Foundations — Work done by variable force — integration

1.3.2 · D1 · Physics › Work, Energy & Power › Work done by variable force — integration

Parent note Work done by a variable force — integration padhne se pehle, tumhe us mein aane wale har symbol ko achhi tarah se jaanna hoga. Neeche, har ek ko zero se build kiya gaya hai: plain words → ek picture → yeh topic uske bina kyon nahi chal sakta. Inhe is tarah order kiya gaya hai ki har brick usse pehle wali brick par tikti hai.

1. Position $x$ — "main path par kahan hoon?"

Ise imagine karo. Ek horizontal line draw karo, left mein $0$ marked ek tick lagao. Right ki taraf har point ko ek bada number milta hai: $1$ m, $2$ m, $3$ m. Woh number hi $x$ hai.

Topic ko yeh kyun chahiye. Ek variable force woh hoti hai jiska strength is baat par depend karta hai ki tum kahan ho. Yeh kehne ke liye bhi ki "force position par depend karti hai", humein pehle position ke liye ek naam chahiye — woh naam $x$ hai.

2. Displacement $d$ aur tiny step $dx$

Ise imagine karo. Number line par, $x=1$ se $x=3$ tak ek arrow ki length $2$ hai: yeh $2$ m ka displacement hai. Ab zoom in karo jab tak arrow ek baal ki motaai ka na ho jaaye — woh microscopic arrow $dx$ hai.

Topic ko yeh kyun chahiye. Parent note "path ko tiny pieces $dx$ mein kaatta hai". $dx$ ek sliver ki width hai.

3. Force $F$ aur force-as-a-function $F(x)$

Ise imagine karo. Ek shopping cart imagine karo ek aise floor par jो jaise-jaise aage badhte hai chipchipa hota jaata hai. $x=0$ par ek force-meter ki needle small padhti hai; $x=3$ par woh large padhti hai. Un readings ko plot karna $F$ against $x$ ka ek curve deta hai — har jagah force alag hai.

Topic ko yeh kyun chahiye. "Variable force" ka matlab hi yeh hai ki $F$ position ka function hai, $F(x)$. Agar $F$ har jagah same number hoti toh woh constant hoti — dekho Work done by a constant force.

4. Work $W$ — distance par spend ki gayi force

Ise imagine karo. Force-vs-position graph par, ek constant force height $F$ par ek flat horizontal line hai. Ise distance $d$ par push karo aur work rectangle hai jिski height $F$ aur width $d$ hai — area $=F\times d$.

Topic ko yeh kyun chahiye. $W$ wahi quantity hai jo hum compute kar rahe hain. Parent ka mission: $W$ nikalna jab graph ka top flat na ho.

5. Slivers ko multiply-and-add karna: sum $\sum$

Ise imagine karo. Wiggly area ko $N$ thin vertical strips mein kaato. Strip $i$ ki height $F(x_i)$ hai (wahan ki force) aur width $\Delta x_i$ hai (uski thodi si width), toh uska area hai $F(x_i)\Delta x_i$. Har strip ka area add karo: $W\approx {\sum }_{i=1}^{N}F(x_i)\Delta x_i$

Topic ko yeh kyun chahiye. Rectangles ki yeh staircase curved area ka approximation hai. Yeh "rectangle" aur "integral" ke beech ka bridge hai.

6. The limit — "steps ko zero tak shrink karo"

Ise imagine karo. Kuch fat strips → rectangles ke tops curve ke upar/neeche nikle hue hain, area rough hai. Bahut thin strips → jagged tops curve mein smooth ho jaate hain. Limit mein, staircase ban jaati hai curve.

Topic ko yeh kyun chahiye. Approximation $\approx$ exact $=$ mein tabhi banta hai jab limit ho. Yahi ek sum ko integral mein upgrade karta hai.

7. The integral ${\int }_{x_A}^{x_B}F(x)dx$

Ise imagine karo. Symbol ko left se right padho ek sentence ki tarah: "Add up ($\int$), start $x_A$ se end $x_B$ tak, quantity height-times-width $F(x)dx$." Yeh curve ke neeche ka exact area hai.

Topic ko yeh kyun chahiye. Yeh parent note ka punchline formula hai: $\boxed{W={\int }_{x_A}^{x_B}F(x)dx}$ Area = integral kyun hota hai uski maths Area under curves and the definite integral mein develop ki gayi hai.

8. Dot product $\vec{F}\cdot d\vec{r}$ aur angle $\theta$

Ise imagine karo. Agar tum ek sled ko upar ki taraf tili rope se kheench rahe ho, sirf tumhara forward part sled ko move karta hai; upar wala part waste hota hai (koi work nahi karta). $\cos \theta$ exactly woh forward fraction measure karta hai: $\theta =0^{\circ }$ → $\cos \theta =1$ (sab kuch count hota hai); $\theta =90^{\circ }$ → $\cos \theta =0$ (kuch bhi count nahi hota, e.g. normal force); $\theta =180^{\circ }$ → $\cos \theta =-1$ (force motion se ladti hai, work negative hoti hai).

Topic ko yeh kyun chahiye. Sabse general work formula $W=\int \vec{F}\cdot d\vec{r}$ hai. 1-D mein, force aur motion align karte hain ($\theta =0$, $\cos \theta =1$) toh yeh $\int Fdx$ mein simplify ho jaata hai — lekin $\cos \theta$ hamesha secretly wahan hota hai. Poori baat Dot product and components of vectors mein hai.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Khud ko test karo — right side cover karo aur reveal karne se pehle answer do.

Connections

• Work done by a constant force — flat-line, rectangle wala case jise yeh foundations generalise karte hain.
• Area under curves and the definite integral — woh maths jo "integral = area" ko rigorous banati hai.
• Dot product and components of vectors — jahan se $\cos \theta$ aur $\vec{F}\cdot d\vec{r}$ aate hain.
• Hooke's law and spring potential energy — pehli jagah jahan variable $F(x)=kx$ appear hoti hai.
• Work–Energy Theorem — yeh saara work actually kya change karta hai (kinetic energy).
• Parent topic — in bricks ko $W=\int Fdx$ mein assemble karta hai.