Rotating frames — centrifugal force, Coriolis force

1. What is a rotating frame?

WHAT we need first: how a vector's time-derivative looks different to an observer rotating with the frame versus a fixed observer.

WHY (derive it). Imagine $\vec A$ written in the rotating basis $\{\hat e_1,\hat e_2,\hat e_3\}$: $\vec A = A_1\hat e_1 + A_2\hat e_2 + A_3\hat e_3.$ The components $A_i$ are numbers; the basis vectors themselves rotate. Differentiating in the inertial frame using the product rule: $\left(\frac{d\vec A}{dt}\right)_{\rm in} = \underbrace{\sum_i \dot A_i\hat e_i}_{(d\vec A/dt)_{\rm rot}} + \sum_i A_i \frac{d\hat e_i}{dt}.$ A unit vector rigidly rotating with $\vec\omega$ obeys $\dfrac{d\hat e_i}{dt} = \vec\omega\times\hat e_i$ (its tip traces a circle of radius $\sin\theta$). So the second sum is $\vec\omega\times\sum_i A_i\hat e_i = \vec\omega\times\vec A$. Why this step? Because rotation only reorients basis vectors — that reorientation rate is precisely $\vec\omega\times$. $\blacksquare$

2. Deriving the fictitious forces

Apply the transport theorem to the position vector $\vec r$ to get velocity, then again to get acceleration.

Step 1 — velocity. Set $\vec A=\vec r$: $\vec v_{\rm in} = \vec v_{\rm rot} + \vec\omega\times\vec r.$ Why? The inertial velocity = velocity you measure inside the frame + the velocity the frame drags you with.

Step 2 — acceleration. Apply the operator $\left(\tfrac{d}{dt}\right)_{\rm in}=\left(\tfrac{d}{dt}\right)_{\rm rot}+\vec\omega\times$ to $\vec v_{\rm in}$ (taking $\vec\omega$ constant): $\vec a_{\rm in}=\left(\frac{d}{dt}\right)_{\rm rot}\!\big(\vec v_{\rm rot}+\vec\omega\times\vec r\big) + \vec\omega\times\big(\vec v_{\rm rot}+\vec\omega\times\vec r\big).$ Expand term by term:

• $\left(\tfrac{d\vec v_{\rm rot}}{dt}\right)_{\rm rot}=\vec a_{\rm rot}$
• $\left(\tfrac{d}{dt}\right)_{\rm rot}(\vec\omega\times\vec r)=\vec\omega\times\vec v_{\rm rot}$ (since $\dot{\vec\omega}=0$)
• $+\,\vec\omega\times\vec v_{\rm rot}$
• $+\,\vec\omega\times(\vec\omega\times\vec r)$

Collecting: $\boxed{\;\vec a_{\rm in}=\vec a_{\rm rot}+2\,\vec\omega\times\vec v_{\rm rot}+\vec\omega\times(\vec\omega\times\vec r)\;}$

Step 3 — put Newton in. True forces obey $\vec F_{\rm real}=m\vec a_{\rm in}$. Solve for what the rotating observer sees, $m\vec a_{\rm rot}$: $\boxed{\;m\vec a_{\rm rot}=\vec F_{\rm real}\underbrace{-\,2m\,\vec\omega\times\vec v_{\rm rot}}_{\text{Coriolis}}\underbrace{-\,m\,\vec\omega\times(\vec\omega\times\vec r)}_{\text{centrifugal}}\;}$

3. Worked examples

4. Common mistakes

5. Active recall

Flashcards

Connections

• Newton's Second Law — fictitious forces are the price of keeping $\vec F=m\vec a$ in non-inertial frames.
• Circular Motion & Centripetal Force — centrifugal is the rotating-frame "image" of centripetal.
• Cross Product & Right-Hand Rule — every direction here comes from $\vec\omega\times$.
• Coriolis Effect in Weather — geophysical application.
• Foucault Pendulum — Coriolis force makes the swing plane precess.
• Inertial vs Non-inertial Frames — the conceptual home of this topic.

Concept Map

Hinglish (regional understanding)