1.2.16 · D1Centripetal force — what provides it in various situations
Jo bhi circle mein move kar raha hai woh turn kar raha hai, aur turn karne ka matlab hai ki uski direction of motion continuously change ho rahi hai — woh change hi ek acceleration hai jo hamesha middle ki taraf point karti hai. Us turn ko make karne ke liye, koi real force (ek string, gravity, friction, ek magnet…) zaroor woh inward pull supply karni chahiye; "centripetal force" bas ek job-title hai jo hum us real force ko dete hain jo woh pulling kar rahi hai. (Hum neeche har symbol build karte hain pehle koi bhi formula likhne se.)
Pehle tum parent note Centripetal force — what provides it padh sako, tumhare paas har woh symbol hona chahiye jo woh tumhare saamne phenkta hai. Hum unhe ek-ek karke build karte hain, har ek us picture se jo uske saath rehti hai.
Definition Moving body ko ekpoint maano
Hum poore object (patthar, car, planet) ko ek single point mein shrink kar dete hain jo ek path trace karta hai. Hum jo bhi bolte hain woh is baare mein hai ki woh point kahan hai aur woh kitni tez move karta hai .
Ek dot ko ek curved track par slide karte hue imagine karo. Woh dot hi abhi ke liye hamaari poori duniya hai.
r r center se moving point tak ki fixed distance hai. Uski picture: ek wheel ke hub se rim tak ek seedha spoke.
Figure dekho: orange dot moving point hai, magenta spoke r circle banati hai na ki koi wobbly loop. Is topic ko r kamzor hoti jaati hai jab circle wider hoti jaati hai (bada r
Intuition Arrow kyun, sirf number kyun nahi?Speed jaise "5 m/s" sirf ek size hai. Lekin kis taraf point move karta hai yeh utna hi matter karta hai. Ek arrow — ek vector — dono store karta hai size (uski length) aur direction (jahan woh point karta hai). Hum vectors ke upar ek chhota arrow likhte hain: v
v v point se chipka hua, track ke along point karta hua — woh direction jisme woh iss instant release hone par fly off kar jaata.
Figure dekho: circle par har jagah violet velocity arrow tangent lie karta hai — woh circle ko bas graze karta hai, hamesha magenta spoke r 9 0 ∘
∣ v ∣ v ∣ v ∣ length hai = plain speed, jise simply v v
Δ Δ delta ) kisi bhi cheez ke aage matlab hai "final minus initial" — yeh kitna badla. Toh Δ t Δ v = v final − v initial
Intuition Do arrows subtract karna — picturev 2 − v 1 v 1 v 2 Δ v v 1 v 2 length same ho (same speed), agar woh alag taraf point karte hain toh yeh connecting arrow zero nahi hoga.
Figure dekho: do violet arrows length mein equal hain (speed unchanged) lekin ek chhote angle se rotated apart hain. Orange arrow jo unki tips ko bridge karta hai woh Δ v original circle ke center ki taraf point karta hai. Yahi centripetal acceleration ka beej hai: constant speed par bhi velocity mein non-zero change.
a a Δ t Δ v Δ v Δ t direction same rakhta hai (center ki taraf) lekin rescale ho jaata hai.
Kyunki Δ v a centripetal ("center-seeking") acceleration hai.
θ θ theta , seedhe alfazon mein: "kitna around, ya kitna tilted") rotation ya tilt measure karta hai. Δ θ Δ t chhota turn hai.
radians mein measure kyun karein
Ek radian aise define hota hai ki arc length = × . Picture: length r 1 = r θ bina kisi extra conversion factor ke true banate hain — aur derivation tiny angles ke liye arc ≈
Definition Small-angle rulesin x ≈ x tiny angle x sin x ≈ x ( for small x ) . Half-chord r sin ( Δ θ /2 ) half-arc r ( Δ θ /2 ) sin ( Δ θ /2 ) ≈ Δ θ /2 chord = 2 r sin ( 2 Δ θ ) ≈ 2 r ⋅ 2 Δ θ = r Δ θ . Swap kitna accurate hai? Error x 3 Δ θ = 0.1 ≈ 5. 7 ∘ sin ( 0.05 ) = 0.0499979 0.05 0.01% Δ θ → 0 chord / ( r Δ θ ) → 1
Figure dekho: magenta arc ki length r Δ θ Δ θ ≈ sin x ≈ x ∣Δ v ∣ ≈ v Δ θ
Δ v ∣Δ v ∣ ≈ v Δ θ v 1 , v 2 same length v Δ v isosceles triangle banate hain — length v Δ θ Δ v v ∣Δ v ∣ = 2 v sin ( 2 Δ θ ) ≈ v Δ θ . v Δ θ a = v 2 / r
ω ω omega , seedhe alfazon mein: "kitne radians turn per second") Δ t Δ θ ω
See Uniform Circular Motion for more on the v = ω r
m F m F
F c F c "required net inward force" ka shorthand hai — saari real forces ke inward slices ka total, kuch nahi. Yeh mass times centripetal acceleration ke equal hai:
F c = m a c = m r v 2 = m ω 2 r . Dhyan se padho: F c requirement hai jo poori karni hai , koi extra force nahi jo tum draw karo. Kisi bhi problem mein tum neeche real providers dhundhte ho aur unka combined inward pull F c Newton's Second Law .
Tum inhe situation-by-situation tour mein miloge. Har ek ek real force hai jo shayad inward point kar rahi ho — yaani jo F c
Definition Providers ki quick glossary
T tension , ek string ka inward pull (Friction ise rubbing se contrast karta hai). Picture: ek taut rope stone ko tumhare haath ki taraf kheench rahi hai.g m g weight , Earth ka downward pull.G , M r 2 GM m Gravitation and Orbits .N normal force , ek surface jo apne aap ke perpendicular push deti hai. Ek tilted road par uska sideways slice turning karta hai. See Banking of Roads .f s , μ s static friction aur uska coefficient; friction at most μ s N Friction .q , B q v B Magnetic Force on Moving Charges ."centrifugal" — spinning frame mein feel hone wali fake outward force; ground frame mein real nahi. See Pseudo-forces and Non-inertial Frames .
Velocity v tangent to circle
Delta v change of velocity
Small angle rule sin x approx x
Delta v approx v times delta theta
Acceleration a points inward
Centripetal a equals v squared over r
Newton second law F equals m a
Required inward force Fc equals m v squared over r
What real force provides it
Ek vector kaunsi do cheezein store karta hai? Uski size (length) aur uski direction.
∣ v ∣ Velocity arrow ki length — plain speed v
Circle par velocity arrow kis taraf point karta hai? Circle ke tangent, radius r 9 0 ∘
Kisi quantity ke aage Δ Usmein change: final minus initial.
Δ v Kyunki
v ki direction change hoti hai, toh do arrows equal length ke bawajood differ karte hain.
Small-angle rule batao aur kyun yeh chord ≈ sin x ≈ x x = 2 r sin ( Δ θ /2 ) ≈ r Δ θ
∣Δ v ∣ ≈ v Δ θ Equal-length velocity arrows ek isosceles triangle banate hain;
∣Δ v ∣ = 2 v sin ( Δ θ /2 ) ≈ v Δ θ .
Circular motion mein a Center ki taraf (centripetal).
Angle radians mein kyun measure karte hain? Taaki arc length = r θ
ω v Radians turned per second; v = ω r
Newton's Second Law ko arrow equation ke roop mein batao. F c Net inward force required , F c = m v 2 / r
