1.2.11 · D1Inclined planes — with and without friction
Inclined plane ka problem sirf itna hai: gravity do perpendicular tukdon mein split hoti hai — ek tukda block ko ramp ke saath slide karata hai, aur doosra usse ramp mein press karta hai. Sirf yeh seekh lo ki ek seedha neeche wala arrow un do tukdon mein kaise banta hai, aur har incline formula apne aap samajh aa jaayega.
Gravity ko split karne se pehle, tumhe symbols aur pictures ka ek toolkit chahiye. Neeche har woh symbol aur idea diya gaya hai jis par parent note Inclined Planes tika hua hai — bilkul zero se shuru karke, har ek pehle wale ke baad.
m m ek object mein kitna matter hai, kilograms (kg
Picture: m dot ki size ki tarah socho jo hum object ke liye draw karte hain. Badi dot = zyada mass = hilana mushkil.
Topic ko yeh kyun chahiye: incline par har force (weight, friction) m
Definition Gravitational accelerationg g = 9.8 m/s 2 har second 9.8 metres-per-second aur tez ".
Picture: ek girta hua pathar, jiska speed arrow har second ek hi chunk se bada hota hai. Neeche figure dekho.
Topic ko yeh kyun chahiye: incline sirf gravity ko "dilute" karta hai, isliye g Free Fall & Gravity mein.
Figure 1 (upar): ek kala girta hua dot t = 1 , 2 , 3 red velocity arrow har second ek hi chunk se lamba hota jaata hai — yahi steady growth g speed badlti hai, aur woh constant rate par badlti hai.
W Weight woh force hai jisse Earth kisi object ko neeche kheenchti hai. Force matlab "ek dhakka ya kheench". Jitna stuff hai (m g W = m g newton (N 1 N = 1 kg ⋅ m/s 2
Common mistake Mass aur weight ek nahi hainKyun ek jaisa lagta hai: Earth par yeh dono saath saath chalte hain.
Fark yeh hai: m kitna stuff hai ; W = m g gravity us stuff ko kitna kheenchti hai . Moon par m W force m g m
Vector ek aisi quantity hai jisme dono chahiye: size (kitna bada) aur direction (kidhar). Hum isse ek arrow ki tarah draw karte hain: length = size, arrowhead = direction.
Picture: weight woh arrow hai jo seedha neeche point karta hai, jiski length m g
Topic ko yeh kyun chahiye: gravity, normal force aur friction sab vectors hain. "Gravity split karna" literally ek arrow ko do arrows mein kaatna hai. Dekho Vector Resolution & Components .
Do lines perpendicular hoti hain jab woh right angle (9 0 ∘
Topic ko yeh kyun chahiye: hum do perpendicular directions chunte hain (slope ke saath, slope mein) kyunki ek direction ki force ka doosri direction par zero effect hota hai. Isi wajah se split clean hoti hai.
θ θ flat ground se tilt hai. θ = 0 θ = 9 0 ∘
Picture: neeche figure mein wedge, jisme θ
Figure 2 (upar): ek wedge. Horizontal floor aur vertical side kaale hain; red line ramp ki surface hai — yahi key object hai. Bottom-left corner mein chhota arc θ θ → 0 θ → 9 0 ∘
Yahi woh tool hai jo actual kaam karta hai. Trig kyun, kuch aur kyun nahi? Kyunki "ek tilted arrow ka kitna hissa kisi given direction mein point karta hai" — yahi sawaal ek right triangle answer karta hai, aur sine/cosine us triangle ke do ratios hain.
Definition Sine aur cosine, triangle parWeight arrow (length m g θ
cos θ = longest side ( m g ) side next to θ ( adjacent ) andar jaata hai.sin θ = longest side ( m g ) side across from θ ( opposite ) saath jaata hai.
Rearrange karo, do tukde milte hain:
m g ⊥ = m g cos θ , m g ∥ = m g sin θ
Figure 3 (upar): block ramp par baitha hai (dot). Red arrow pura weight m g slope ke neeche jaata hai (m g sin θ surface ke andar point karta hai (m g cos θ θ
Definition Sign convention — yeh magnitudes hain fixed directions ke saathm g ∥ = m g sin θ m g ⊥ = m g cos θ positive magnitudes hain (arrows ki lengths), kabhi negative nahi. Unki directions geometry se fix hain, isliye hum unhe haath se track karte hain:
m g ∥ = m g sin θ slope ke neeche act karta hai — hum slope ke neeche ko positive x lete hain.m g ⊥ = m g cos θ surface ke andar act karta hai — surface wapas N y
Toh jab hum force equations likhte hain, sign hum khud insert karte hain based on kis taraf har arrow point karta hai; trig sirf size deta hai.
Intuition Triangle mein YEH angle kyun aata haiSlope ground se θ θ equal hote hain — toh wahi θ
N perpendicular . N
Picture: ek arrow jo ramp ki surface se bahar point karta hai (positive y m g cos θ
Topic ko yeh kyun chahiye: N N = m g cos θ m g kam hai — ek key surprise jo parent note baar baar batata hai.
μ s μ k μ grippy hain. μ s pehle ki grip rate karta hai; μ k dauran ki grip rate karta hai. Rubber-on-road mein bada μ μ
a a velocity kitni tezi se badal rahi hai — speed arrow ke barne ki rate, m/s 2
F F = ma F net force hai: woh single arrow jo tumhe tab milta hai jab ek axis par object ke saare force arrows (apne signs ke saath) add karo. Yeh koi ek particular force nahi hai — yeh bacha hua hissa hai.
Topic ko yeh kyun chahiye: a F m
Definition Free body diagram (FBD)Ek stripped-down sketch jo sirf object ko ek dot ki tarah aur har force ko us dot se nikalta hua arrow dikhata hai. Koi ramp drawing nahi, koi clutter nahi.
Topic ko yeh kyun chahiye: yahan weight (m g N μ N Free Body Diagrams .
mg sin theta down the slope
mg cos theta into surface
Newtons Second Law F equals ma
Free body diagram ties it together
Khud test karo — right side cover karo aur zor se jawab do.
m Matter ki miqdar, kilograms (kg) mein.
g Gravitational acceleration, lagbhag 9.8 m/s 2
m g W = m g
Vector mein hamesha kaun si do cheezein hoti hain? Ek size (length) aur ek direction.
"Perpendicular" ka matlab kya hai aur hum perpendicular axes kyun chahte hain? 9 0 ∘
Angle θ Flat ground se; θ = 0 θ = 9 0 ∘
Gravity ka kaun sa tukda sine use karta hai, aur kaun sa cosine, aur dono kidhar point karte hain? Sine → slope ke neeche (slides); cosine → surface ke andar (jise N
Kya m g sin θ m g cos θ Nahi — yeh positive magnitudes hain; direction/sign hum khud lagate hain.
Limits se check karo: θ = 0 sin θ cos θ sin 0 = 0 cos 0 = 1
"Normal" ka matlab kya hai aur slope par N Perpendicular; N m g cos θ
F = ma F Ek axis par net (total) force, saare force arrows ko unke signs ke saath add karne ke baad.
Slope ke neeche frictionless acceleration derive karo. ma = m g sin θ ⇒ a = g sin θ
Static friction: inequality ya equality, aur uski ceiling kya hai? Inequality, F s ≤ μ s N μ s N ≥ m g sin θ
Kinetic friction: uska exact size aur direction kya hai? F k = μ k N
