Acceleration Addition. Coriolis Acceleration

In relative particle motion with a rotating reference frame, absolute acceleration is not merely the sum of relative and transport accelerations. An additional term appears: Coriolis acceleration.

Acceleration-addition theorem

For a particle moving relative to a moving reference frame, absolute acceleration is $\vec a_a=\vec a_r+\vec a_e+\vec a_C$, where $\vec a_r$ is relative acceleration, $\vec a_e$ is transport acceleration, and $\vec a_C$ is Coriolis acceleration.

Coriolis acceleration

For a particle moving relative to a reference frame rotating with angular velocity $\vec\omega_e$, the Coriolis acceleration is:

$$\vec a_C=2\vec\omega_e\times\vec v_r.$$

Its magnitude is:

$$a_C=2\omega_e v_r\sin\theta,$$

where $\theta$ is the angle between $\vec\omega_e$ and relative velocity $\vec v_r$. Its direction follows from the cross-product right-hand rule.

When Coriolis acceleration is zero

The term $\vec a_C$ vanishes if the moving frame does not rotate ($\omega_e=0$), if the particle has no relative velocity ($v_r=0$), or if $\vec v_r$ is parallel or antiparallel to $\vec\omega_e$.

Transport acceleration

For a moving frame with origin $O'$, the transport acceleration of the coincident frame point contains origin acceleration, tangential rotational acceleration, and centripetal acceleration: $\vec a_e=\vec a_{O'}+\vec\varepsilon_e\times\vec r+\vec\omega_e\times(\vec\omega_e\times\vec r)$.

Direction of Coriolis acceleration

The direction of $\vec a_C$ follows from $2\vec\omega_e\times\vec v_r$. In planar problems, first determine the direction of $\vec\omega_e$ by the right-hand rule and then take its cross product with $\vec v_r$.

Example

A disk rotates with $\omega_e=4$ rad/s while a slider moves radially in a slot at $v_r=0.5$ m/s. Here $\theta=90^\circ$, so $a_C=2\cdot4\cdot0.5=4$ m/s². Its direction lies in the disk plane and is perpendicular to the radial relative-velocity direction.

Common mistakes

• omitting the factor 2 in the Coriolis formula;
• using absolute velocity instead of relative velocity $\vec v_r$;
• adding acceleration magnitudes without accounting for direction;
• including a Coriolis term when the moving frame undergoes pure translation;
• reversing the cross-product order $\vec\omega_e\times\vec v_r$.