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Guide 10-1. Rotational Kinematics

 Part A. Summary of Linear and Rotational Definitions and Relationships

The linear kinematics of Chapter 2 deals with uniformly-accelerated motion in one dimension. The rotational kinematics of Chapter 10 deals with uniformly-accelerated motion in a circle. The new things that you need to learn are a) the definitions of the rotational motion quantities that correspond to the linear quantities, and b) the relationships between the linear and rotational quantities. One other thing you need to be able to do is deal with angles in radian measure, but you may already know that from a math class. The conversion factor is 2π radians = 360°.

First, here are the definitions of the rotational quantities, the linear quantities that they correspond to, and the SI units of measure.

Linear Rotational
Name and symbol Definition SI Units Name and symbol Definition SI Units
Position, x displacement measured from the origin along a defined axis m Angular position, θ angle measured from a reference line* radian
Displacement, Δx change in position between any two points m Angular displacement, Δθ change in angle between any two radius vectors radian
Average velocity, vav m/s Average angular velocity, ωav 1/s+
Instantaneous velocity, v m/s Instantaneous angular velocity, ω 1/s
Average acceleration, aav m/s² Average angular acceleration, αav 1/s²
Instantaneous acceleration, a m/s² Instantaneous angular acceleration, α 1/s²


*By convention, positive angular positions are measured counterclockwise from a reference line. This also means that counterclockwise angular velocities and accelerations are positive.
+
The unit of radian generally isn't expressed.

With these symbols and definitions, one can quickly convert a linear dvat equation into the corresponding rotational equation.

Linear equation
(a is constant)
Angular equation
(α  is constant)

 Part B. Connections Between Linear and Rotational Quantities

In order to make connections between the rotational quantities in circular motion and the corresponding tangential quantities, we start by looking at the connection between displacement and angular displacement. We refer to the diagram to the right. An object initially at point A moves counterclockwise in a circular path of radius r. The object travels an arc length Δs corresponding to an angular displacement Δθ. The symbol s is commonly used to represent distance along a circular arc.) If the object travels around the entire circumference C of the circle, the corresponding angle is 2π. (We must use radian measure for this argument.) We set up a proportion between linear displacements and angular displacements.

Substituting C = 2πr,

and solving for Δs,

We can now use this relationship together with the definitions of angular velocity and angular acceleration to obtain the tangential velocity and acceleration. We summarize the results in the table below together with the equation for centripetal acceleration in terms of ω.

Linear
quantity
= Radius × Rotational
quantity
s = r × θ
vt = r × ω
at = r × α
ac* = r × ω²

*Note that centripetal acceleration is sometimes called radial acceleration.

Here is one additional relationship that relates angular velocity to period T or frequency f:

An object in circular motion can in general have both tangential and centripetal (radial) acceleration. The magnitude and direction of the total acceleration vector is found by adding the tangential and centripetal acceleration components vectorially.

 


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