# Understanding Angular Velocity: Rotations and Accelerated Frames

• CrusaderSean
In summary, the conversation covers the topic of accelerated frames and rotations, specifically focusing on angular velocity and its relationship to different frames of reference. The concept of an inertial frame and a rotating frame is introduced, along with the equations for velocity and angular velocity in these frames. The conversation also delves into the concept of uniform angular velocity in rotating bodies and its relevance in calculating moments of inertia. The conversation concludes with a clarification on the relationship between omega and the base vectors of the two frames of reference.
CrusaderSean
i'm doing accelerated frames and rotations in class right now. I'm not sure if i understand angular velocity correctly so i hope someone can correct me.

Lets say there is an inertial frame and rotating frame.
r' = distance from origin of inertial to rotating frame
P = point in rotating frame
x = P measured from rotating frame
r = P measured from fixed frame

to describe velocity of P from inertia frame, it's
$$\frac{dr}{dt} = \frac{dr'}{dt} + \omega \times x$$

the way my textbook defined is $$\omega = \frac{d \phi}{dt} \hat{n}$$ , it points in normal direction of rotation axis. this normal direction is measured from inertial frame right? if you define the following as operator and apply it to omega:
$$\frac{d}{dt} = \frac{d}{dt} + \omega \times$$
$$\frac{d \omega}{dt} = \frac{d \omega'}{dt}$$
omega is observed from inertial while omega' from rotating frame. does this mean angular acceleration has same magnitude and direction in both frames, but angular velocity does not necessarily have to be the same?... kind of an odd question i guess, but i can't see how omega would have the same direction as if you measure it from different frames.

my textbook only emphasized omega is uniform for rotating (rigid) body because it does not depend on where the origin is in the rotating frame.

Last edited:
You have one frame that is 'fixed' and one frame that rotates along with the object (this does NOT necessarily have to be the case ofcourse). So when looking in the fixed frame, the object rotates but when looking in the rotating frame, the object does not move. The omega expresses the rotation of the object but the clue is that the rotating frame has the exact same angular velocity omega, because it rotates along with the object. This way of working is especially usefull when calculating the moments of inertia for rigid rotators (ie not pointlike objects but solid objects like an apple). Why ? Well, because when you are calculating this tensor in the rotating frame (the object does not move here) this tensor will reduce to a diagonal matrix.

marlon

Besides, there is only one omega which is the instantaneous rotation vector of the rotating frame wtr to the 'fixed' inertial frame. This omega really connects the base-vectors of these two frames.

marlon

marlon said:
Besides, there is only one omega which is the instantaneous rotation vector of the rotating frame wtr to the 'fixed' inertial frame. This omega really connects the base-vectors of these two frames.

marlon

i see. that makes more sense then. thanks for the clarification.

## What is angular velocity?

Angular velocity is a measure of the rate of change of angular displacement over time. It is a vector quantity, meaning it has both magnitude and direction, and is typically represented by the Greek letter omega (ω).

## How is angular velocity different from linear velocity?

Angular velocity describes the rotation of an object around a fixed axis, while linear velocity describes the straight-line motion of an object. Angular velocity is measured in radians per second, while linear velocity is measured in meters per second. Angular velocity also takes into account the direction of rotation, while linear velocity only considers the direction of linear motion.

## What is the relationship between angular velocity and angular acceleration?

Angular velocity and angular acceleration are related by the equation ω = ω0 + αt, where ω is the final angular velocity, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time interval. In other words, angular velocity is the integral of angular acceleration over time.

## How does angular velocity change in an accelerated frame of reference?

In an accelerated frame of reference, the angular velocity of an object will also change. This is because the frame of reference is accelerating, which affects the rotation of the object. In this case, the angular velocity will change not only in magnitude but also in direction.

## What are some real-world applications of understanding angular velocity?

Understanding angular velocity is crucial in many fields, including physics, engineering, and astronomy. It is used to describe the motion of rotating objects, such as wheels, gears, and celestial bodies. It is also essential in the design and analysis of machines and structures that involve rotational motion, such as turbines and satellites.

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