Exercises — Newton's first law — inertia, operational definition of force

Law kya kehta hai: velocity constant $⟺$ net force zero. Puck seedhi line mein fixed speed se chal raha hai, toh uski velocity change nahi ho rahi. KYUN: "Koi touch nahi kar raha" aur "frictionless" dono milke matlab hai kuch bhi nahi bacha jo motion ko change kare. Isliye ${\vec{F}}_{\text{net}}=0$, aur speed $4\text{ m/s}$ hi rehti hai. Das seconds baad bhi $4\text{ m/s}$ hi hai.

Test lagao: kya velocity vector constant hai?

• (a) speed up ho raha hai → speed change ho rahi hai → velocity change ho rahi hai → force act kar raha hai.
• (b) rest mein hai → velocity constant hai (zero hai, jo phir bhi constant hai) → ${\vec{F}}_{\text{net}}=0$. ✓
• (c) constant speed hai lekin circular hai → direction change ho raha hai → velocity change ho rahi hai → force act kar raha hai (yeh classic trap hai; neeche dekho).
• (d) straight line, steady speed → velocity constant → ${\vec{F}}_{\text{net}}=0$. ✓ Answer: (b) aur (d).

WHAT: block rest mein hai, toh $\vec{v}$ constant hai, toh first law se ${\vec{F}}_{\text{net}}=0$. KYUN vertical balance: sirf do vertical forces hain — gravity ($19.6\text{ N}$ neeche) aur normal push $N$ (upar). Unka sum zero hone ke liye: $N-19.6=0\Rightarrow N=19.6\text{ N (upar)}.$ First law ne humein $N$ directly nahi bataya — usne bataya ki forces cancel hone chahiye, aur uss se humne $N$ solve kiya.

Constant velocity $⟹{\vec{F}}_{\text{net}}=0⟹$ net force $=0$. Horizontally sirf do forces hain — thrust (forward) aur drag (backward). Unhe cancel hona chahiye: $\text{drag}=\text{thrust}=40,000\text{ N}.$ Bahut saari forces act karti hain, phir bhi net zero hai — yahi parent note ke example 3 ka message hai.

Kaunsa force act karta hai: jab cloth slide karti hai, kinetic friction plate ko sideways drag karti hai force ke saath $f=\mu mg=0.2\times 0.5\times 9.8=0.98\text{ N}.$ KYUN sirf thodi der: yeh force sirf uss chhoti si $\Delta t=0.05\text{ s}$ ke dauran act karta hai jab cloth neeche hoti hai. Sideways acceleration hai $a=f/m=\mu g=0.2\times 9.8=1.96{\text{ m/s}}^2$. Speed gained: $v=a\Delta t=1.96\times 0.05=0.098\text{ m/s}$. Toh plate lagbhag $0.098\text{ m/s}\approx 10\text{ cm/s}$ pe creep karti hai — barely move kar rahi hai. Uski inertia ne change ko resist kiya kyunki force ke paas act karne ke liye almost koi time nahi tha. Practically stay put karti hai. Dekho Friction.

WHAT: koi horizontal force torso ko forward push nahi karta. First law se, torso apni original $10\text{ m/s}$ pe chalte rehna jaari rakhta hai. KYUN lurch hota hai: bus (aur passenger ke pair, floor se pakde hue) $5{\text{ m/s}}^2$ se slow ho jaate hain, lekin torso, jiske paas koi forward-changing force nahi, continue karta hai. Decelerating bus ke relative, torso forward move karta hai — yahi lurch hai. Torso pe forward force: koi nahi. Forward motion inertia hai, koi force nahi. (Bus ke khud ke accelerating frame mein tum ek forward pseudo-force invent karte — ek signal ki bus ek non-inertial frame hai.)

(a) Speed constant hai lekin car turn kar rahi hai, toh uski direction change ho rahi hai → velocity change ho rahi hai → ${\vec{F}}_{\text{net}}\ne 0$. (b) Radius $r$ aur speed $v$ ke circle pe ek body ko ek net inward (centripetal) force chahiye jiska magnitude hai $F=\frac{m{v}^2}{r}=\frac{1200\times 15^2}{50}=\frac{1200\times 225}{50}=5400\text{ N},$ circle ke centre ki taraf point karta hai. (c) Tyres aur road ke beech static Friction ise provide karta hai. Friction ke bina (icy track) car first law ko follow karti aur straight line mein (tangent pe) chali jaati, curve nahi karti. Figure dekho: velocity arrow tangent hai (green), net force arrow inward point karta hai (red).

(a) Dono forces alag directions mein point karte hain aur cancel nahi hote, toh ${\vec{F}}_{\text{net}}\ne 0$ → equilibrium nahi hai. (b) Forces perpendicular hain (east ⟂ north), toh unhe joḍne ke liye right-triangle rule (Pythagoras) use karo: $|{\vec{F}}_{\text{net}}|=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ N}.$ (c) Nahi — nonzero net force matlab velocity change hoti hai, toh speed constant nahi hai. First law sirf "constant velocity" kehta hai jab net exactly zero ho, jo yahan nahi hai.

(a) Ground frame — honest view. Ball ko train ke saath forward $2{\text{ m/s}}^2$ pe accelerate karna hai. Sirf horizontal cheez jo use forward push kar sakti hai woh string tension ka horizontal part hai. Toh string peeche tilt ho jaati hai taaki uske tension ka ek forward component ho. Balance karne se, horizontal aur vertical force equations se, $\tan \theta =\frac{a}{g}=\frac{2}{9.8}=0.2041,$ jahan $\theta$ vertical se angle hai. Toh $\theta =\arctan (0.2041)\approx 11.5^{\circ }.$ Yahan $\arctan$ jawaab deta hai "kis angle ka yeh tangent hai?" — hum ratio jaante hain (opposite $a$ over adjacent $g$ chhote force triangle pe) aur angle vaapas chahiye. Figure dekho: tilt (orange), forward tension component (blue), gravity (red). (b) Train frame. Train non-inertial hai (woh accelerate karti hai). Wahan ball ek angle pe still hang hoti hai, toh passenger ek backward pseudo-force $ma=0.5\times 2=1\text{ N}$ invent karta hai tilt explain karne ke liye. Woh force real nahi hai — yeh ek dishonest (accelerating) frame use karne ki cost hai. First law ek litmus test hai jo ise expose karta hai.

(a) Koi net force nahi (kuch push ya pull karne ke liye nahi), toh first law se velocity constant hai → acceleration $=0$. (b) Zero. Yahi Galileo/Newton ka punchline hai: friction aur drag ki absence mein, motion maintain karne ke liye koi force nahi chahiye. Aristotle ne fuel hamesha ke liye jalaya hota; Newton kuch nahi jalata. (c) Pen ship ki velocity pe move kar raha tha. Uske upar koi force nahi, toh woh wohi velocity maintain karta hai — toh cabin ke relative woh motionless float karta hai (ya ek nudge milne par constant speed pe straight line mein drift karta hai). Yahi inertia visible form mein hai.

(a) East aur west cancel ho jaate hain ($10-10=0$). Bacha: $10\text{ N}$ north. Net force $=10\text{ N}$ north, zero nahi. (b) Haan — first law kabhi nahi kehta "force zero matlab motionless." Ek moving body jiske upar nonzero net force hai simply changing velocity hoti hai. Motion aur force alag alag questions hain. (c) Object apni northwest velocity maintain karta hai aur ab northward $a=F/m=10/2=5{\text{ m/s}}^2$ pe accelerate karta hai, apna path curve karta hai. First law ka equilibrium toot gaya hai, toh hum Newton's second law — F=ma ko hand off karte hain exactly yeh jaanne ke liye ki velocity kaise evolve hogi.

Har problem ke peeche ek hi question ::: Kya velocity vector (speed aur direction) constant hai? Agar haan, ${\vec{F}}_{\text{net}}=0$; agar nahi, toh net force act kar raha hai. Plane aur resting block dono ka ${\vec{F}}_{\text{net}}=0$ kyun tha? ::: Unki velocities constant thi; bahut saari individual forces cancel hokar zero net ho gayi. Orbiting satellite aur circling car ka kyun nahi tha? ::: Unki directions change hui, toh velocity change hui, jisne ek net centripetal force maanga. Non-inertial frame ko hamesha kya signal karta hai? ::: Ek free body bina kisi real force ke accelerate karta hua dikhta hai, tumhe $ma$ size ka pseudo-force invent karne par majboor karta hai.