Centripetal force — what provides it in various situations

1.2.16 · Physics › Newton's Laws & Dynamics

KYUN chahiye circular motion ko ek inward force?

Centripetal acceleration ki derivation (scratch se)

Formula

$a = v^{2} / r$ ko geometrically derive karna Ek particle ko do instants par consider karo jo $Δ t$ time se alag hain, aur jo ek chota angle $Δ θ$ sweep karta hai.

Step 1 — position vectors. Dono velocity vectors ka magnitude $v$ same hai. Jab particle $Δ θ$ se turn karta hai, velocity vector bhi $Δ θ$ se turn karta hai (velocity hamesha radius ke perpendicular hoti hai). Yeh step kyun? Velocity circle ki tangent hoti hai; radius ko $Δ θ$ se rotate karo toh tangent bhi utne hi $Δ θ$ se rotate hoti hai.

Step 2 — velocities ka chota triangle. Do velocity vectors $v_{1}, v_{2}$ (har ek ki length $v$ ) aur unka difference $Δ v$ ek isosceles triangle banate hain. Chote angle ke liye, choti side ≈ arc: $∣Δ v ∣ \approx v Δ θ$ Yeh step kyun? Chote angles ke liye, chord ≈ arc = (is triangle ki radius, jo $v$ hai) × angle.

Step 3 — time se divide karo. $a = \frac{∣Δ v ∣}{Δ t} = v \frac{Δ θ}{Δ t} = v ω$ Yeh step kyun? $Δ θ /Δ t$ angular velocity $ω$ hoti hai.

Step 4 — $v = ω r$ use karo. Kyunki arc speed $v = ω r$ hai, yaani $ω = v / r$ : $a_{c} = \frac{v ^{2}}{r} = ω^{2} r$ Direction: center ki taraf (centripetal = "center-seeking").

KAUNSA force centripetal force provide karta hai? (situation by situation)

Figure — Centripetal force — what provides it in various situations

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek ball ko string se baandho aur apne sar ke upar ghuma lo. String ball ko inward teri haath ki taraf kheenchti rehti hai — woh inward kheench hi ball ko circle mein rakhti hai straight fly off hone ki jagah. Jis second tune chhoda, ball seedhi line mein shoot kar jaati hai, kyunki koi cheez usse inward nahi kheench rahi. "Centripetal force" sirf ek fancy naam hai jo bhi inward kheechna kar raha ho uske liye: ek string, gravity, car tyres ki grip, ya ek magnet. Yeh kabhi brand-new force nahi hota — yeh ek kaam hai jo koi real force kar raha hota hai.

Flashcards

Centripetal force physically kya hai?

Koi naya force nahi — jo bhi real force (ya net inward component) center ki taraf point kare aur

m v^{2} / r

supply kare, uska naam hai.

Centripetal acceleration ka magnitude derive karo.

∣Δ v ∣ = v Δ θ

, toh

a = v Δ θ /Δ t = v ω = v^{2} / r = ω^{2} r

, center ki taraf directed.

Satellite ke liye centripetal force kya provide karta hai?

Gravity:

GM m / r^{2} = m v^{2} / r

Flat curve par car ke liye kya provide karta hai, aur speed ko kya limit karta hai?

Static friction;

v_{ma x} = μ_{s} g r

Banking angle formula (frictionless) aur kaunsa force

F_{c}

provide karta hai?

tan θ = v^{2} / (r g)

; normal force ka horizontal component.

Vertical loop ke top par force equation aur minimum speed likho.

T + m g = m v^{2} / r

;

v_{min} = g r

jab

T = 0

Turning car mein bahar ki taraf throw hone ka feel kyun hota hai?

Inertia (body seedha-line motion chahta hai); outward "centrifugal" force ek pseudo-force hai, ground frame mein real nahi.

Magnetic field mein charge ke liye circular path ki radius?

q v B = m v^{2} / r \Rightarrow r = m v / (q B)

Connections

Newton's Second Law — $F = ma$ inward direction par apply hota hai.
Uniform Circular Motion — $v = ω r$ ki kinematics.
Gravitation and Orbits — gravity provider ke roop mein.
Friction — static friction ceiling $μ_{s} N$ .
Banking of Roads — normal force geometry.
Pseudo-forces and Non-inertial Frames — centrifugal force.
Magnetic Force on Moving Charges — $q v B$ centripetal ke roop mein.