10.13

Vector Transformation in Rotating Coordinate Systems

2.9K views01:16 min

Consider a vector rotating about an axis with an angular velocity such that its tip sweeps a circular path.

The time derivative of this vector equals its position change during an infinitesimal time interval. This expression is rewritten in terms of the cross product of the angular velocity and the vector.

Consider an arbitrary vector and its time derivative in an inertial frame. When observed from a rotating frame, the scalar components and the unit vectors change.

The vector's time derivative in the rotational frame yields two terms. The first term is the time derivative of the vector measured in the rotating frame.

As the unit vectors rotate, their rate of change expression is used to simplify the second term.

So, the time derivative of a vector in the inertial frame equals its time derivative in the rotating frame plus the cross product of the angular velocity and the vector.

The transformation equation for a position vector gives the velocity, while that for a velocity vector yields the acceleration in a rotating system.

Overview

Loading...

$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

Consider a vector rotating about an axis with an angular velocity, such that its tip sweeps a circular path.

Rotational motion diagram with vectors, angular velocity, magnetic field change; physics concept.

In time Δt, the vector moves through an angle given by the product of the angular velocity and time. So, the tip of the vector moves by a magnitude ΔB.

Magnetic field change equation, ΔB=(B sin α)ωΔt, formula representation for physics study.

Here, α is the angle between the vector and the angular velocity vector.

The rate of change of the vector $Magnetic field vector diagram; formula: \(\vec{B}\); illustrates magnetic flux direction and magnitude.$ equals the cross-product of the angular velocity and the vector itself.

$Magnetic field change equation \( \frac{d\vec{B}}{dt} = \vec{\omega} \times \vec{B} \), vector math formula.$

The above relation is valid for any vector that undergoes pure rotation at a constant angular velocity along a fixed rotational axis.

Now, consider an arbitrary vector in an inertial frame of reference. Its time derivative in this frame can be written considering the time derivative of its scalar components along the three coordinate axes.

$Vector differentiation formula, \(d\vec{C}/dt\), demonstrating component analysis in physics equations.$

When viewed from a rotating frame of reference, the scalar components and the unit vectors change. The magnitude and direction of the vector remain the same, regardless of the coordinate system chosen to assign its components.

Vector equation, vector notation formula, diagram for vector components in Cartesian coordinates.

Taking the time derivative of the vector in the rotational frame of reference gives two terms.

Vector calculus equation; time derivative of vector C; mathematical symbol; education, research.

The first term is essentially the time derivative of the vector measured in the rotating frame of reference. The unit vectors in the rotating system rotate with a constant angular velocity. So, the time derivative of the unit vectors along the three coordinate axes is equal to the cross product of the angular velocity and the unit vectors in the rotating frame. Substituting the expression of the rate of change of unit vectors, the second term can be written as the cross product of the angular velocity and the vector.

So, the time derivative of any arbitrary vector in the inertial frame is equal to its time derivative in the rotating frame plus the cross product of the angular velocity and the vector.

Time derivative of vectors in rotating reference frame equation, related to angular velocity dynamics.

When the arbitrary vector is replaced with a position vector, the transformation equation for the velocity vector is obtained. The velocity vector in the inertial frame equals the velocity vector in the rotational frame plus the cross product of angular velocity and the position vector.

Similarly, replacing the arbitrary vector with a velocity vector yields the transformation equation for the acceleration vector.

Transcript

Explore More Videos

Vector Transformation

Rotating Coordinate Systems

Angular Velocity

Time Derivative

Position Vector

Velocity Vector

Acceleration Vector