Linear and Angular Quantity Relationships
Simple Explanation
Every point on a rotating rigid object shares the same angular velocity Ο, but points farther from the axis move through more distance in the same time β their linear (tangential) speed is v=rΟ, growing with radius.
Why Do We Need It?
This relationship is the bridge between rotational motion (described by ΞΈ, Ο, Ξ±) and the everyday linear motion (described by s, v, a) of any specific point on a spinning object.
Formula
Linear Speed from Angular Velocity
v = rΟ
The linear (tangential) speed of a point on a rotating object equals its distance from the axis times the angular velocity.
- v
- β linear (tangential) speed, in metres per second (m/s)
- r
- β distance from the axis of rotation, in metres (m)
- Ο
- β angular velocity, in radians per second (rad/s)
When to use it: Whenever converting between how fast something spins (angular velocity) and how fast a specific point on it moves through space (linear speed).
Worked Example
Find a linear speed from angular velocity
A point on a wheel of radius 0.5 m rotates with the wheel at Ο=8 rad/s. Find the point's linear speed.
Why Does This Work?
In one full rotation, a point at radius r travels a distance equal to the circle's circumference, 2Οr, while the whole object turns through 2Ο radians β so distance traveled is always r times the angle swept, and dividing both sides by time gives v=rΟ.
Real-Life Example
Why the outer edge of a spinning record moves faster
A vinyl record spins at a constant angular velocity, but a speck of dust near the outer edge and a speck near the center both complete one rotation in exactly the same time.
Since the outer speck has a larger radius r, its linear speed v=rΟ is greater β the outer edge physically travels farther (and faster) than a point near the center, even though both share the same Ο.
Practice
A point on a wheel of radius 0.2 m rotates at Ο=15 rad/s. Find its linear speed.
MediumCommon mistake
Assuming every point on a spinning object moves at the same speed β only Ο is shared by every point; linear speed v=rΟ is different for every different radius.
Quick Review
- v = rΟ β linear speed depends on both angular velocity and distance from the axis.
- Every point on a rigid rotating object shares the same Ο.
- Points farther from the axis move faster in a straight-line sense.