Foundations — Rotating frames — centrifugal force, Coriolis force

1.2.14 · D1 · Physics › Newton's Laws & Dynamics › Rotating frames — centrifugal force, Coriolis force

Is page par assume kiya gaya hai ki tumhe kuch nahi pata. Hum parent note ke har symbol ko build karenge, ek aisi order mein jahan har brick pehle wali brick par tikti hai.

0. "Frame" hota kya hai

Socho do log ek hi ball dekh rahe hain: ek zameen par khada hai, ek spinning platform par sawaar hai. Ek hi ball, do alag kahaniyan. Yeh poora topic un dono kahanion ke beech translate karne ke baare mein hai.

Topic ko yeh kyun chahiye: Newton ka rule sirf inertial frames mein hi guarantee se kaam karta hai. Ek rotating frame non-inertial hai, isliye humein rule ko patch karna hoga — wahi patch hai centrifugal + Coriolis.

1. Vectors aur chhota arrow $\stackrel{⃗}{}$

• $\vec{r}$ — position vector: origin se object ki jagah tak ka arrow. Iska length origin se door ki hai.
• $\vec{v}$ — velocity vector: object kitni tez aur kis direction mein move kar raha hai.
• $\vec{a}$ — acceleration vector: velocity kitni tez change ho rahi hai.
• $\vec{F}$ — force vector: ek push ya pull, strength aur direction ke saath.

2. Magnitude bars $|\vec{A}|$ aur unit vectors $\hat{e}$

Koi bhi vector = (uski length) × (uski direction signpost): $\vec{A}=|\vec{A}|\hat{A}$.

Parent note ek spinning disk par teen special signposts use karta hai:

• $\hat{r}$ — radially outward, centre se seedha door ki taraf point karta hai.
• $\hat{\theta }$ — sideways, spin ki direction mein (circle ka tangent).
• $\hat{z}$ — spin axis ke upar, disk se bahar ki taraf.

Topic ko yeh kyun chahiye: centrifugal force $\hat{r}$ ke along point karta hai; radially-thrown ball par Coriolis $\hat{\theta }$ ke along point karta hai. Named signposts hone se hum bol sakte hain ki har fictitious force kis taraf push karta hai.

3. Angular velocity $\vec{\omega }$

Topic ko yeh kyun chahiye: $\vec{\omega }$ woh input hai jo rotating frame ko describe karta hai. Dono fictitious forces $\vec{\omega }$ se bani hain.

4. Cross product $\vec{\omega }\times \vec{A}$

Yeh poore topic ka engine hai, isliye hum ise dheere dheere build karte hain. Full treatment: Cross Product & Right-Hand Rule.

Do facts jo parent use karta hai, ab mysterious nahi:

• $\hat{z}\times \hat{r}=\hat{\theta }$ — spin axis ko "outward" se cross karo to "sideways" milta hai. (Dono ke perpendicular, right-hand rule.)
• $\sin \varphi =0$ jab $\vec{A}\parallel \vec{B}$: agar do arrows parallel hain, unka cross product zero hai. Isliye axis ke along velocity ko koi Coriolis feel nahi hota.

5. Newton's Second Law $\vec{F}=m\vec{a}$

Topic ko yeh kyun chahiye: yeh woh rule hai jise hum chhodna nahi chahte. Kyunki yeh sirf inertial frame mein hold karta hai, rotating observer ka kaam hai ise $m{\vec{a}}_{\text{rot}}={\vec{F}}_{\text{real}}+(\text{fictitious terms})$ mein rearrange karna taaki wahi jaana-pehchaana equation spin ke andar bhi kaam kare.

6. Centripetal force (woh asli inward pull)

Yahan $r_{\perp }$ (padho "r-perp") ka matlab hai spin axis se perpendicular distance — tum pole se kitni door ho, origin se kitni door nahi. Flat disk par dono same hain; globe par alag hain.

7. Derivative aur $d/dt$ — "rate of change"

Parent likhta hai ${\left(\frac{d\vec{A}}{dt}\right)}_{\text{inertial}}$ aur ${\left(\frac{d\vec{A}}{dt}\right)}_{\text{rot}}$: wahi arrow, ground observer vs spinning observer ki nazar se uski change-per-second. Yeh alag hote hain kyunki spinning observer ke apne signposts ghoom rahe hain.

Yeh sab topic mein kaise feed hota hai

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