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KMC
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The$ \require{enclose} $ The relation between angular velocity and linear velocity is given by this equation (from openstax: angular velocity):

$$ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} $$

I can rearrange $\boldsymbol{\omega}$ to the left while keeping the right-hand-rule:

$$ \boldsymbol{\omega} = \mathbf{r} \times \mathbf{v} $$I can rearrange $\boldsymbol{\omega}$ to the left while keeping the right-hand-rule: $$ \enclose{horizontalstrike}{\boldsymbol{\omega} = \mathbf{r} \times \mathbf{v}} $$

But Wikipedia defines $\boldsymbol{\omega}$ in three-dimension to be this instead:

$$ \boldsymbol{\omega} = \frac{\mathbf{r} \times \mathbf{v}}{r^2} $$

Why is the $ \boldsymbol{\omega} $ scaled down by a factor of $ \frac{1}{r^2} $ in three-dimension?

There is a somewhat related question in Physics SE: Is Wikipedia's definition of angular velocity incorrect? but it doesn't address the additional $\frac{1}{r^2}$ factor.

The relation between angular velocity and linear velocity is given by this equation (from openstax: angular velocity):

$$ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} $$

I can rearrange $\boldsymbol{\omega}$ to the left while keeping the right-hand-rule:

$$ \boldsymbol{\omega} = \mathbf{r} \times \mathbf{v} $$

But Wikipedia defines $\boldsymbol{\omega}$ in three-dimension to be this instead:

$$ \boldsymbol{\omega} = \frac{\mathbf{r} \times \mathbf{v}}{r^2} $$

Why is the $ \boldsymbol{\omega} $ scaled down by a factor of $ \frac{1}{r^2} $ in three-dimension?

There is a somewhat related question in Physics SE: Is Wikipedia's definition of angular velocity incorrect? but it doesn't address the additional $\frac{1}{r^2}$ factor.

$ \require{enclose} $ The relation between angular velocity and linear velocity is given by this equation (from openstax: angular velocity):

$$ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} $$

I can rearrange $\boldsymbol{\omega}$ to the left while keeping the right-hand-rule: $$ \enclose{horizontalstrike}{\boldsymbol{\omega} = \mathbf{r} \times \mathbf{v}} $$

But Wikipedia defines $\boldsymbol{\omega}$ in three-dimension to be this instead:

$$ \boldsymbol{\omega} = \frac{\mathbf{r} \times \mathbf{v}}{r^2} $$

Why is the $ \boldsymbol{\omega} $ scaled down by a factor of $ \frac{1}{r^2} $ in three-dimension?

There is a somewhat related question in Physics SE: Is Wikipedia's definition of angular velocity incorrect? but it doesn't address the additional $\frac{1}{r^2}$ factor.

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Qmechanic ♦
Qmechanic ♦
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What is this $\frac{1}{r^2}$ factor added to the definition of $ \boldsymbol{\omega} = \mathbf{r} \times \mathbf{v} $?

Source Link
KMC
KMC
  • 401
  • 5
  • 14

What is this $\frac{1}{r^2}$ factor added to the definition of $ \boldsymbol{\omega} = \mathbf{r} \times \mathbf{v} $

The relation between angular velocity and linear velocity is given by this equation (from openstax: angular velocity):

$$ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} $$

I can rearrange $\boldsymbol{\omega}$ to the left while keeping the right-hand-rule:

$$ \boldsymbol{\omega} = \mathbf{r} \times \mathbf{v} $$

But Wikipedia defines $\boldsymbol{\omega}$ in three-dimension to be this instead:

$$ \boldsymbol{\omega} = \frac{\mathbf{r} \times \mathbf{v}}{r^2} $$

Why is the $ \boldsymbol{\omega} $ scaled down by a factor of $ \frac{1}{r^2} $ in three-dimension?

There is a somewhat related question in Physics SE: Is Wikipedia's definition of angular velocity incorrect? but it doesn't address the additional $\frac{1}{r^2}$ factor.