10.14:
Coriolis Force
A ball sliding on a turntable follows a straight-line trajectory. The force acting on the object in this inertial frame equals the product of mass and acceleration.
When the turntable starts rotating counterclockwise, the sliding ball's trajectory changes. The equation of motion in this non-inertial frame is modified to include the particle's acceleration in the rotating reference frame.
Fictitious forces, including Coriolis and centrifugal forces, act additionally in a rotating frame modifying the equation of motion.
The Coriolis force is proportional to the cross-product of the rotating frame's angular velocity and the object's velocity.
The cross-product right-hand rule gives the direction of the Coriolis force. Consider the palm of the right hand aligning along the rotating frame's angular velocity, and the fingers curl toward the object's velocity. The thumb points opposite the Coriolis force, as indicated by the negative sign in the Coriolis force expression.
So, a ball in a counterclockwise rotating turntable follows a circular trajectory toward the right. If the table rotates clockwise, the ball deflects toward the left.
10.14:
Coriolis Force
An accelerating particle experiences a force equal to the mass multiplied by the acceleration in an inertial frame of reference. Consider a particle in a non-inertial frame of reference, such as a sliding ball on a rotating table. The acceleration of the ball in this rotating reference frame is different than in the intertial frame, which modifies its equation of motion. The fictitious forces acting additionally on a rotating frame of reference alter Newton's Second Law expression. Centripetal and Coriolis forces are the fictitious or pseudo forces that act on a particle in a non-inertial frame of reference.
The discovery of the Coriolis force dates back to 1651 when Italian military officers reported that, during artillery practice, the cannon balls always landed to the right of where the calculations predicted. The Coriolis force is named after a French mathematician Gaspard Gustave Coriolis (1792–1843) for his work in the field of rotating reference frames.
The Coriolis force is proportional to the cross-product of the rotating frame's angular velocity vector and the object's velocity vector. Its direction can be estimated using the right-hand rule for the cross-product. Suppose the index finger and thumb are directed toward the object's velocity and the rotating frame's angular velocity, respectively. The middle finger points opposite to the Coriolis force, as indicated by the negative sign in the Coriolis force expression. The work done by Coriolis force is zero since it always acts perpendicular to the object's motion.
Due to the Earth's rotation, the Coriolis force tends to deflect moving bodies toward the right in the northern hemisphere and toward the left in the southern hemisphere. As a result, long-range snipers aim to the left of their target in the northern hemisphere.
The Coriolis force also results in cyclone formation. In the northern hemisphere, air rushing out of a high-pressure system swirls clockwise around the pressure center. Similarly, air flowing into a low-pressure region swirls counterclockwise. The turn direction of the swirling is opposite in the southern hemisphere due to the Coriolis force. The Coriolis force is zero at the equator, and as a result, no cyclones occur at the equatorial axis.
Suggested Reading
- Kleppner, Daniel and Kolenkow, Robert, 2nd Edition. An Introduction to Mechanics. San Francisco, Cambridge University Press pp. 359