GRAVITATION
Chapter 9, Gravitation, is a cornerstone of physics, explaining why objects fall, how planets orbit, and the phenomenon of tides. You'll learn about Newton's universal law of gravitation, the concept of acceleration due to gravity (g), and the distinction between mass and weight. The chapter also delves into fluid mechanics, covering thrust, pressure, buoyancy, and Archimedes' principle, explaining why objects float or sink. Understanding these concepts is crucial for building a strong foundation in science.
Gravitation: Introduction
Newton ne observe kiya ki ek apple neeche girta hai, aur Moon Earth ke around revolve karta hai. Unhone conjecture kiya ki dono cases mein same type ki force responsible hai.
- Centripetal Force: Circular path mein move karne ke liye ek object ko center ki taraf force chahiye hoti hai. Is force ko centripetal force kehte hain.
- Example: Stone ko thread se ghumane par, thread ki tension centripetal force provide karti hai. Agar thread chhod de toh stone tangential direction mein seedha chala jaata hai.
- Moon Earth ke around revolve karta hai, toh usko bhi ek centripetal force chahiye. Ye force Earth ki gravitational attraction se milti hai.
- Agar Earth Moon ko attract karti hai, toh Moon Earth ki taraf girta kyu nahi?
- Moon Earth ki taraf har point par girta hai, but uski orbital velocity usko straight line mein jaane se rok kar circular path mein maintain karti hai. Isliye Moon Earth par crash nahi karta.
Gravitational force ek attractive force hai jo universe mein har do objects ke beech act karti hai.
Centripetal Force: Woh force jo kisi object ko circular path mein move karne ke liye center ki taraf act karti hai. Latin mein 'centre-seeking' matlab 'centripetal'.
Universal Law of Gravitation
Newton ne ye law diya jo batata hai ki universe mein har do object ek dusre ko attract karte hain. Ye attraction force:
- Unke masses ke product ke proportional hoti hai.
- Unke beech ki distance ke square ke inversely proportional hoti hai.
- Force dono objects ke centers ko join karne wali line ke along act karti hai.
Mathematical Form: Consider two objects with masses \(M\) and \(m\), separated by a distance \(d\).
- Force \(F \propto M \times m\)
- Force \(F \propto \frac{1}{d^2}\)
Combining these, we get: \(F \propto \frac{M m}{d^2}\)
Proportionality sign hatane ke liye, hum ek constant use karte hain, jise Universal Gravitational Constant (G) kehte hain: \(F = G \frac{M m}{d^2}\)
- G ki Value: \(G = 6.673 \times 10^{-11} \text{ Nm}^2 \text{ kg}^{-2}\)
- G ki SI Unit: \(Nm^2 kg^{-2}\)
- G ki Nature: Ye ek universal constant hai, iski value medium, temperature, ya objects ke nature par depend nahi karti.
Universal Law of Gravitation: \(F = G \frac{M m}{d^2}\)
Jahan:
- \(F\) = Gravitational Force
- \(G\) = Universal Gravitational Constant
- \(M, m\) = Masses of the two objects
- \(d\) = Distance between their centers
Henry Cavendish ne 1798 mein sensitive torsion balance ka use karke G ki value find ki thi.
Importance of Universal Law of Gravitation
Ye law bahut saare phenomena ko explain karta hai jo pehle unconnected lagte the:
- Humein Earth se bind karna: Jis force se hum Earth par khade hain ya koi cheez neeche girti hai, woh gravitational force hai.
- Moon ka Earth ke around motion: Moon Earth ke gravitational pull ki wajah se hi uske around revolve karta hai.
- Planets ka Sun ke around motion: Saare planets Sun ke gravitational force ki wajah se hi uske around elliptical orbits mein ghoomte hain.
- Tides ka formation: Ocean mein tides (high and low) Moon aur Sun ki gravitational pull ki wajah se aati hain.
- Rivers ka flow: Nadiyon ka paani bhi Earth ki gravity ki wajah se hi neeche ki taraf flow karta hai.
Ye points direct question mein pooche jaate hain. Minimum 3-4 points yaad rakho.
Free Fall
Jab koi object sirf gravitational force ke under girta hai (air resistance ko ignore karte hue), toh uski motion ko Free Fall kehte hain.
- Free fall mein object ki velocity continuously change hoti hai, yaani usmein acceleration hota hai.
- Ye acceleration Earth ki gravitational force ki wajah se hota hai, isliye ise acceleration due to gravity (g) kehte hain.
- Value of g: Earth ki surface par \(g \approx 9.8 \text{ m/s}^2\).
- Direction of g: Always vertically downward, towards the center of the Earth.
Calculation of 'g' (Acceleration due to Gravity)
Hum Newton's Universal Law of Gravitation aur Newton's Second Law of Motion ko combine karke 'g' ki value derive kar sakte hain.
- Consider an object of mass \(m\) on or near the Earth's surface.
- Earth ka mass \(M_e\) aur radius \(R_e\) hai.
- Gravitational force \(F = G \frac{M_e m}{R_e^2}\)
- Newton's Second Law ke according, \(F = m a\). Free fall mein acceleration \(a = g\), toh \(F = m g\).
Equating both forces: \(m g = G \frac{M_e m}{R_e^2}\) \(g = G \frac{M_e}{R_e^2}\)
Values substitute karne par:
- \(G = 6.67 \times 10^{-11} \text{ Nm}^2 \text{ kg}^{-2}\)
- \(M_e = 6 \times 10^{24} \text{ kg}\)
- \(R_e = 6.4 \times 10^6 \text{ m}\)
\(g = (6.67 \times 10^{-11}) \times \frac{(6 \times 10^{24})}{(6.4 \times 10^6)^2}\) \(g \approx 9.8 \text{ m/s}^2\)
Variation of 'g'
- Height ke saath: Earth ki surface se upar jaane par \(d\) increase hota hai, toh \(g\) decrease hota hai.
- Depth ke saath: Earth ke andar jaane par bhi \(g\) decrease hota hai (mass of Earth attracting the object decreases).
- Shape of Earth: Earth perfect sphere nahi hai, poles par thodi flat aur equator par bulge hai.
- Poles par radius kam hai, isliye \(g\) zyada hai (approx \(9.83 \text{ m/s}^2\)).
- Equator par radius zyada hai, isliye \(g\) kam hai (approx \(9.78 \text{ m/s}^2\)).
Motion of Objects under Gravity
- Free fall mein, air resistance ko ignore karein toh saare objects same rate se girte hain, irrespective of their mass, shape, ya size.
- Equations of motion (v = u + at, s = ut + 1/2 at², v² = u² + 2as) ko modify karke use karte hain:
- \(a\) ko \(g\) se replace karte hain.
- Upward motion: \(g\) ko negative lete hain (gravity ke against).
- Downward motion: \(g\) ko positive lete hain (gravity ki direction mein).
| Equation | General Form | Under Gravity (Downward) | Under Gravity (Upward) | |---|---|---|---| | Velocity-Time | \(v = u + at\) | \(v = u + gt\) | \(v = u - gt\) | | Position-Time | \(s = ut + \frac{1}{2}at^2\) | \(s = ut + \frac{1}{2}gt^2\) | \(s = ut - \frac{1}{2}gt^2\) | | Position-Velocity | \(v^2 = u^2 + 2as\) | \(v^2 = u^2 + 2gs\) | \(v^2 = u^2 - 2gs\) |
Free Fall: Jab koi object sirf Earth ki gravitational force ke influence mein move karta hai, tab uski motion ko free fall kehte hain.
Acceleration due to Gravity (g): Gravitational force ke karan produce hone wala acceleration. Iski average value Earth par \(9.8 \text{ m/s}^2\) hai.
Students aksar \(G\) aur \(g\) ko confuse karte hain. Yaad rakho, \(G\) universal constant hai, \(g\) acceleration hai jo place to place vary karta hai.
Mass
Mass kisi object mein present matter ki quantity ka measure hai.
- Inertia ka measure: Jis object ka mass zyada hota hai, uski inertia bhi zyada hoti hai (yani uski state of motion change karna mushkil hota hai).
- Constant quantity: Mass ek object ki intrinsic property hai. Ye universe mein kahin bhi change nahi hota, chahe object Earth par ho, Moon par ho, ya outer space mein ho.
- Scalar quantity: Iski sirf magnitude hoti hai, direction nahi.
- SI Unit: kilogram (kg).
Mass: Kisi object mein maujood matter ki quantity. Ye uski inertia ka measure bhi hai.
Weight
Weight woh force hai jis se Earth kisi object ko apni taraf attract karti hai.
- Formula: \(W = m \times g\)
- \(W\) = Weight
- \(m\) = Mass of the object
- \(g\) = Acceleration due to gravity
- Vector quantity: Iski magnitude aur direction dono hoti hain (direction always towards the center of Earth).
- SI Unit: Newton (N), kyuki ye ek force hai. (Kgf bhi use hota hai, \(1 \text{ kgf} = 9.8 \text{ N}\)).
- Variable quantity: Weight place to place vary karta hai kyuki \(g\) ki value vary karti hai. Example: Poles par weight zyada hoga, equator par kam. Moon par weight Earth se kam hoga.
Weight of an Object on the Moon
Moon ka mass aur radius Earth se kam hai, isliye Moon par \(g\) ki value bhi Earth se kam hoti hai.
- \(g_{moon} = G \frac{M_{moon}}{R_{moon}^2}\)
- \(g_{earth} = G \frac{M_{earth}}{R_{earth}^2}\)
Calculations se pata chalta hai ki \(g_{moon} \approx \frac{1}{6} g_{earth}\).
Isliye, kisi object ka weight Moon par Earth par uske weight ka 1/6th hota hai. \(W_{moon} = m \times g_{moon} = m \times \frac{1}{6} g_{earth} = \frac{1}{6} W_{earth}\)
Example: Agar ek object ka weight Earth par 60 N hai, toh Moon par uska weight \(10 \text{ N}\) hoga. Lekin uska mass dono jagah \(60/9.8 \approx 6.12 \text{ kg}\) hi rahega.
Weight: Woh force jis se Earth (ya koi celestial body) kisi object ko apne center ki taraf attract karti hai. \(W = mg\).
Weight of an object on the Moon is 1/6th of its weight on the Earth.
Thrust and Pressure
Thrust
- Jab koi force kisi surface par perpendicularly act karti hai, toh use Thrust kehte hain.
- Thrust ki SI unit Newton (N) hai, kyuki ye ek type ki force hi hai.
- Example: Jab hum drawing pin ko board mein push karte hain, toh hamari finger ki force pin ke head par perpendicular act karti hai, aur pin ka tip board par thrust apply karta hai.
Pressure
- Pressure per unit area par lagne wala Thrust hai.
- Formula: \(P = \frac{\text{Thrust}}{\text{Area}} = \frac{F}{A}\)
- SI Unit: Pascal (Pa). \(1 \text{ Pa} = 1 \text{ N/m}^2\).
- Dependence: Pressure do factors par depend karta hai:
- Thrust (Force): Zyada thrust, zyada pressure (area constant ho toh).
- Area: Kam area, zyada pressure (thrust constant ho toh). Isliye, knives sharp hote hain (kam area, zyada pressure), aur school bags ki straps broad hoti hain (zyada area, kam pressure).
Pressure in Fluids (Liquids and Gases)
- Fluids (liquids aur gases) mein particles freely move karte hain, isliye woh container ki walls aur base par pressure exert karte hain.
- Fluid ke andar, pressure har direction mein equally transmit hota hai (Pascal's Law).
- Depth ke saath pressure increase hota hai.
Thrust: Kisi surface par perpendicular direction mein act karne wali force. Unit: Newton (N).
Pressure: Per unit area par lagne wala thrust. Formula: \(P = \frac{F}{A}\). Unit: Pascal (Pa) ya \(N/m^2\).
Pressure ke applications par reason-based questions bahut common hain (e.g., why camels have broad feet, why buildings have wide foundations).
Buoyancy
Jab koi object kisi fluid (liquid ya gas) mein partially ya fully immerse kiya jaata hai, toh fluid us object par upward direction mein ek force exert karta hai. Is upward force ko Buoyant Force ya Upthrust kehte hain.
- Reason: Fluid ke andar, depth ke saath pressure increase hota hai. Object ke bottom surface par top surface se zyada pressure lagta hai, jiski wajah se net upward force generate hoti hai.
- Effects of Buoyant Force:
- Object ka apparent weight kam ho jaata hai jab use fluid mein immerse karte hain.
- Objects float ya sink karte hain buoyant force aur object ke weight ke comparison par.
Factors Affecting Buoyant Force
Buoyant force do factors par depend karti hai:
- Volume of the fluid displaced: Jitna zyada fluid displace hoga, utni zyada buoyant force lagegi.
- Density of the fluid: Jitni zyada fluid ki density hogi, utni zyada buoyant force lagegi.
- Object ki density ya mass par directly depend nahi karti, sirf displaced fluid par.
Buoyant Force (Upthrust): Woh upward force jo ek fluid kisi object par exert karta hai jab object usmein partially ya fully immerse hota hai.
Buoyant force hamesha upward direction mein act karti hai, gravity ke opposite.
Archimedes' Principle
Archimedes' Principle states karta hai ki jab koi body kisi fluid mein fully ya partially immerse hoti hai, toh woh ek upward buoyant force experience karti hai jo uske dwara displaced fluid ke weight ke equal hoti hai.
- Mathematical Form: \(F_b = W_{displaced\_fluid} = m_{displaced\_fluid} \times g\)
- Hum jante hain ki \(\text{Density} (\rho) = \frac{\text{Mass}}{\text{Volume}}\), toh \(\text{Mass} = \text{Density} \times \text{Volume}\).
- So, \(F_b = \rho_{fluid} \times V_{displaced\_fluid} \times g\)
Applications of Archimedes' Principle
- Hydrometers: Liquids ki density measure karne ke liye use hote hain.
- Lactometers: Milk ki purity check karne ke liye use hote hain.
- Ships and Submarines: Inka design Archimedes' Principle par based hota hai. Ships float karte hain kyuki woh apne weight ke barabar paani displace karte hain, aur unki average density paani se kam hoti hai.
Why Objects Float or Sink?
Kisi object ka float ya sink karna uski density aur fluid ki density par depend karta hai:
- Float: Agar object ki density (\(\rho_{object}\)) fluid ki density (\(\rho_{fluid}\)) se kam hai, toh object float karega. (\(\rho_{object} < \rho_{fluid}\))
- Is case mein, object ka weight buoyant force se kam hota hai.
- Sink: Agar object ki density (\(\rho_{object}\)) fluid ki density (\(\rho_{fluid}\)) se zyada hai, toh object sink karega. (\(\rho_{object} > \rho_{fluid}\))
- Is case mein, object ka weight buoyant force se zyada hota hai.
- Just Float (Suspended): Agar object ki density fluid ki density ke equal hai, toh object fluid ke andar float karega (na upar aayega na neeche jayega). (\(\rho_{object} = \rho_{fluid}\))
Example: Iron nail sinks (density > water), cork floats (density < water).
Archimedes' Principle: Jab koi object kisi fluid mein fully ya partially immerse hota hai, toh woh ek upward buoyant force experience karta hai jo uske dwara displaced fluid ke weight ke equal hoti hai.
Buoyant Force: \(F_b = \rho_{fluid} \times V_{displaced\_fluid} \times g\)
Archimedes' Principle aur uske applications (ships, hydrometers) par conceptual aur numerical questions frequently aate hain.
