Torque τ = r × F — definition, physical meaning

1.5.4 · Physics › Rotational Mechanics

Torque exist hi kyun karta hai?

Torque KYA hai (definition)

Formula ko kaise padhein — do pictures derive karna

Picture 1: "Effective force" (lever arm × force)

Hum $F$ ko $r$ ke relative do parts mein split karte hain:

$r$ ke along component (radial): $F cos θ$
$r$ ke perpendicular component (tangential): $F sin θ$

Picture 2: "Moment arm" (perpendicular distance)

Usi product ko regroup karo: $τ = (r sin θ) F = d_{⊥} \cdot F$ jahan $d_{⊥} === r sin θ ==$ hai force ki line of action se pivot ki perpendicular distance (lever arm / moment arm).

Figure — Torque τ = r × F — definition, physical meaning

Magnitude ko scratch se derive karna (cross-product first principles)

Components se Direction (taaki tumhe trust ho)

Agar $r = (x, y, z)$ aur $F = (F_{x}, F_{y}, F_{z})$ hain:

= (yF_z - zF_y)\,\hat i + (zF_x - xF_z)\,\hat j + (xF_y - yF_x)\,\hat k.$$ $xy$-plane mein ek planar problem ke liye ($z=0,\ F_z=0$): $$\vec\tau = (xF_y - yF_x)\,\hat k.$$ Isliye 2D torque bas ek **signed scalar** hai: $+$ page se bahar (anticlockwise), $-$ page ke andar (clockwise). --- ## Worked examples > [!example] 1 — Wrench > Tum ek $0.30\ \text{m}$ wrench ke end par $F = 50\ \text{N}$ apply karte ho, uske perpendicular push karte hue. > **Step 1:** $\theta = 90^\circ$ identify karo → *Kyun?* Perpendicular push = maximum twist. > **Step 2:** $\tau = rF\sin\theta = 0.30 \times 50 \times \sin 90^\circ = 0.30\times50\times1$. > **Result:** $\tau = 15\ \text{N·m}$, bolt axis ke along directed hai. > [!example] 2 — Angle par push > Same wrench, same force, lekin ab wrench ke saath $30^\circ$ par push kar rahe hain. > **Step 1:** $\sin 30^\circ = 0.5$ → *Kyun?* Sirf sideways component $F\sin\theta$ twist karta hai. > **Step 2:** $\tau = 0.30 \times 50 \times 0.5 = 7.5\ \text{N·m}$. > **Insight:** Same effort ke liye aadha twist — angle bahut matter karta hai. > [!example] 3 — Moment arm use karna > Ek force $\vec F = (4,3,0)\ \text{N}$, $\vec r = (2,0,0)\ \text{m}$ par act kar rahi hai. > **Step 1:** Component formula use karo: $\tau_z = xF_y - yF_x = (2)(3) - (0)(4) = 6$. *Kyun component form?* $\theta$ dhundhne se zyada fast hai. > **Step 2:** $\vec\tau = 6\,\hat k\ \text{N·m}$ → anticlockwise. > **Magnitude se check:** $r=2$, $F=5$, $(2,0,0)$ aur $(4,3,0)$ ke beech angle: $\cos\theta = \frac{8}{2\cdot5}=0.8$, toh $\sin\theta=0.6$. Phir $rF\sin\theta = 2\cdot5\cdot0.6 = 6$. ✓ Same. --- ## Forecast-then-Verify > [!recall]- Answer padhne se pehle forecast karo > Fixed magnitude $F$ ki ek force fixed distance $r$ par act kar rahi hai. Jaise tum force ko $\theta=0^\circ$ se $90^\circ$ se $180^\circ$ rotate karte ho, check karne se pehle $\tau$ ka behavior sketch karo. > > **Verify:** $\tau = rF\sin\theta$, $0$ se $90^\circ$ par max tak badhta hai, phir $180^\circ$ par wapas $0$ par aa jaata hai. Maximum twist hamesha tab hoti hai jab force radius ke perpendicular ho; pivot ki taraf push karna YA usse door ($0^\circ$ ya $180^\circ$) zero torque deta hai. --- ## Common mistakes (Steel-manned) > [!mistake] "Zyada force hamesha zyada torque deti hai." > **Kyun sahi lagta hai:** Roz ki pushing mein, zyada force = zyada effect. **Flaw:** torque $r$ aur $\sin\theta$ par bhi depend karta hai. Ek bahut badi force pivot *par* aim ki gayi **zero** torque deti hai. **Fix:** judge karne se pehle hamesha pucho "lever arm $r\sin\theta$ kya hai?" > [!mistake] "$\cos\theta$ use karo kyunki force $\theta$ angle par hai." > **Kyun sahi lagta hai:** Hum $\cos$ ko "axis ke along components" ke liye hamesha dekhte hain. **Flaw:** yahan hume $\vec r$ ke **perpendicular** component chahiye, jo $F\sin\theta$ hai, $F\cos\theta$ nahi. **Fix:** torque ko *sideways* part chahiye; $\sin$ zero hota hai jab vectors parallel hoon (sahi tarike se zero torque deta hai). > [!mistake] "$\vec r$ wahan se measure hota hai jahan force act karti hai." > **Kyun sahi lagta hai:** $\vec r$ force ke point of application par hai. **Flaw:** $\vec r$ **pivot/axis** se shuru hona chahiye aur application point ki taraf point karna chahiye. Galat origin → galat torque. **Fix:** pehle apna axis fix karo, phir $\vec r$ ko axis → contact point se draw karo. > [!mistake] "Torque ek scalar hai (woh N·m mein hai jaise energy)." > **Kyun sahi lagta hai:** Joules jaise hi units. **Flaw:** torque ek **vector** hai (iska ek rotation axis aur sense hota hai). **Fix:** direction report karo (right-hand rule / 2D mein sign). Torque ke liye N·m, energy ke liye J — kabhi mix mat karo. --- ## Mnemonic > [!mnemonic] > **"Sin to spin."** Torque $\sin\theta$ use karta hai (perpendicular force *spins* karta hai). > Right-hand rule: fingers $\vec r \to \vec F$ curl karti hain, thumb = $\vec\tau$. > Door rule: *Far + Sideways = strong twist.* --- ## Feynman (ek 12-saal ke bachche ko explain karo) > [!recall]- Simplicity se explain karo > Imagine karo ek bhaari darwaza kholna. Agar tum door edge par push karo, woh aasaani se swing ho jaata hai. Agar tum hinges ke bilkul paas push karo, toh bahut mushkil hai. Aur agar tum darwaze ko uske hinges mein (wall mein sideways) push karo, toh woh bilkul nahi khulta — tum bas use squish kar rahe ho. **Torque** bas ek number hai jo batata hai ki tumhari push kisi cheez ko *spin* karne mein kitni achi hai. Spinning point se **door** push karo aur **sideways** push karo (uski taraf nahi), aur tumhe sabse zyada spin milega. --- ## #flashcards/physics Torque ki definition as a vector kya hai? ::: $\vec\tau = \vec r \times \vec F$, jahan $\vec r$ pivot se force application ke point tak hai. Torque ki magnitude kya hai? ::: $\tau = rF\sin\theta$, jahan $\theta$, $\vec r$ aur $\vec F$ ke beech ka angle hai. Torque $\sin\theta$ kyun use karta hai $\cos\theta$ nahi? ::: Sirf force ka $\vec r$ ke perpendicular component ($F\sin\theta$) rotation cause karta hai; radial part kuch bhi spin nahi kar sakta. Moment (lever) arm kya hai? ::: $d_\perp = r\sin\theta$, pivot se force ki line of action tak ki perpendicular distance. Nonzero force ke liye torque zero kab hota hai? ::: Jab $\vec F$, $\vec r$ ke parallel ya antiparallel ho ($\theta=0$ ya $180^\circ$), yaani force pivot ki taraf line ke along aim ho. Torque ki SI unit, aur woh energy se alag kaise hai? ::: N·m; joule jaise hi dimensions lekin torque ek vector quantity hai (energy scalar hai). xy-plane mein components se 2D torque kya hai? ::: $\tau_z = xF_y - yF_x$; positive = anticlockwise (page se bahar). Torque vector ki direction kya hai? ::: $\vec r$ aur $\vec F$ ke plane ke perpendicular, right-hand rule se ($\vec r$ ko $\vec F$ mein curl karo). $|\vec r \times \vec F|$ ka geometric meaning kya hai? ::: $\vec r$ aur $\vec F$ se bane parallelogram ka area. $\vec r$ kahan se shuru hona chahiye? ::: Chosen pivot/axis se, force apply hone wale point ki taraf point karte hue. --- ## Connections - [[Cross Product (Vector Algebra)]] — torque iska prototypical physical example hai. - [[Moment of Inertia]] aur [[Newton's Second Law for Rotation]] — $\vec\tau = I\vec\alpha$. - [[Angular Momentum]] — $\vec\tau = \dfrac{d\vec L}{dt}$. - [[Equilibrium of Rigid Bodies]] — $\sum\vec\tau = 0$ condition. - [[Work-Energy Theorem (Rotational)]] — rotational work $= \tau\,\theta$. - [[Center of Mass & Gravity]] — line of action aur lever arms. ## 🖼️ Concept Map ```mermaid flowchart TD F[Force F] R[Position vector r from pivot] ANGLE[Angle theta between r and F] CROSS[Cross product r x F] TAU[Torque vector tau] MAG[Magnitude rF sin theta] FPERP[Tangential component F sin theta] LEVER[Moment arm d_perp = r sin theta] DIR[Direction by right-hand rule] AREA[Parallelogram area] DOOR[Door example] F -->|acts at| R R -->|separated by| ANGLE R -->|combined via| CROSS F -->|combined via| CROSS CROSS -->|defines| TAU CROSS -->|gives| MAG MAG -->|read as| FPERP MAG -->|read as| LEVER ANGLE -->|only sideways part twists| FPERP ANGLE -->|sets perpendicular distance| LEVER CROSS -->|equals| AREA AREA -->|explains| MAG TAU -->|orientation from| DIR DOOR -->|motivates| TAU ```