Non-Inertial Frames and Fictitious Forces

The Intuition of the Rotating Frame

Imagine you are standing on a merry-go-round. If you throw a ball straight ahead relative to yourself, an observer standing still on the ground sees the ball move in a straight line. However, because you are rotating, you see the ball curve away from its straight path — to the right if the merry-go-round turns anticlockwise when seen from above. This apparent curvature is not caused by a physical force pushing the ball sideways; it is a consequence of your changing perspective. In a rotating frame, the axes themselves are turning. Therefore, a vector that is constant in the inertial frame appears to change direction in the rotating frame. To make Newton's Second Law, \( \vec{F} = m\vec{a} \), hold true in this rotating frame, we must account for these apparent accelerations by adding fictitious forces to the right-hand side of the equation.

Deriving the Fictitious Forces

To derive the equations of motion in a rotating frame, we relate the time derivative of a vector in the inertial frame \( S \) to its derivative in the rotating frame \( R \). For any vector \( \vec{A} \), the relationship is given by:

Applying this to the position vector \( \vec{r} \), we find the velocity in the inertial frame is \( \vec{v}_S = \vec{v}_R + \vec{\omega} \times \vec{r} \). Differentiating again to find the acceleration \( \vec{a}_S \), and substituting into Newton's Second Law \( m\vec{a}_S = \vec{F} \), we isolate the acceleration in the rotating frame \( \vec{a}_R \). This yields the equation of motion in the rotating frame:

Here, \( \vec{F} \) represents the sum of all real physical forces acting on the particle. The three additional terms on the right-hand side are the fictitious forces.

Centrifugal and Euler Forces

The term \( -m\,\vec{\omega}\times(\vec{\omega}\times\vec{r}) \) is the centrifugal force. Using the vector triple product identity, this can be rewritten as \( m\,\omega^2 \vec{r}_{\perp} \), where \( \vec{r}_{\perp} \) is the component of the position vector perpendicular to the axis of rotation. This force always points radially outward from the axis of rotation. It is independent of the particle's velocity in the rotating frame. If the angular velocity \( \vec{\omega} \) is constant, this is the only fictitious force present (besides Coriolis if moving).

The term \( -m\,\dot{\vec{\omega}}\times\vec{r} \) is the Euler force. It arises only when the angular velocity of the frame is changing with time. It is analogous to the tangential force experienced in linear acceleration. If the rotation rate is constant, \( \dot{\vec{\omega}} = 0 \), and this force vanishes. In most steady-state rotating systems, such as the Earth's rotation, this term is negligible.

The Coriolis Force

The term \( -2m\,\vec{\omega}\times\vec{v}_R \) is the Coriolis force. Unlike the centrifugal force, this force depends on the velocity of the particle relative to the rotating frame. It is always perpendicular to both the angular velocity vector \( \vec{\omega} \) and the relative velocity \( \vec{v}_R \). Consequently, the Coriolis force does no work on the particle, as the power delivered is \( \vec{F}_{cor} \cdot \vec{v}_R = 0 \). It acts to deflect the trajectory of moving objects. In the Northern Hemisphere, where \( \vec{\omega} \) has a component pointing upward, a moving object is deflected to the right of its direction of motion. In the Southern Hemisphere, the deflection is to the left. This effect is crucial in meteorology and oceanography, governing the formation of cyclones and the direction of ocean currents.