, and angular acceleration, providing a framework for understanding rotational motion. In this lesson, we explore the practical application of these concepts by introducing two crucial equations: S = rθ and v = rω.
i. Angular Arc Length: Tracing the Path of Rotation
The angular arc length (S) represents the distance traveled along a circular path by an object moving in a circular motion. It is measured in the same units as the radius (r), typically meters, and is related to the angular displacement (θ) by the equation:
S = rθ
This equation highlights the direct proportionality between angular arc length and angular displacement: a larger angular displacement corresponds to a longer angular arc length.
ii. Linear Velocity in Circular Motion: Tangential Velocity
The linear velocity (v) of an object moving in a circular path represents its tangential velocity, the component of its velocity tangent to the circular path. It is measured in meters per second (m/s) and is related to the angular velocity (ω) and radius (r) by the equation:
v = rω
This equation demonstrates that linear velocity is directly proportional to both radius and angular velocity: a larger radius or angular velocity results in a higher linear velocity.
iii. Applications of S = rθ and v = rω: Solving Problems in Rotational Motion
The equations S = rθ and v = rω provide powerful tools for solving problems involving circular motion. Let's consider two examples:
Calculating Angular Arc Length: A car travels along a circular track with a radius of 100 meters. If the car turns through an angle of 45 degrees, what is the angular arc length it has covered?
Using S = rθ, we can calculate the angular arc length:
S = 100 meters * 45 degrees * (π/180 degrees) ≈ 157 meters
Determining Linear Velocity: A Ferris wheel rotates with an angular velocity of 0.5 radians per second and has a radius of 20 meters. What is the linear velocity of a passenger on the Ferris wheel?
Applying v = rω, we can calculate the linear velocity:
v = 20 meters * 0.5 radians/second ≈ 10 meters/second
The equations S = rθ and v = rω provide essential tools for analyzing and solving problems involving circular motion. By understanding the relationships between angular arc length, radius, angular displacement, linear velocity, and angular velocity, we can effectively navigate the world of rotational motion and gain insights into various physical phenomena.
