Foundations — Circular motion — centripetal acceleration derivation

1.2.15 · D1 · Physics › Newton's Laws & Dynamics › Circular motion — centripetal acceleration derivation

Yeh page assume karta hai ki tumne kuch bhi nahi dekha. Parent Circular motion — centripetal acceleration derivation note mein jo bhi letter aur symbols hain, unhe yahan build-order mein khola gaya hai, taaki jab tak tum $a_c=v^2/r$ tak pahuncho, koi bhi cheez anjaan na lage.

Building blocks, order mein

1. Point, path, aur circle

Figure dekho: black dot centre hai, blue line ek radius hai jiska length $r$ hai, aur object (orange dot) rim par chal raha hai. Radius same rehta hai chahe object rim par kaheen bhi ho — yahi constancy baad mein maths ko clean banati hai.

Topic ko yeh kyun chahiye: answer $a_c=v^2/r$ mein $r$ denominator mein hai. Tight circle (chhota $r$) matlab sharp turn matlab zyada acceleration. Bina yeh jaane ki $r$ kya measure karta hai, woh formula padha hi nahi ja sakta.

2. Vector — size aur direction wala arrow

Topic ko yeh kyun chahiye: parent note ka core sentence hai "speed constant hai lekin velocity change hoti hai." Yeh sentence tabhi meaningful hai jab tum jaante ho ki ek vector direction change kar sakta hai jabki same length rakh sake.

3. Position vector $\vec{r}$ aur velocity vector $\vec{v}$

Figure mein blue arrow $\vec{r}$ object ki taraf bahar point karta hai; orange arrow $\vec{v}$ rim ke saath-saath point karta hai — yeh circle ka tangent hai.

Topic ko yeh kyun chahiye: dono derivation methods $\vec{r}$ likh kar aur $\vec{v}$ padhkar shuru karte hain. Yeh fact ki $\vec{v}\perp \vec{r}$ woh single hinge hai jis par geometry method turn karta hai.

4. Speed $v$ vs. velocity $\vec{v}$

5. Change $\Delta$ (Greek "delta")

Figure woh trick dikhata hai jo poori derivation ko kaam karti hai. Old aur new velocity arrows ko slide karo taaki unki tails mile. Red arrow $\Delta \vec{v}$ jo gap band karta hai — old ki tip se new ki tip tak — woh change in velocity hai. Dhyan do ki yeh motion ke saath-saath nahi point karta; yeh andar ki taraf jhukta hai. Woh inward lean, limit mein, centripetal acceleration ban jaata hai.

Topic ko yeh kyun chahiye: acceleration defined hai $\Delta \vec{v}/\Delta t$ ke roop mein. $\Delta$ ke bina tum likh bhi nahi sakte ki acceleration ka kya matlab hai.

6. Angle $\theta$ aur radian

7. Angular velocity $\omega$ aur period $T$

Topic ko yeh kyun chahiye: parent note mein Worked Example 3 $\omega$ ko $2\pi /T$ mein convert karta hai taaki $a_c=4{\pi }^{2}r/T^2$ mile. Woh step invisible hai jab tak tum yeh dono definitions na jaano. Dekho Angular velocity and period.

8. Derivative $\frac{d}{dt}$ aur limit

Yeh tool kyun, koi doosra nahi? Humein ek instant par exact acceleration chahiye, path ke visible slice par average nahi. Sirf limit/derivative jawab deta hai "rate of change abhi is waqt."

Calculus route (Method 2) ke liye dekho Vectors — derivative of a unit vector, jahan $\vec{r}=r(\cos \omega t,\sin \omega t)$ ko do baar differentiate karne par $-{\omega }^2\vec{r}$ nikalta hai.

9. Acceleration $a_c$ aur Newton's laws

$a_c$ mein subscript $c$ sirf ise centripetal ("centre-seeking") label karta hai: woh acceleration jo inward point karti hai. Yeh exist karti hai kyunki Newton's First Law (Inertia) kehta hai ki akela chhoda gaya object seedha chalta hai — isliye use circle mein mod'ne ke liye ek real inward force chahiye, jo $a_c$ se Newton's Second Law ke through $F_c=m{a}_c$ ke roop mein judi hai.

Foundations topic ko kaise feed karte hain

Upar se neeche padho: geometry (circle, vectors) do arrows banata hai; $\Delta$ inward change banata hai; radians aur $\omega$ timing dete hain; limit approximation ko exact rate mein sharpen karta hai; Newton's laws explain karte hain ki inward acceleration kyun exist karni chahiye.

Equipment checklist

Khud test karo — right side cover karo, jawab do, phir reveal karo.