Visual walkthrough — Non-inertial reference frames — pseudo forces

1.2.13 · D2 · Physics › Newton's Laws & Dynamics › Non-inertial reference frames — pseudo forces

Step 1 — Do watchers, ek ball

KYA. Ek smooth (frictionless) bus floor imagine karo jisme ek choti ball rakhi hai. Ek watcher, G, ground par khada hai. Doosra watcher, P, bus ke andar baitha hai. Bus abhi speed up karne wali hai.

KYUN. Kisi bhi algebra se pehle hume bilkul clear hona chahiye ki kaun kya measure karta hai. Ek "reference frame" bas ek chosen watcher plus unka ruler aur stopwatch hota hai. Alag-alag watchers, alag-alag tarike se move karte hue, ek cheez ki motion ke baare mein alag-alag bol sakte hain — aur yahi disagreement pseudo force ka poora beej hai.

PICTURE. Ground-watcher G road pe fixed hai. Bus-watcher P bus pe fixed hai. Ball dono ke beech baitha hai, abhi kuch nahi kar raha.

Step 2 — Positions ko arrows se naam do

KYA. Ground origin se teen arrows draw karo:

• $\vec{R}$ — G ke origin se P ke origin tak (jahan bus hai).
• $\vec{r}{}^{\prime }$ — P ke origin se ball tak (jo P measure karta hai).
• $\vec{r}$ — G ke origin se ball tak (jo G measure karta hai).

KYUN. Do watchers ko compare karne ke liye humein ek honest bookkeeping equation chahiye jo unke measurements ko link kare. Ek position "arrow" (ek vector: ek arrow jisme length aur direction dono hain) humein journeys ko tip-to-tail add karne deta hai. G→bus→ball jaana us same ball par land karna chahiye jaise G→ball seedha jaana.

PICTURE. Do chote arrows $\vec{R}$ aur $\vec{r}{}^{\prime }$ tip-to-tail rakhne par bade arrow $\vec{r}$ ke barabar hain.

Step 3 — Acceleration ke baare mein pucho, aur kyun do baar

KYA. Arrow equation ko time $t$ ke saath respect mein do baar differentiate karo: $\vec{r}=\vec{R}+\vec{r}{}^{\prime }\stackrel{\frac{d^2}{d{t}^2}}{\to }\vec{a}={\vec{a}}_0+\vec{a}{}^{\prime }$ jahan $\vec{a}=\ddot{\vec{r}}$ ball ka acceleration hai jaise G dekhta hai, ${\vec{a}}_0=\ddot{\vec{R}}$ bus ka acceleration hai, aur $\vec{a}{}^{\prime }=\ddot{\vec{r}}{}^{\prime }$ ball ka acceleration hai jaise P dekhta hai.

KYUN yeh tool — second derivative? Newton ka law acceleration ki baat karta hai, position ki nahi. Tool "$\frac{d}{dt}$" yeh sawaal ka jawaab deta hai: "abhi yeh kitni tezi se change ho raha hai?" Position batata hai kahan; uski first derivative velocity deti hai (position ka change); uski second derivative acceleration deti hai (velocity ka change). Hume acceleration chahiye, isliye hum change-question do baar poochte hain. Hum derivatives use karte hain naa ki, say, ek simple difference, kyunki bus ka speed-up ek instant se doosre instant tak vary kar sakta hai — hume instantaneous rate chahiye.

KYUN yeh sirf ek sum hai. Differentiation linear hai: ek sum ka change, changes ka sum hota hai. Toh teen arrows do derivatives ke baad bhi wohi tip-to-tail shape rakhte hain. Velocity term $\dot{\vec{r}}$ aur uske constant pieces drop hote hain ya cleanly carry through karte hain, sirf pure accelerations bachti hain.

PICTURE. Wohi tip-to-tail triangle, ab har arrow ko acceleration ke roop mein relabel kiya gaya hai.

Step 4 — Sirf G ko Newton use karne do (usse allowed hai)

KYA. G inertial hai, isliye G ke liye honest law sirf real forces ke saath valid hai: ${\vec{F}}_{\text{real}}=m\vec{a}.$ Yahan ${\vec{F}}_{\text{real}}$ un forces ka sum hai jo actually ball ko touch kar rahe hain ya pull kar rahe hain (gravity, floor, string...), aur $m$ ball ka mass hai — uski accelerate hone ki resistance.

KYUN. "Inertial" hone ka poora matlab yahi hai ki $\vec{F}=m\vec{a}$ wahan bina kisi fudge ke sach hai. Ek smooth floor par koi horizontal real force nahi hai, toh G predict karta hai $\vec{a}=0$ horizontally: ball wahi raha. Yahi hamara truth ka anchor hai; P ki duniya isse consistent honi chahiye.

PICTURE. G ka view: ball still hai, bus uske neeche se aage nikal rahi hai. Ball ko koi horizontal arrow touch nahi kar raha.

Step 5 — Substitute karo, taaki P ki acceleration aaye

KYA. Step 3 ka $\vec{a}={\vec{a}}_0+\vec{a}{}^{\prime }$ Step 4 ke law mein daalo: ${\vec{F}}_{\text{real}}=m({\vec{a}}_0+\vec{a}{}^{\prime })=m{\vec{a}}_0+m\vec{a}{}^{\prime }.$

KYUN. Hum $\vec{a}{}^{\prime }$ ke terms mein likhi ek equation chahte hain — woh acceleration jo P actually dekhta hai — kyunki P hi confused hai. Toh hum G ke measure ki hui quantity ($\vec{a}$) ko un pieces se swap karte hain jo P ki quantity include karti hain.

PICTURE. Real-force arrow ko do contributions mein split hota dikhaya gaya hai: ek jo bus ki motion "pay karta hai" ($m{\vec{a}}_0$) aur ek jo P ke ball-motion ke liye bacha hai ($m\vec{a}{}^{\prime }$).

Step 6 — Rearrange karo, aur bache hue ko naam do

KYA. $m{\vec{a}}_0$ ko left side le jao: $m\vec{a}{}^{\prime }={\vec{F}}_{\text{real}}-m{\vec{a}}_0.$ P chahta hai yeh Newton's law jaisa dikhe: "(mass)×(jo acceleration main dekhta hun) = (total force jo main feel karta hun)". Right side aise hi padha jaata hai agar hum extra piece ko naam den: $\boxed{{\vec{F}}_{\text{pseudo}}=-m{\vec{a}}_0}$

KYUN. P insist karta hai bus ke andar $\vec{F}=m\vec{a}{}^{\prime }$ use karne par — yeh aadat hai aur convenient bhi hai. Iss aadat ko honest rakhne ka ek hi tarika hai ki ek extra term maano. Yeh term naya physics ka discovery nahi hai; yeh ek accelerating watcher use karne ki keemat hai. Minus sign iski core hai: bus jis bhi direction mein accelerate kare (${\vec{a}}_0$), yeh invented force opposite direction mein point karti hai.

PICTURE. P ki nazar mein ball peeche slide karti hai; hum backward pseudo-force arrow $-m{\vec{a}}_0$ seedha ball par draw karte hain, bilkul forward bus-acceleration arrow ke opposite.

Step 7 — Smooth-bus numbers check karo

KYA. Smooth floor ⇒ ${\vec{F}}_{\text{real}}=0$ horizontally. $a_0=2{\text{m/s}}^2$ aur $m=0.5\text{kg}$ ke saath: $F_{\text{pseudo}}=-m{a}_0=-(0.5)(2)=-1\text{N},a{}^{\prime }=\frac{F_{\text{pseudo}}}{m}=\frac{-1}{0.5}=-2{\text{m/s}}^2.$

KYUN. P predict karta hai ki ball $2{\text{m/s}}^2$ peeche accelerate karti hai — exactly woh peeche ka slide jo P dekhta hai. Aur yeh G se match karta hai, jo kehta hai ball still hai jabki bus $2{\text{m/s}}^2$ aage surges karti hai (toh bus relative mein ball $-2{\text{m/s}}^2$ par peechhe jaati hai). Do watchers, ek reality. ✓

PICTURE. Side-by-side: G ka frame (ball fixed, bus arrow aage) versus P ka frame (ball arrow peeche), dono accelerations magnitude mein equal aur opposite.

Step 8 — Har case: ${\vec{a}}_0$ ka sign, aur degenerate frame

KYA. Har situation ke liye ${\vec{F}}_{\text{pseudo}}=-m{\vec{a}}_0$ padho:

KYUN. Formula ek hi rule hai jo sab cover karta hai kyunki har case bas ek alag arrow ${\vec{a}}_0$ hai jo $-m{\vec{a}}_0$ mein jaata hai. Degenerate case notice karo: agar bus accelerate karna band kar de (${\vec{a}}_0=0$), toh invented force gayab ho jaati hai aur P ka repaired law ordinary Newton par wapas aa jaata hai. Yahi sanity check hai ki humne kuch spurious invent nahi kiya.

PICTURE. ${\vec{a}}_0$ arrows ka ek dial (aage, peeche, upar, neeche, zero) har ek ke saath uska opposite pseudo-force arrow, aur zero case empty dikhaya gaya.

Ek picture mein poori summary

KYA. Ek figure poori kahani rakhti hai: tip-to-tail acceleration triangle ($\vec{a}={\vec{a}}_0+\vec{a}{}^{\prime }$), G ka honest ${\vec{F}}_{\text{real}}=m\vec{a}$, aur woh flip jo $-m{\vec{a}}_0$ ko P ki duniya mein ball par backward pseudo force mein badalta hai.

Active recall

Connections

• Parent — pseudo forces — yeh page uski visual derivation hai.
• Newton's Second Law — woh law jise hum rakhne par insist karte hain, jo invention ko force karta hai.
• Newton's Third Law — pseudo force ise tod deti hai (Step 6: koi partner nahi).
• Apparent weight & normal force — Step 8 table ki up/down rows.
• Centrifugal force — same trick with ${\vec{a}}_0$ centre ki taraf point karte hue.
• Coriolis force — rotating frames mein velocity-dependent cousin.
• Galilean relativity — constant-velocity family define karta hai jahan ${\vec{a}}_0=0$.