Foundations — Work — definition, dot product F·d, sign convention

1.3.1 · D1 · Physics › Work, Energy & Power › Work — definition, dot product F·d, sign convention

Yeh D1 Foundations child hai Work topic ka. Parent note mein arrows, angles, $\cos \theta$, dot products, aur $F_x{d}_x+F_y{d}_y$ freely use hote hain. Yahan hum har ek cheez zero se build karte hain, ek aisi order mein jahan har piece apne pehle wale piece par lean karta hai. Jo reader ne kabhi arrow-with-a-hat nahi dekha, woh is page ke baad ready ho jayega.

0. Number vs. arrow — scalar aur vector

Kisi bhi physics se pehle, ek split matter karti hai.

Figure dekho. Pale-yellow arrow ka ek tail (jahan se shuru hota hai) aur ek head (pointy end) hota hai. Uski length woh yellow bar hai jis par "magnitude" likha hai; uska tilt direction hai. Do arrows same vector hain agar unki length aur tilt same hain — chahe alag jagah draw kiye gaye hon.

Hum ek vector ko upar ek chota arrow laga ke likhte hain: $\vec{F}$, $\vec{d}$. Sirf uska size (ek scalar) usi letter se, bina arrow ke: $F$, $d$. To $F$ hamesha $\ge 0$ hota hai; yeh sirf $\vec{F}$ ki length hai.

1. Displacement $\vec{d}$ — start se finish tak ka arrow

Figure mein ek ghoomta hua chalk path (actual route) dikhta hai versus seedha blue arrow $\vec{d}$ (start-to-finish). Yeh alag hain:

• Path length / distance = aapne actually kitna squiggle par chale (hamesha scalar, hamesha badhta hai).
• Displacement $\vec{d}$ = shortcut arrow (ek vector, zero bhi ho sakta hai agar aap ghar wapas aa jao).

Isliye parent insist karta hai "displacement, path length nahi". Uss sentence ka matlab tabhi ban sakta tha jab hum vector $\vec{d}$ samajh lein.

2. Components — ek arrow ko $x$ aur $y$ pieces mein kaatna

Arrows ke saath arithmetic karne ke liye hum do reference directions banate hain: right ($x$) aur up ($y$). Poori story ke liye Vectors & Components dekho.

Figure mein, arrow ke head se seedhi line $x$-axis par daalo: woh ground-shadow $F_x$ hai (pink). Ek $y$-axis par daalo: woh side-shadow $F_y$ hai (blue). Arrow, uska $x$-shadow, aur uska $y$-shadow milke ek right triangle banate hain — ek triangle jisme ek $90°$ corner hota hai. Us triangle ko yaad rakho; pura $\cos \theta$ story usi ke andar rehti hai.

Hum vector ko uske components se unit vectors $\hat{ı}$ (ek step right) aur $\hat{ȷ}$ (ek step up) ke saath likhte hain: $\vec{F}=F_x\hat{ı}+F_y\hat{ȷ}$ "Hat" ka matlab hai "length exactly 1, sirf pure direction." To $3\hat{ı}+4\hat{ȷ}$ ka matlab hai "3 steps right, 4 steps up" — exactly wahi notation jo parent ke Example 2 mein hai.

3. Angle $\theta$ — aur kyun $\cos$ sahi tool hai

Ab woh key question jo work ka jawab dhundhna hai: $\vec{F}$ ka kitna hissa $\vec{d}$ ke along point karta hai? Woh "along-$\vec{d}$ shadow" hi woh piece hai jo motion ke saath ride karti hai. Figure mein, $\vec{F}$ ka shadow $\vec{d}$ ki direction par daalo (yellow segment $F_{\parallel }$).

Sine nahi, cosine kyun? $\vec{F}$ aur uske shadow se bane right triangle mein, shadow $F_{\parallel }$ woh side hai jo angle $\theta$ ke saath hai (adjacent side), aur $F$ sabse lambi side (hypotenuse) hai. Right triangle par cosine ki definition se: $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{F_{\parallel }}{F}\Rightarrow F_{\parallel }=F\cos \theta$

To $\cos \theta$ precisely "full force ka woh fraction hai jo road ke along point karne par bachta hai." Sine perpendicular leftover $F_{\perp }=F\sin \theta$ deta — woh steering part jo koi work nahi karta. $\cos$ choose karna matlab hai "motion ke along" choose karna. Yahi pura reason hai ki parent ka formula $Fd\cos \theta$ hai, kabhi $Fd\sin \theta$ nahi.

4. Dot product — do-vectors-in, ek-scalar-out machine

Same number ke do faces kyun hain?

• $Fd\cos \theta$ geometry face hai: "(along-force) × (displacement)." Tab use karo jab angle pata ho.
• $F_x{d}_x+F_y{d}_y$ component face hai: matching shadows multiply karo, add karo. Tab use karo jab arrows $\hat{ı},\hat{ȷ}$ lists ke roop mein diye gaye hon aur $\theta$ kabhi nahi bataya gaya.

Dono hamesha agree karte hain. Parent ke Example 2 se quick check, $\vec{F}=(3,4)$, $\vec{d}=(2,-1)$: $\vec{F}\cdot \vec{d}=(3)(2)+(4)(-1)=6-4=2\text{J}$ Dot product sign free mein de deta hai — positive ya negative decide karne ka alag step nahi.

5. Units — the joule

Kyunki work ek dot product hai, uska output ek scalar hai, isliye ek joule mein koi direction nahi — sirf ek size aur ek sign ($+$ energy in, $-$ energy out).

Prerequisite map

Equipment checklist

Right side cover karo aur khud ko test karo.

Connections

• Parent: Work definition — jahan yeh foundations assemble hoti hain.
• Vectors & Components — arrow-chopping jis par yeh page lean karta hai.
• Dot Product (Scalar Product) — two-in-one-out machine poori tarah se.
• Kinetic Energy & Work-Energy Theorem — jahan positive/negative work aage jaata hai.
• Power — work karne ki rate, ek aur dot product $P=\vec{F}\cdot \vec{v}$.
• Friction — ek force jo reliably negative work karti hai.
• Potential Energy & Conservative Forces — kyun loop displacement ka zero hona matter karta hai.