Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²): Foundations

1.3.5 · D1 · Physics › Work, Energy & Power › Potential energy — definition, gravitational (mgh and −GMm -

Hum end mein ek single master sentence padhenge jo symbols mein likhi hogi. Lekin hum use tab tak nahi dikhayenge jab tak uske andar ka har ingredient define aur draw na ho jaaye. Toh is page mein pieces ek ek karke introduce kiye jayenge — pehle plain words mein, phir ek picture, aur tab jaake symbol. Master sentence poori tarah section 6 mein aayegi, aur jo teen concrete PE formulas woh produce karta hai woh section 6b mein likhi aur derive ki gayi hain.

1. Position, height, distance — "kahan" ke symbols

Picture: charon sirf arrows hain jo kitni door measure karte hain. $h = Δ y$ aur $x$ ek natural "zero" se start karte hain (ground $y_{0}$ ; rest par spring). $r$ planet ke core se start karta hai.

Figure — Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²)

Figure 1 — Teen "kahan" ke measures. Left: $h = y - y_{0}$ raised object aur floor $y_{0} = 0$ ke beech ka vertical gap (lavender arrow) hai. Middle: $r$ (lavender arrow) planet ke centre se object tak jaata hai. Right: $x$ (lavender arrow) spring ki relaxed length $x = 0$ se stretch hai. Dhyan do ki har ek apne natural zero se start karta hai.

Topic ko inki zaroorat kyun hai: potential energy sirf position par depend karta hai. Agar hum position ko naam nahi de sakte, toh PE nahi likh sakte. Har formula woh position label leta hai jo uski geometry se fit ho: $m g h$ mein $h$ use hota hai, $- GM m / r$ mein $r$ , aur $\frac{1}{2} k x^{2}$ mein $x$ .

2. Force aur arrow $F$ — "push" ka symbol

Picture: $F = - m g \overset{y}{^}$ padho as "length $m g$ ka arrow $- \overset{y}{^}$ direction (neeche) mein point karta hua." Aage ka minus sirf arrow ko flip karta hai.

3. Dot product $F \cdot d r$ — "push ka kitna hissa motion ke saath hai"

Neeche di gayi picture dikhati hai kyun: force ka sirf woh hissa jo tiny step $d r$ ke saath hai count karta hai.

Figure 2 — Dot product padhna. Coral arrow force $F$ hai (neeche point karta hua); mint arrow tiny step $d r$ hai (upar-aur-daayein point karta hua). Lavender arrow $F$ ka woh tukda hai jo $d r$ ke saath lie karta hai — woh projected piece, step ki length se multiply karke, $F \cdot d r$ hai. Agar $F$ $d r$ se right angle par hota, toh lavender arrow kuch bhi nahi hota aur dot product zero hota.

Yeh tool kyun use karein aur plain multiplication kyun nahi? Kyunki "work" (aage define hota hai) ka matlab hai force jo motion mein madad kare. Agar tum ek book ko side mein le jaao, gravity (neeche) tumhare step (sideway) se right angle par hai — woh zero work karta hai. Sirf dot product hi us "sirf aligned part count karta hai" rule ko capture karta hai. Dekho Work done by a force.

4. Work $W$ , tiny step $d r$ , aur integral $\int$ — "poore path par add up karo"

Picture: path ko lakhon dots mein kaato; har dot par little work $F \cdot d r$ nikalo; unhe sab add karo.

5. Reference point aur $Δ U$ — "kis jagah se measure kiya?"

Figure 3 — Reference ek free choice hai. Lavender curve $U = - GM m / r$ hai jisme $U = 0$ infinity par rakha gaya hai; dashed coral curve same physics hai jisme zero kahin aur move kar diya gaya — poora curve upar slide karta hai. Do vertical slate arrows un same pair of points ke beech gap dikhate hain dono curves par: individual $U$ values alag hain, lekin $Δ U$ (arrow ki length) identical hai. Sirf $Δ U$ physical hai.

6. Master sentence — aur iska minus sign kahan se aata hai

Ab har mark define ho gaya. Padho: $Δ U = - W = - \int_{i}^{f} F \cdot d r$

Intuition Leading minus sign kyun hai?

"Mehnat bank karna" socho. Do cases, aur sign dono se fit hona chahiye:

Tum ek book ko gravity ke khilaf uthate ho. Gravity neeche point karti hai, tumhara step upar point karta hai, toh $F \cdot d r < 0$ har grain par → force negative work karta hai, $W < 0$ . Lekin tumne clearly energy store ki (book ab gir ke speed up kar sakti hai), toh $U$ upar jaana chahiye. " $U$ upar" aur " $W$ negative" ke saath agree karne ka ek hi tarika hai $Δ U = - W$ : minus negative $W$ ko positive $Δ U$ mein flip karta hai.
Book girti hai. Ab gravity motion mein madad karta hai, $W > 0$ , aur stored energy speed mein spend hoti hai, toh $U$ neeche jaata hai. Phir bhi $Δ U = - W$ negative change deta hai. ✓

Ek line mein: positive work karne wali force bank spend kar rahi hai, toh $U$ girta hai; force ke khilaf kaam karna bank fill karta hai, toh $U$ badhta hai. Woh opposite-ness hi minus sign hai.

Sirf conservative forces (section 8) hi hume $U$ likhne deti hain — wahi upar ki equation ka license hai.

6b. Master sentence ko feed karna — teen concrete $U$ formulas

Master sentence ek machine hai: isme force daalo, $U$ nikalo. Yahan hum iska crank teen baar ghumate hain taaki tum actually dekh sako har woh formula jo topic tumse apply karaayega.

Teen kyun? Har ek ek alag position label ( $h$ , $r$ , $x$ ) aur alag force shape (constant, inverse-square, linear) use karta hai — milke woh "flat ground," "deep space," aur "stretchy things" cover karte hain, jo ki parent topic mein maange jaane wale har case hain. Dekho Conservation of mechanical energy for what these $U$ 's then do.

7. Constants $m$ , $g$ , $k$ , $G$ , $M$ , aur radius $R$ — "kitna strong" ke numbers

Topic ko inki zaroorat kyun hai: woh "kitni door" ko "kitni energy" mein convert karte hain. Bada $m$ , stiffer $k$ , heavier $M$ → same move ke liye zyada banked energy.

8. "Conservative" force — gatekeeper concept

Kyun matter karta hai: PE sirf tab position ki clean function ho sakti hai jab banked energy route par depend na kare. Yahi poora license hai $U (position)$ likhne ka. Dekho Conservative vs non-conservative forces.

9. Slope of the landscape — integral se $\frac{d U}{d x}$ wapas

Section 6 ne $U$ paane ke liye force add up ki. Ab hum machine ulta chalate hain: diye gaye $U$ landscape se force recover karte hain. "Add up" ka reverse hai "slope dekho."

Picture: $U$ ko hills aur valleys ke roop mein draw karo. Ek ball downhill roll karta hai, lower $U$ ki taraf. Force minus the slope hai: steeper hill, harder shove.

Figure 4 — Force $U$ ki slope ka minus hai. Lavender curve valley-shaped $U (x)$ hai. Coral dot par butter dashed line local slope $d U / d x$ dikhati hai (positive: $U$ daayein badhta hai). Mint arrow force $F = - d U / d x$ hai, opposite direction mein point karta hua — downhill, valley floor ki taraf wapas. Jahan curve flat hai (bilkul neeche) wahan slope zero hai aur force bhi zero hai.

Intuition Edge case: jab position ek number se zyada ho

$\frac{d U}{d x}$ sirf ek line ke saath motion handle karta hai. Real space mein tum $x$ , $y$ aur $z$ mein move kar sakte ho, toh $U$ teeno par depend karta hai. Partial derivative $\frac{\partial U}{\partial x}$ (curly $\partial$ ka matlab hai " $x$ mein derivative, $y$ aur $z$ ko frozen rakhke") sirf ek axis ke saath slope measure karta hai. Teeno slopes ko ek arrow mein stack karo aur tumhe gradient milta hai, jo upside-down triangle $\nabla$ ("del" ya "grad") se likha jaata hai: $\nabla U = (\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}) .$ Yeh arrow woh direction dikhata hai jis mein $U$ sabse tezi se badhta hai, aur iski length woh steepest slope hai. Poora law tab hai: $F = - \nabla U,$ minus ka matlab "force downhill point karta hai, steepest increase ke opposite." Ek single axis ke saath sirf $\partial U / \partial x$ bachta hai aur yeh $F_{x} = - \frac{d U}{d x}$ par collapse ho jaata hai — toh 1-D rule sirf real cheez ka flat-ground version hai, exactly jaisa $m g h$ $- GM m / r$ ke liye hai. Force from potential — F = -dU/dx mein aage build hota hai.

Equipment checklist

F = - m g \overset{y}{^}

ko plain words mein zor se padho.

"Size

m g

ki force bilkul seedha neeche point karti hui."

\overset{r}{^}

mein hat ka kya matlab hai?

Ek unit arrow — exactly length 1 — jo sirf direction carry karta hai.

h

y

aur

y_{0}

ke terms mein likho aur batao kis

Δ

use karta hai.

h = y - y_{0} = Δ y

— wahi "final minus initial"

Δ

jaise

Δ U

mein.

F \cdot d r

kab zero hota hai?

Jab force aur step right angle par hon (force koi work nahi karta).

Work

W

ko ek integral ke roop mein likho.

W = \int_{i}^{f} F \cdot d r

— aligned-push × tiny-step ka path par sum.

Integral kyun use karein force × distance ki jagah?

Kyunki force path ke saath change karta hai (door kamzor, stretch par stronger); sirf slicing-and-adding hi use honestly total karta hai.

U

aur

W

ke beech master relation batao.

Δ U = - W

; minus isliye flip karta hai kyunki positive work bank spend karta hai jabki force ke khilaf kaam bank fill karta hai.

Teen concrete PE formulas aur unke references likho.

U = m g h

(zero at floor);

U = - GM m / r

(zero at infinity);

U = \frac{1}{2} k x^{2}

(zero at relaxed length).

Δ U = - W

mein minus sign kyun?

Gravity ke khilaf uthane par

W < 0

milta hai phir bhi energy store hoti hai (

U

upar); sirf

- W

inka agreement karaata hai.

Δ U

mein

Δ

ka kya matlab hai?

"Change in" — final minus initial.

R

define karo aur

g

G

M

R

ke terms mein batao.

R

= planet ka radius (centre se surface); surface par

g = GM / R^{2}

Force conservative kab hoti hai?

Jab uska work path-independent ho, toh PE position ki ek single-valued function ho sake.

Dikhao ki

F = - d U / d x

master integral se kaise aata hai.

Tiny step par

d U = - F d x

;

d x

se divide karo

F = - d U / d x

milta hai.

\partial U / \partial x

ka kya matlab hai aur yeh

\nabla U

kaise banata hai?

Slope of

U

x

with

y, z

frozen;

\partial U / \partial x, \partial U / \partial y, \partial U / \partial z

stack karne se

\nabla U

milta hai, aur

F = - \nabla U

Connections

Parent: 1.3.05 Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²) (Hinglish)
Work done by a force
Conservative vs non-conservative forces
Conservation of mechanical energy
Hooke's Law
Gravitation — Newton's law
Escape velocity
Force from potential — F = -dU/dx