Understanding Coriolis Acceleration Magnitude

The Coriolis acceleration is a component of the acceleration experienced by an object moving within a rotating frame of reference. It arises due to the rotation of the coordinate system itself.

Coriolis Acceleration Formula

The Coriolis acceleration vector ($ \vec{a}_c $) is mathematically defined as:

$$ \vec{a}_c = -2 (\vec{\omega} \times \vec{v}) $$

Where:

• $ \vec{\omega} $ represents the angular velocity vector of the rotating frame.
• $ \vec{v} $ represents the linear velocity vector of the object relative to the rotating frame.
• $ \times $ denotes the vector cross product.

Calculating the Magnitude

The magnitude of the Coriolis acceleration is found by taking the magnitude of the cross product:

$$ |\vec{a}_c| = | -2 (\vec{\omega} \times \vec{v}) | $$

Using the property of the cross product magnitude, $ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta $, where $ \theta $ is the angle between the vectors:

$$ |\vec{a}_c| = 2 |\vec{\omega}| |\vec{v}| \sin \theta $$

In many contexts, especially when discussing the primary effect or when the velocity is perpendicular to the axis of rotation ($ \theta = 90^\circ $), $ \sin \theta = 1 $. The question asks for the magnitude, often represented in its simplified or maximum form using the given variables $ v $ for velocity magnitude ($ |v| $) and $ \omega $ for angular velocity magnitude ($ |\omega| $). Therefore, the magnitude simplifies to:

$$ |\vec{a}_c| = 2 v \omega $$

Matching with Options

Comparing this result, $ 2v\omega $, with the provided options:

• Option 1: $ 2v\omega $
• Option 2: $ 2v^2 \omega $
• Option 3: $ v\omega $
• Option 4: $ 2v\omega^2 $

The calculated magnitude $ 2v\omega $ directly matches Option 1.