Conservation of mechanical energy — derivation: Question bank

1.3.8 · D5 · Physics › Work, Energy & Power › Conservation of mechanical energy — derivation

Neeche har line Prompt ::: Answer format mein hai — answer mein hamesha kyun hota hai, sirf haan/nahi nahi.

Symbol glossary — pehle yeh padho

Kisi law par bahas karne se pehle, aapko pata hona chahiye ki har letter ka matlab kya hai. Neeche ke har trap mein yahi symbols reuse hote hain; is table ko samne rakhna.

Definition Is page par use hone wale har symbol ka matlab

$m$ — object ka mass (kilograms). Kitna "stuff" hai jo speed badhne se resist karta hai.
$v$ — object ki speed (metres per second). Abhi kitni tezi se chal raha hai.
$h$ — chosen reference level se upar height (metres). "Kitna upar hai."
$x$ — spring ka apni relaxed length se stretch ya compression (metres).
$g$ — Earth ke paas gravitational field strength ( $\approx 9.8 m/s^{2}$ ). Gravity har kilogram ko kitni force se kheenchti hai.
$k$ — spring constant (newtons per metre). Spring kitna stiff hai. Dekho Spring potential energy (Hooke's law).
$F$ — ek force (newtons): ek dhakka ya kheench, direction ke saath.
$d$ — displacement: woh seedha arrow jahan se object shuru hua wahan se jahan khatam hua.
$K$ — kinetic energy $= \frac{1}{2} m v^{2}$ : motion mein stored energy.
$U$ — potential energy: position mein stored energy (jaise gravitational $m g h$ , spring $\frac{1}{2} k x^{2}$ ). Dekho Potential energy.
$Δ$ (delta) — "change in": $Δ U = U_{after} - U_{before}$ .

Definition Mechanical energy, aur teen "work" wale words

Kisi system ki Mechanical energy $E$ uski kinetic aur potential energy ka sum hai: $E = K + U .$ Yeh ek number — motion-energy plus position-energy — woh cheez hai jiske baare mein hum claim karte hain ki yeh constant rehti hai.
Kisi force dwara kiya gaya Work $W$ hota hai $W = F \cdot d$ : force ko displacement ke saath cover ki gayi distance se multiply karo. Perpendicular force $\Rightarrow$ zero work. Dekho Work-energy theorem.
$W_{cons}$ woh work hai jo ek conservative force (gravity, spring) karta hai; yeh obey karta hai $W_{cons} = - Δ U$ .
$W_{friction}$ woh work hai jo friction karta hai, jo ek non-conservative force hai; yeh hamesha negative hota hai (yeh mechanical energy ko heat mein drain kar deta hai). Dekho Energy lost to friction.
$P$ — power $= \frac{d W}{d t}$ : woh rate jis par work hota hai. Dekho Power.

Ek picture — energy do swapping tanks ki tarah

Figure — Conservation of mechanical energy — derivation

Recall

$F d x = m v d v$ kahan se aata hai? (woh gap jo derivation fill karta hai) Newton ka 2nd law kehta hai $F = m \frac{d v}{d t}$ . Dono sides ko ek tiny displacement $d x$ se multiply karo: $F d x = m \frac{d v}{d t} d x$ . Ab $d x = v d t$ use karo (displacement = speed × time): $d t$ cancel ho jata hai, aur milta hai $F d x = m v d v$ . Left side ko start se finish tak integrate karne par total work $W$ milta hai; right side ko integrate karne par, $\int_{v_{1}}^{v_{2}} m v d v = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} = Δ K$ . Yahi work–energy theorem hai $W_{net} = Δ K$ .

True ya false — justify karo

Bramhand ki total energy hamesha conserved hoti hai, isliye mechanical energy

K + U

bhi hamesha conserved hoti hai.

False — total conserved hoti hai, lekin friction aur doosre non-conservative forces energy ko heat/sound mein le jaate hain, jo

K + U

ke bahar rehti hai. Dekho Energy lost to friction.

Agar koi force badi hai, toh usse zaroor bada kaam (work) karna chahiye.

False — work hai

W = F \cdot d

, isliye motion ke perpendicular force zero work karta hai chahe woh kitna bhi bada ho (jaise normal force, string tension).

Seedha upar phenki gayi ball ke liye, uski mechanical energy top par launch se kam hoti hai kyunki woh ruk gayi hai.

False — top par kinetic energy

K

zero hai, lekin KE jo gayi woh gravitational PE

U

ban gayi; sum

K + U

unchanged hai (air ignore karte hue).

Potential energy ek aisa property hai jo ek object "khud se" rakhta hai.

False — PE ek system ki hoti hai (object + field/spring jo conservative force produce karta hai); bina field ke akela mass mein koi PE nahi hoti. Dekho Potential energy.

Aap gravity dwara kiya gaya work aur

m g h

term ko ek energy equation mein add kar sakte ho.

False — yeh double-counting hogi.

m g h

hi gravity ke work ka bookkeeping hai; force ya PE use karo, dono kabhi nahi.

Mechanical energy conserve ho sake iske liye sirf conservative forces hi present ho sakti hain.

False — non-conservative forces present ho sakti hain jab tak woh koi net work nahi karti (jaise frictionless slope par perpendicular normal force).

Ek circular orbit mein satellite mechanical energy conserve karta hai chahe gravity constantly usse kheenchti ho.

True — gravity conservative hai, aur koi non-conservative force act nahi karti, isliye

K + U

constant rehta hai (speed aur radius dono yahan constant hain).

Kisi object ki speed double karne par uski kinetic energy double ho jaati hai.

False —

K = \frac{1}{2} m v^{2}

v^{2}

par depend karta hai, isliye speed double karne par KE char guna ho jaati hai.

Gravitational PE ka zero level ground par rakhna zaroori hai.

False —

U

ka zero choose karna free hai; sirf differences

Δ U

physics mein aate hain, isliye koi bhi reference height same speeds deta hai.

Error dhundo

"Ek block rough ramp se slide karta hai; main neeche ki speed dhundne ke liye

m g h = \frac{1}{2} m v^{2}

use karunga."

Error — friction non-conservative hai aur negative work karta hai, isliye tumhe likhna hoga

m g h + W_{friction} = \frac{1}{2} m v^{2}

jahan

W_{friction} < 0

. Dekho Energy lost to friction.

"Pendulum string mein tension sabse badi force hai, isliye zaroor bob ko energy de rahi hai."

Error — tension string ke saath point karti hai, hamesha bob ki velocity ke perpendicular, isliye uska work

F \cdot d = 0

hai; sirf gravity bob ki energy change karti hai.

"Swing ke lowest point par bob ki PE maximum hai kyunki woh sabse tezi se chal raha hai."

Error — sabse tezi ka matlab maximum kinetic energy

K

hai; PE lowest point par apne minimum par hoti hai. Dono ek saath nahi badhte, balki trade off karte hain.

"Spring PE hai

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

Error — yeh hai

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

; force linearly stretch

x

ke saath badhti hai, isliye stored energy (force ka integral)

x^{2}

ke roop mein badhti hai. Dekho Spring potential energy (Hooke's law).

"Gravity ball girtey waqt negative work karti hai, isliye PE badhti hai."

Error — gravity ek girtey ball par positive work karti hai (force aur displacement dono neeche), isliye

W_{cons} > 0

aur PE ghatti hai. Work ka sign ulta ho gaya tha.

"Kyunki energy conserved hai, spring dwara launch kiya gaya block woh speed hamesha rakhega."

Error — conservation sirf launch instant par speed deta hai; baad mein woh constant rahegi ya nahi yeh baad ki forces (gravity, friction) par depend karta hai, jo single energy balance promise nahi karta.

"Maine ek ball

h

se drop ki; woh ground par

v = g h

se takraati hai."

Error —

m g h = \frac{1}{2} m v^{2}

se milta hai

v = 2 g h

; 2 ka factor chhoot gaya.

Why questions

W_{cons} = - Δ U

mein minus sign kyun aata hai?

Taaki force ka work aur stored PE opposite directions mein move karein: jab force positive work kare (

W_{cons} > 0

), stored PE ko girna chahiye (

Δ U < 0

Energy method aksar kinematics se zyada aasaan kyun hota hai?

Yeh sirf do states compare karta hai aur time aur force direction ignore karta hai, isliye aapko poori trajectory solve karne ki zaroorat nahi. Dekho Work-energy theorem.

Friction ka koi potential energy function kyun nahi ho sakta?

Kyunki friction ka work

W_{friction}

path length par depend karta hai (lamba path = zyada heat), isliye yeh path-independence test fail karta hai jo ek conservative force define karta hai. Dekho Conservative and non-conservative forces.

Work–energy theorem tab bhi kyun valid rehta hai jab forces complicated hon?

Yeh seedha Newton ke 2nd law se aata hai jo displacement par integrate kiya gaya hai; algebra

F d x = m v d v

(upar glossary derivation dekho) kabhi assume nahi karta ki force simple hai.

Energy conservation invoke karne se pehle kyun check karna chahiye ki tension/normal forces koi work nahi kar rahi?

Mechanical energy ka conservation tabhi valid hai jab non-conservative ya constraint forces zero net work karein; yeh confirm karna hi woh cheez hai jo shortcut ko valid banata hai.

Ek simple harmonic oscillator bina total energy khoye KE aur PE swap kyon karta rehta hai?

Spring force conservative hai aur (ideally) koi aur force work nahi karta, isliye

K + U

fixed hai jabki har ek baari baari utha-girta hai. Dekho Simple harmonic motion energy.

Power energy se alag quantity kyun hai, sirf uska bada version nahi?

Power

P = \frac{d W}{d t}

energy transfer ki rate hai; do processes same energy move kar sakti hain lekin bahut alag powers par. Dekho Power.

Edge cases

Ek object ko rest se drop kiya — release ke instant par uski KE aur total mechanical energy kya hai?

KE zero hai (rest mein) lekin total energy uski PE

m g h

ke barabar hai, jo nonzero hai; "rest mein" ka matlab "koi energy nahi" nahi hota.

Ek pendulum ko lowest point se zero speed ke saath release kiya — kya hota hai?

Kuch nahi hota — woh PE minimum par hai zero KE ke saath, ek stable equilibrium; ek chhoti si push sirf uss lowest point ki taraf gentle restoring swings produce karta hai.

Ek spring apni natural length par hai aur block ek pal ke liye rest mein hai — energy kahan hai?

Us instant par spring PE aur KE dono zero hain; energy ya toh poori gravitational PE mein hai ya pehle hi dissipate ho chuki hai — tumhe track karna hoga ki woh kahan gayi, yeh assume nahi karna ki woh gayab ho gayi.

Ek ball upar phenki jaati hai aur launch height par wapas aati hai — launch aur return speeds compare karo (bina air ke).

Woh magnitude mein equal hain, kyunki

U

us height par same hai isliye

K

(hence speed) match karni chahiye; sirf velocity ki direction ulti ho gayi hai.

Maximum spring compression par block ki speed zero hai — kya us instant par energy conserved hai?

Haan — KE momentarily zero hai aur sab kuch ab spring PE

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

mein hai; total unchanged hai, sirf puri tarah costume change ho gayi hai.

Ek mass table par chupchaap baith hai — kya uski mechanical energy conserved hai?

Trivially haan; KE aur (chosen) PE dono constant hain, aur gravity ka downward pull normal force se balance hai jo zero work karta hai.

Deep space mein bina fields aur forces ke, kya ek drifting probe mechanical energy conserve karta hai?

Haan, lekin trivially — PE constant hai (koi field nahi) aur KE constant hai (koi force nahi), isliye

K + U

trivially frozen hai.

Recall Ek-line self-test

Agar koi tumhe ek problem deta hai aur kehta hai "energy conservation use karo," toh pehli cheez kya check karte ho? ::: Kya har non-conservative force (friction, air, applied pushes) zero net work karti hai — agar nahi, toh uske work ko ek explicit term ke roop mein add karo instead of yeh assume karne ke ki $K + U$ constant hai.

Question bank — Conservation of mechanical energy — derivation