6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force

Fictitious Forces and Non-inertial Frames

Fictitious forces show up when you try to apply Newton's laws inside an accelerating or rotating reference frame. They aren't caused by any physical interaction between objects, but you need them to correctly describe what you observe from within that frame. The Coriolis force is the most well-known example, and it explains phenomena from hurricane rotation to the deflection of long-range projectiles.

Inertial vs. Non-inertial Frames

An inertial frame is any reference frame where Newton's laws work as written, with no modifications needed. That means no net force acts on the frame itself. A lab bench sitting on the ground or a train cruising at constant velocity both count as inertial frames.

A non-inertial frame is one that's accelerating or rotating relative to an inertial frame. Inside these frames, you'll observe motions that can't be explained by real forces alone. To make Newton's second law work, you have to introduce fictitious forces that account for the frame's own acceleration.

Common non-inertial frames:

• A car that's speeding up or braking
• A spinning merry-go-round
• Earth's surface (because Earth rotates about its axis)

The core distinction: in an inertial frame, every observed acceleration traces back to a real force (gravity, friction, tension, etc.). In a non-inertial frame, some of the observed acceleration comes from fictitious forces that exist only because the frame itself is accelerating. These fictitious forces are not caused by any object pushing or pulling on anything.

Origin of the Coriolis Force

The Coriolis force is a fictitious force that appears in rotating reference frames. It acts perpendicular to both the object's velocity and the rotation axis, making moving objects appear to curve even when no real sideways force is acting on them.

On Earth's surface, this plays out in a specific pattern. Earth rotates from west to east, so:

• In the Northern Hemisphere, moving objects are deflected to the right of their direction of travel.
• In the Southern Hemisphere, moving objects are deflected to the left.

The magnitude of the Coriolis force depends on three things: the object's speed, the latitude (the effect is strongest at the poles and zero at the equator), and Earth's angular rotation rate ($\omega = 7.29 \times 10^{-5}$ rad/s).

The Coriolis force is given by:

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

where:

• $m$ = mass of the moving object
• $\vec{\omega}$ = angular velocity vector of the rotating frame (points along the rotation axis, toward the North Pole for Earth)
• $\vec{v}$ = velocity of the object relative to the rotating frame
• $\times$ = the cross product, which is why the resulting force is perpendicular to both $\vec{\omega}$ and $\vec{v}$

The negative sign and cross product together determine the direction of deflection. You don't need to compute full cross products for this course, but you should understand that the force always acts sideways relative to the object's motion.

Effects of the Coriolis Force

Weather patterns. The Coriolis force shapes large-scale wind circulation on Earth:

• Near the equator, trade winds are deflected westward (toward the west).
• At mid-latitudes, prevailing westerlies are deflected eastward.
• Cyclones and hurricanes rotate counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. This is why Hurricane Katrina (Northern Hemisphere) and Cyclone Yasi (Southern Hemisphere) spun in opposite directions.

Projectile motion. For short-range projectiles, the Coriolis deflection is negligible. But for long-range projectiles like artillery shells or ballistic missiles, the deflection becomes significant. A shell fired northward in the Northern Hemisphere will drift to the right of its intended target. The amount of deflection depends on the projectile's speed, latitude, and total flight time. Military targeting systems routinely correct for this.

Ocean currents and navigation. The Coriolis force helps drive major ocean currents like the Gulf Stream. It also creates geostrophic flow, where pressure gradients and the Coriolis force balance each other, producing the large-scale circulation patterns in both the atmosphere and the oceans. Ships and aircraft on long-distance routes (great circle routes) must account for Coriolis effects in their navigation.

Demonstrating and Measuring Coriolis Effects

The Foucault pendulum is the classic demonstration of Earth's rotation and the Coriolis effect. A long pendulum, once set swinging, gradually appears to change its plane of oscillation over the course of hours. The pendulum's swing plane actually stays fixed in an inertial frame; it's the Earth rotating underneath it that creates the apparent rotation. At the poles, the plane completes a full rotation in 24 hours. At other latitudes, it takes longer.

The underlying concept is relative motion: what looks like a deflection from within the rotating frame is really just straight-line (inertial) motion viewed from a rotating perspective. Objects tend to maintain their velocity due to inertia, and when the frame rotates beneath them, the result looks like a sideways push. That "push" is the Coriolis force.