Non-inertial reference frames — pseudo forces

1.2.13 · Physics › Newton's Laws & Dynamics

Pseudo forces ki zaroorat kyun hai?

Kya problem hai? Socho ek ball ek bus ke smooth floor par rakhi hai. Bus achanak ${\vec{a}}_0$ se aage ki taraf accelerate karti hai. Zameen par khada ek aadmi (inertial) dekhta hai: ball par koi horizontal force nahi $\Rightarrow$ ball wahi ruki rehti hai. Lekin bus ke andar baitha passenger dekhta hai ki ball peeche ki taraf slide karti hai — woh unke relative accelerate karti hai! Agar woh naively $\vec{F}=m\vec{a}$ apply kare, toh unhe acceleration milta hai lekin koi real force nahi jo usse cause kare. Newton toot-ta hua lagta hai.

Hum ise kaise theek karte hain? Hum ek fake force add karte hain taaki math close ho sake. Chaliye ise derive karte hain.

Scratch se Derivation

Maano frame $S$ inertial hai aur frame $S^{\prime }$, $S$ ke relative ${\vec{a}}_0$ se accelerate karta hai. Mass $m$ ke ek particle ki positions $\vec{r}$ ($S$ mein) aur $\vec{r}{}^{\prime }$ ($S^{\prime }$ mein) hain, $S^{\prime }$ ka origin $\vec{R}$ par hai:

$\vec{r}=\vec{R}+\vec{r}{}^{\prime }$

Time ke saath do baar differentiate karo:

${\vec{a}}_{\text{in }S}={\vec{a}}_0+{\vec{a}}_{\text{in }{S}^{\prime }}$

Yeh step kyun? Acceleration simply add hoti hai; frame ki apni acceleration ${\vec{a}}_0=\ddot{\vec{R}}$ shift karti hai ki har observer kya measure karta hai.

Inertial frame mein Newton sirf real forces ke saath hold karta hai:

${\vec{F}}_{\text{real}}=m{\vec{a}}_{\text{in }S}=m({\vec{a}}_0+{\vec{a}}_{\text{in }{S}^{\prime }})$

Ab passenger kya measure karta hai, ${\vec{a}}_{\text{in }{S}^{\prime }}$, uske liye solve karo:

$m{\vec{a}}_{\text{in }{S}^{\prime }}={\vec{F}}_{\text{real}}-m{\vec{a}}_0$

Yeh step kyun? Humne rearrange kiya taaki right side dekhe "total force = mass × (woh acceleration jo passenger dekhta hai)". Extra term $-m{\vec{a}}_0$ ek force ka role play karta hai.

Worked examples

Common mistakes

Active recall

Connections

• Newton's Second Law — pseudo forces exist karte hain $\vec{F}=m\vec{a}$ ko accelerating frames mein preserve karne ke liye.
• Newton's Third Law — exception: pseudo forces ise break karte hain (koi reaction nahi).
• Centrifugal force — rotating frame mein pseudo force (${\vec{a}}_0$ centripetal).
• Coriolis force — rotating frames mein velocity-dependent pseudo force.
• Apparent weight & normal force — lift problems pseudo-force problems hain.
• Galilean relativity — inertial frames ki family define karta hai.

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