Spring-mass system — horizontal, vertical — Hinglish notes

1.6.8 · Physics › Oscillations & Waves

WHAT hai yeh system?

Horizontal: spring frictionless surface pe hoti hai; equilibrium spring ki natural length hoti hai.
Vertical: spring latki hoti hai; gravity pehle usse stretch karti hai, isliye equilibrium natural length ke neeche hoti hai.

WHY oscillate karta hai? (first-principles derivation)

Horizontal case — equation of motion derive karo

HOW — mass pe Newton's second law lagao. Maano $x$ natural length se displacement hai.

Sirf ek horizontal force hai: spring force (Hooke's law): $F = - k x$

Minus sign ka matlab: right stretch karo ( $x > 0$ ) → force left taraf. Yahi iska restoring nature hai.

Newton's law $F = ma = m \overset{x}{¨}$ : $m \overset{x}{¨} = - k x \Rightarrow \overset{x}{¨} + \frac{k}{m} x = 0$

Yeh hai SHM equation $\overset{x}{¨} + ω^{2} x = 0$ jisme $ω = \frac{k}{m}$

Vertical case — kya gravity kuch badlati hai?

WHY yeh poochna? Intuitively gravity ek constant downward pull add karti hai, toh lagta hai oscillation slow ya fast hogi. Chalo derive karte hain aur forecast-then-verify karte hain.

Mass latkaao. Do forces: spring $+ k$ direction mein aur gravity $m g$ neeche. Maano $y$ neeche ki taraf point karta hai, natural length se measure kiya gaya.

Equilibrium pe mass extension $y_{0}$ pe baithti hai jahan forces balance hon: $k y_{0} = m g \Rightarrow y_{0} = \frac{m g}{k}$

Yeh $y_{0}$ hai static extension — purely rest point ka ek shift.

Ab general $y$ pe displace karo. Net downward force: $F = m g - k y$

Newton: $m \overset{y}{¨} = m g - k y$ . Naya coordinate $x = y - y_{0}$ (displacement naye equilibrium se) substitute karo, toh $\overset{y}{¨} = \overset{x}{¨}$ : $m \overset{x}{¨} = m g - k (x + y_{0}) = m g - k x - k y_{0} = = 0 (m g - k y_{0}) - k x$ $\Rightarrow m \overset{x}{¨} = - k x$

Bonus trick: kyunki $y_{0} = m g / k$ hai, toh $k = m g / y_{0}$ milta hai, isliye $T = 2 π \frac{m}{k} = 2 π \frac{y _{0}}{g}$ Static stretch $y_{0}$ maap lo → $k$ ya $m$ jaane bina period mil jaata hai!

Figure — Spring-mass system — horizontal, vertical

Energy view (alternate derivation)

HOW se $ω$ derive karein energy se: $E$ ko time ke respect mein differentiate karo (energy constant hai toh $d E / d t = 0$ ): $\frac{d E}{d t} = m v \overset{v}{˙} + k xv = v (m \overset{x}{¨} + k x) = 0$ Kyunki generally $v \neq = 0$ , toh $m \overset{x}{¨} + k x = 0$ — same SHM equation. ✓ Do raaste, ek answer.

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek toy car jo ek smooth table pe stretchy rubber band se bandhi ho. Kheencho aur choddo: band usse wapas kheenchti hai, woh beech se guzar jaati hai, doosri taraf stretch karti hai, phir wapas kheenchi jaati hai — hamesha aage-peechhe (agar kuch slow na kare). Band jitni stiff, utni jaldi wiggle; car jitni heavy, utni lazy wiggle. Ab wohi band ceiling se weight ke saath latkaao: gravity sirf weight ko ek naye resting spot pe drop karti hai, lekin bounce karo toh bilkul wahi speed pe bounce karta hai jaise table pe — gravity ne sirf "home base" move ki, bounce nahi badla.

Connections

Simple Harmonic Motion — yeh uska sabse clean example hai.
Hooke's Law — $F = - k x$ provide karta hai jo SHM banata hai.
Simple Pendulum — compare karo: pendulum $T = 2 π L / g$ mein $g$ depend karta hai, spring mein nahi.
Energy in SHM — $\frac{1}{2} k A^{2}$ conservation argument.
Springs in Series and Parallel — effective stiffness.
Damped Oscillations — jab friction add ho toh kya hota hai.

Flashcards

Horizontal spring-mass system ki equation of motion kya hai?

m \overset{x}{¨} = - k x

, yaani

\overset{x}{¨} + (k / m) x = 0

Spring-mass oscillator ki angular frequency kya hai?

ω = k / m

Spring-mass system ka period

T

kya hai?

T = 2 π m / k

Kya horizontal aur vertical spring-mass systems ka period alag hota hai?

Nahi — dono mein

T = 2 π m / k

milta hai; gravity sirf equilibrium shift karti hai.

Gravity vertical spring ka period kyun affect nahi karti?

Yeh constant hai, isliye sirf equilibrium relocate karti hai (

k y_{0} = m g

); naye equilibrium ke baare mein restoring force abhi bhi

- k x

hai.

Vertical spring ki static extension kya hai?

y_{0} = m g / k

Sirf static stretch use karke vertical spring ka period express karo.

T = 2 π y_{0} / g

Horizontal SHM mein total mechanical energy kya hai?

E = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2} = \frac{1}{2} k A^{2}

Spring SHM mein maximum speed kya hai?

v_{ma x} = A ω = A k / m

, equilibrium pe hoti hai.

Do springs parallel mein effective stiffness?

k_{e f f} = k_{1} + k_{2}

(stiffer, chota

T

Do springs series mein effective stiffness?

1/ k_{e f f} = 1/ k_{1} + 1/ k_{2}

(softer, bada

T

Mass badhane se period pe kya asar padta hai?

T

badhta hai (slower oscillation), kyunki

T \propto m

Spring-mass system — horizontal, vertical