Rotating frames — centrifugal force, Coriolis force

1.2.14 · Physics › Newton's Laws & Dynamics

1. Rotating frame kya hota hai?

WHAT jo pehle chahiye: kisi vector ki time-derivative, frame ke saath rotate karne wale observer aur fixed observer ko alag-alag kaise dikhti hai.

WHY (derive karo). Socho $\vec{A}$ ko rotating basis $\{{\hat{e}}_1,{\hat{e}}_2,{\hat{e}}_3\}$ mein likha gaya hai: $\vec{A}=A_1{\hat{e}}_1+A_2{\hat{e}}_2+A_3{\hat{e}}_3.$ Components $A_i$ numbers hain; basis vectors khud rotate karte hain. Inertial frame mein product rule se differentiate karne par: ${\left(\frac{d\vec{A}}{dt}\right)}_{\mathrm{in}}=\underset{(d\vec{A}/dt{)}_{\mathrm{rot}}}{\underbrace{\sum _i{\dot{A}}_i{\hat{e}}_i}}+{\sum }_i{A}_i\frac{d{\hat{e}}_i}{dt}.$ $\vec{\omega }$ ke saath rigidly rotate karta hua ek unit vector $\frac{d{\hat{e}}_i}{dt}=\vec{\omega }\times {\hat{e}}_i$ satisfy karta hai (uski tip radius $\sin \theta$ ka ek circle trace karti hai). Toh doosra sum $\vec{\omega }\times {\sum }_i{A}_i{\hat{e}}_i=\vec{\omega }\times \vec{A}$ hai. Yeh step kyun? Kyunki rotation sirf basis vectors ko reorient karta hai — woh reorientation rate precisely $\vec{\omega }\times$ hai. $■$

2. Fictitious forces derive karna

Transport theorem ko position vector $\vec{r}$ par apply karo velocity pane ke liye, phir dobara acceleration ke liye.

Step 1 — velocity. $\vec{A}=\vec{r}$ set karo: ${\vec{v}}_{\mathrm{in}}={\vec{v}}_{\mathrm{rot}}+\vec{\omega }\times \vec{r}.$ Kyun? Inertial velocity = woh velocity jo tum frame ke andar measure karte ho + woh velocity jisse frame tumhe kheenchta hai.

Step 2 — acceleration. Operator ${\left(\frac{d}{dt}\right)}_{\mathrm{in}}={\left(\frac{d}{dt}\right)}_{\mathrm{rot}}+\vec{\omega }\times$ ko ${\vec{v}}_{\mathrm{in}}$ par apply karo ($\vec{\omega }$ constant maante hue): ${\vec{a}}_{\mathrm{in}}={\left(\frac{d}{dt}\right)}_{\mathrm{rot}}({\vec{v}}_{\mathrm{rot}}+\vec{\omega }\times \vec{r})+\vec{\omega }\times ({\vec{v}}_{\mathrm{rot}}+\vec{\omega }\times \vec{r}).$ Term by term expand karo:

• ${\left(\frac{d{\vec{v}}_{\mathrm{rot}}}{dt}\right)}_{\mathrm{rot}}={\vec{a}}_{\mathrm{rot}}$
• ${\left(\frac{d}{dt}\right)}_{\mathrm{rot}}(\vec{\omega }\times \vec{r})=\vec{\omega }\times {\vec{v}}_{\mathrm{rot}}$ (kyunki $\dot{\vec{\omega }}=0$)
• $+\vec{\omega }\times {\vec{v}}_{\mathrm{rot}}$
• $+\vec{\omega }\times (\vec{\omega }\times \vec{r})$

Collect karne par: $\boxed{{\vec{a}}_{\mathrm{in}}={\vec{a}}_{\mathrm{rot}}+2\vec{\omega }\times {\vec{v}}_{\mathrm{rot}}+\vec{\omega }\times (\vec{\omega }\times \vec{r})}$

Step 3 — Newton lagao. Real forces ${\vec{F}}_{\mathrm{real}}=m{\vec{a}}_{\mathrm{in}}$ satisfy karte hain. Solve karo ki rotating observer kya dekhta hai, $m{\vec{a}}_{\mathrm{rot}}$: $\boxed{m{\vec{a}}_{\mathrm{rot}}={\vec{F}}_{\mathrm{real}}\underset{\text{Coriolis}}{\underbrace{-2m\vec{\omega }\times {\vec{v}}_{\mathrm{rot}}}}\underset{\text{centrifugal}}{\underbrace{-m\vec{\omega }\times (\vec{\omega }\times \vec{r})}}}$

3. Worked examples

4. Common mistakes

5. Active recall

Flashcards

Connections

• Newton's Second Law — fictitious forces woh price hai jo non-inertial frames mein $\vec{F}=m\vec{a}$ rakhne ke liye deni padti hai.
• Circular Motion & Centripetal Force — centrifugal, centripetal ka rotating-frame "image" hai.
• Cross Product & Right-Hand Rule — yahan har direction $\vec{\omega }\times$ se aati hai.
• Coriolis Effect in Weather — geophysical application.
• Foucault Pendulum — Coriolis force swing plane ko precess karta hai.
• Inertial vs Non-inertial Frames — is topic ka conceptual ghar.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — pictures mein derivation
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps