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

1.5.4 · D1 · Physics › Rotational Mechanics › Torque τ = r × F — definition, physical meaning

Is page par assume kiya gaya hai ki aap kuch nahi jaante. Parent page par har squiggle yahan ek ek karke unpack ki gayi hai, har ek pehle waali ke upar lean karti hui. Akhir tak aap $\vec{\tau }=\vec{r}\times \vec{F}$ ka har symbol earn kar chuke honge.

0 · Hat wala arrow, $\stackrel{⃗}{}$, aakhir hai kya?

Isko scalar se contrast karo — ek plain number jisme koi direction nahi hoti, jaise temperature ($20^{\circ }$) ya mass ($3\text{kg}$). Torque ki magnitude ek scalar niklegii, lekin torque khud ek vector hai — yeh distinction yaad rakhna.

1 · Position arrow $\vec{r}$ — "pivot se, jahan push karo, wahan tak"

Kuch bhi twist hone se pehle, hume ek fixed point chahiye jiske around twist ho. Ise pivot kehte hain (ya "axis" ya "hinge" bhi). Socho woh exact line jis par door swing karta hai.

Figure s02 dekho: blue arrow ki tail hinge par hai, head handle par hai. Dur waala handle ek lamba $\vec{r}$ deta hai; hinge ke paas push karne se chhota $\vec{r}$ milta hai. Yahi length reason hai ki handle par door asaani se khulta hai.

2 · Force arrow $\vec{F}$ — "kitna hard aur kis taraf push karo"

Ab crucial move. Ek push $\vec{r}$ ke relative kisi bhi direction mein point kar sakti hai. Hum us relationship ko ek single number se measure karte hain: do arrows ke beech ka ==angle $\theta$==, tails ko tail-to-tail rakh ke.

3 · Push ko split karna: $\sin \theta$ aur $\cos \theta$ kyun chahiye

Ab hum poochte hain: push ka kitna hissa actually "sideways" jaata hai? Jawab ke liye hum $\vec{F}$ ko $\vec{r}$ ke against do perpendicular pieces mein split karte hain.

Ek arrow ko "along" aur "across" pieces mein split karne ke liye hum right triangle use karte hain — aur right triangles exactly woh hain jo $\sin$ aur $\cos$ describe karte hain.

Figure s03 mein pink arrow ($F_{\perp }=F\sin \theta$) sirf wahi hai jo point ko curl karta hai; yellow arrow ($F_{\parallel }=F\cos \theta$) sirf pivot ki line ke along tug karta hai.

4 · Cross product $\times$ — woh machine jo do arrows ko ek twist mein badal deti hai

Ab pure topic ka star symbol: $\vec{r}\times \vec{F}$ mein $\times$.

Yahi reason hai ki parent page ka $\tau =rF\sin \theta$ kabhi guess nahi tha — yeh cross product ke built-in $\sin \theta$ se forced hai. Is machine par aur gehraai se jaane ke liye Cross Product (Vector Algebra) dekho.

5 · Output symbol $\vec{\tau }$ aur iska unit

6 · Unit vectors $\hat{i},\hat{j},\hat{k}$ — space ki teen directions

Parent page forces ko $(x,y,z)$ mein likhta hai aur results $6\hat{k}$ jaisi form mein deta hai. Yeh hats kya matlab rakhte hain.

Prerequisite map

Har arrow ka matlab hai "neeche wala box samajhne se pehle upar wala box chahiye." Notice karo ki sab kuch $\vec{\tau }$ ki taraf flow karta hai — aur sab kuch vector ki single idea se upar flow karta hai.

Yeh aage kahan jaata hai

• Twist $\vec{\tau }$ rotation ko exactly usi tarah drive karta hai jaise force motion ko drive karti hai: dekho Newton's Second Law for Rotation.
• Ek body us twist ko kitna resist karti hai yeh hai Moment of Inertia.
• Twist time ke saath Angular Momentum build karta hai.
• Jab saare twists cancel ho jaate hain, milta hai Equilibrium of Rigid Bodies.
• Gravity ki twist aise act karti hai jaise Center of Mass & Gravity par concentrated ho.
• Twist jo ek angle par rotation kare woh hai Work-Energy Theorem (Rotational).

Jab bhi physics ki puri kahani chahiye parent par wapas jao: Torque — definition & physical meaning.

Equipment checklist

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