Newton's laws hold in inertial frames. In non-inertial frames (rotating or accelerating), an object accelerates with no apparent force — inconsistent with F = ma.
Fix: introduce "fictitious forces" that explain those apparent accelerations.
Linearly accelerating frame — a linear fictitious force
Scenario: a bus accelerating at \vec a_0 (forward). A standing passenger (unbraced). From outside (inertial): passenger stationary on the ground, bus accelerates — passenger moves backward relative to the bus.
From inside the bus: passenger accelerated backward with no visible force. To keep F = ma, we introduce a fictitious force -m\vec a_0.
Interpretation: the fictitious force is always opposite the frame's acceleration, with magnitude m|\vec a_0|.
Rotating frame — centrifugal force
Scenario: a rotating disc with a coin on it. Disc rotates at angular velocity \omega. Coin at distance r.
From external (inertial) frame: coin in circular motion, static friction provides centripetal force (f_s = m\omega^2 r).
From rotating frame (sitting on disc): coin is at rest. Some force must balance friction. That's the "centrifugal force":
\[ \vec F_\text{centrifugal} = m\omega^2 r\ \hat r \]
(pointing outward, opposite the centripetal)
Note: centrifugal force doesn't exist in an inertial frame. It's a computational tool for rotating frames only.
Rotating frame — Coriolis force
If an object is moving in the rotating frame, we have a second fictitious force — the Coriolis:
\[ \vec F_\text{Coriolis} = -2m\vec \omega \times \vec v_\text{in frame} \]
Interpretation: perpendicular to the object's velocity in the rotating frame and perpendicular to the rotation axis.
Earth-scale applications
Earth rotates at \omega = 2\pi/86164\ s \approx 7.29 \times 10^{-5}\ rad/s (small but important).
1. Storms rotate. In the northern hemisphere, winds around low-pressure systems rotate counterclockwise — due to Coriolis on air heading toward the low.
2. Ocean currents. Global currents deflect due to Coriolis (Gulf Stream, etc.).
3. Eastward deflection of falling objects: an object released from a high tower falls slightly east of straight down (the tower rotates faster than the surface).
4. Foucault pendulum: a long pendulum in Paris shows its swing plane gradually rotates — a direct proof of Earth's rotation.
A few notes and common mistakes
1. A fictitious force isn't "real" physically.
No other body exerts it. It's a mathematical trick to preserve F = ma in non-inertial frames.
2. No third-law pair. If a force is fictitious, there's no partner object exerting a reaction.
3. Centrifugal only in rotating frames. From outside, all is explained by the centripetal.
4. Coriolis only for moving objects in a rotating frame. Stationary objects don't feel it.
What you should be able to do
- Distinguish inertial vs non-inertial frames
- Apply the linear fictitious force in accelerating frames
- Apply the centrifugal force in rotating frames
- Apply the Coriolis force to moving objects in a rotating frame
- Recognize Earth-scale examples (storms, ocean currents)
Preview of §6.5
Air resistance — a velocity-dependent force. Terminal velocity, skydiver dynamics, and the Zeno-like paradox.
📚 See also: Halliday Vol 1, Ch 6, §6.6.
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