Atwood machine — derivation — Hinglish notes

1.2.10 · Physics › Newton's Laws & Dynamics

Atwood machine KYA hai?

Idealisations matter karti hain — chaliye batate hain kyun:

Massless string → tension iske saath har jagah same hoti hai. (Ek massive string ko khud ko accelerate karne ke liye net force chahiye hogi, isliye $T$ dono ends par alag hogi.)
Inextensible string → dono masses ki same acceleration magnitude hoti hai. Agar $m_{1}$ $x$ se neeche jaata hai, toh $m_{2}$ $x$ se upar aata hai.
Massless, frictionless pulley → yeh sirf rope ko redirect karta hai; yeh koi bhi force store/rob nahi karta, isliye tension pulley ke across unchanged rehti hai.

Ise (first principles se) KAISE derive karein

Hum sirf Newton's second law, $F_{n e t} = ma$ , ko har mass par alag-alag use karte hain. Yeh Derivation-from-scratch hai.

Step 1 — Choose karein ki yeh kis direction mein accelerate karta hai. Assume karein $m_{1} > m_{2}$ , isliye $m_{1}$ neechay jaata hai aur $m_{2}$ upar jaata hai. Yeh step kyun? Hume ek sign convention commit karni hogi. Hum har mass ke liye "motion ki direction" ko positive define karenge — agar hum galat guess karein, toh $a$ negative aayega, jo khud ko theek kar lega.

Step 2 — Har mass par forces. Har mass do forces feel karta hai: gravity $m g$ (neechay) aur tension $T$ (upar, rope hamesha pulley ki taraf kheenchti hai).

$m_{1}$ ke liye (neechay accelerate karta hai, neechay ko + lein): $m_{1} g - T = m_{1} a (1)$ Yeh step kyun? Yahaan neechay positive hai, isliye neechay wala weight $+ m_{1} g$ hai aur upar wali tension $- T$ hai. Net = mass × acceleration.

$m_{2}$ ke liye (upar accelerate karta hai, upar ko + lein): $T - m_{2} g = m_{2} a (2)$ Yeh step kyun? Yahaan upar positive hai, isliye tension $+ T$ hai, weight $- m_{2} g$ hai. Same $a$ hai kyunki rope inextensible hai.

Step 3 — $T$ eliminate karne ke liye dono equations add karein. $(m_{1} g - T) + (T - m_{2} g) = m_{1} a + m_{2} a$ $m_{1} g - m_{2} g = (m_{1} + m_{2}) a$ Yeh step kyun? Unknown $T$ , $- T$ aur $+ T$ ke roop mein appear hota hai; add karne par yeh cancel ho jaata hai, sirf $a$ bachta hai.

Step 4 — Tension find karein. $a$ ko (2) mein substitute karein: $T = m_{2} (g + a)$ . $T = m_{2} (g + \frac{( m _{1} - m _{2} ) g}{m _{1} + m _{2}}) = m_{2} \cdot \frac{( m _{1} + m _{2} ) g + ( m _{1} - m _{2} ) g}{m _{1} + m _{2}}$

WORKED EXAMPLES

COMMON MISTAKES

Flashcards

Inextensible string do masses ke baare mein kya guarantee deti hai?

Unki equal acceleration magnitude hoti hai (aur equal-but-opposite displacement/velocity).

Massless string kya guarantee deti hai?

Tension rope ke har point par same hoti hai.

Atwood machine ka acceleration?

a = \frac{( m _{1} - m _{2} ) g}{m _{1} + m _{2}}

Atwood machine mein tension?

T = \frac{2 m _{1} m _{2}}{m _{1} + m _{2}} g

Derivation ke dauran do Newton equations kyun add karte hain?

Unknown tension

T

eliminate karne ke liye (yeh

+ T

aur

- T

ke roop mein appear hota hai).

Check: agar

m_{1} = m_{2}

, toh

a

aur

T

kya hain?

a = 0

aur

T = m g

Check: agar

m_{2} = 0

, toh

a

aur

T

kya hain?

a = g

(free fall) aur

T = 0

Tension bhaari mass ke weight se kam kyun hoti hai?

Kyunki

m_{1}

neechay accelerate kar raha hai, isliye uस par net force neechay hai →

T < m_{1} g

System ki driving force kya hai?

Weight difference

(m_{1} - m_{2}) g

Us force ko kaunsi inertia resist karti hai?

Total mass

(m_{1} + m_{2})

Recall Feynman: 12-saal ke bacche ko explain karein

Ek wheel ke upar rope imagine karein jiske dono ends par ek-ek bucket ho. Agar dono buckets ka weight same ho, kuch nahi hoga. Agar aap ek bucket mein patthar daalein, toh wo side neechay jaayegi aur doosri upar aayegi — lekin wo baandhe hue hain, isliye same speed se move karte hain. Rope dono buckets ko same strength se upar kheenchti hai (kyunki rope halki hai aur wheel smooth hai). Yeh kitni tezi se speed up karta hai yeh depend karta hai ki aapne kitna extra weight daala compare kiya total cheez se jo aap move karne ki koshish kar rahe ho. Zyada extra weight → faster; zyada total cheez → slower (start karna mushkil).

Connections

Newton's Second Law — puri derivation ka engine.
Free Body Diagrams — har mass ko kaise isolate karte hain.
Tension in Strings — massless rope mein $T$ uniform kyun hoti hai.
Constraint Relations — inextensible string ⇒ equal accelerations.
Inclined Plane Problems — same method gravity ke component ke saath.
Pulley Systems & Mechanical Advantage — multi-pulley generalisation.
Harmonic Mean — tension expression ki form.

Atwood machine KYA hai?

Ise (first principles se) KAISE derive karein

WORKED EXAMPLES

COMMON MISTAKES

Flashcards

Connections

Concept Map