Apply F = G m1 m2 / r^2
Scenario
ISRO ko satellite launch karte waqt pata hona chahiye ki Earth satellite ko kitni force se kheeche gi har height par — warna satellite ya toh crash kar jaayega ya space mein kho jaayega.
Agar tumhe kisi planet ke surface par gravitational force nikaalini ho, toh kaise karoge?
Mental model
`F = G m₁ m₂ / r²` ek calculator hai — masses aur distance daalo, gravitational force nikal jaayegi.
Socho jaise ek bill calculate karte ho: items ki cost multiply karo, phir kuch divide karo. Yahaan bhi masses multiply karo, `r²` se divide karo, aur `G` se scale karo. Bas powers of `10` carefully handle karna — yahi main skill hai.
Explanation
Ab tumne law padh liya, toh ab usse problems mein apply karna seekho. Sabse pehle samjho ki formula mein kya kya daalna hai: `m₁` aur `m₂` dono objects ke masses (`kg` mein), `r` dono ke CENTRES ke beech ki distance (`m` mein) — surface se surface nahi.
Sabse common mistake yeh hai ki jab object Earth ke surface par ho, toh `r` = Earth ka RADIUS (`6.4 × 10⁶ m`) hota hai, kyunki distance centre se measure hoti hai. Agar object `h` height par ho, toh `r = R + h`.
Powers of `10` handle karna yahan sabse important skill hai. Tip: pehle coefficients multiply/divide karo alag se, phir powers of `10` alag se. Neeche worked example 1 mein isko solve karke dekho.
Kuch important variations jo boards mein aate hain: (a) Agar ek mass double ho jaaye toh force double ho jaayegi. (b) Agar distance half ho jaaye toh force `4` guna ho jaayegi. (c) Agar dono masses triple ho jaayein aur distance bhi triple, toh force same rahegi (`9/9 = 1`). In sab mein formula mein directly substitute karke ratio nikaal lo.
Ek aur important application: `g` derive karna. Earth ke surface par kisi object (mass `m`) par gravitational force hai `mg`. Isko `F = G M m / R²` se equate karo:
mg = G M m / R²
cancel `m`, toh:
g = G M / R²
Isliye `g` alag planets par alag hota hai — `M` aur `R` alag hain.
Worked example
Problem: Calculate the force of gravity on a `50 kg` person standing on Earth's surface. (`M = 6 × 10²⁴ kg`, `R = 6.4 × 10⁶ m`, `G = 6.674 × 10⁻¹¹ N m² kg⁻²`)
- Values note karo — person Earth ke surface par hai toh `r = R` (Earth ka radius).
- Formula lagao.
- Numerator: coefficients = `6.674 × 6 × 50` = `2002.2`, powers = `10⁻¹¹⁺²⁴` = `10¹³`.
- Denominator: `(6.4)²` = `40.96`, `(10⁶)²` = `10¹²`.
- Divide karo: `2002.2 / 40.96` ≈ `48.88`, `10¹³ / 10¹²` = `10¹`.
Answer: `F ≈ 490 N`. Cross-check: `mg` = `50 × 9.8` = `490 N` ✓ — formula gives exactly the weight!
Problem: Two spheres of masses `20 kg` and `30 kg` are placed with their centres `0.5 m` apart. Find the gravitational force between them.
- Chhote objects hain, toh force bahut chhoti hogi — but calculate karte hain.
- Formula mein values daalo.
- Numerator: `6.674 × 600` = `4004.4`, power = `10⁻¹¹`. Denominator: `0.25`.
- Divide: `4004.4 / 0.25` = `16017.6`.
Answer: `F ≈ 1.6 × 10⁻⁷ N` — itni chhoti force ki feel bhi nahi hogi, tabhi daily life mein objects ek doosre ki taraf nahi khisakte.
Key points
- Distance r is centre-to-centre: On Earth's surface, `r = R` (radius of Earth, `6.4 × 10⁶ m`), NOT zero.
- Powers of 10 trick: Handle coefficients and powers separately: multiply/divide coefficients, then add/subtract exponents.
- Deriving g: `g = G M / R²`. This connects the universal law to free-fall acceleration.
- Proportionality shortcuts: `F ∝ m₁m₂` and `F ∝ 1/r²`. Use ratios for 'what happens if...' questions without calculating.
Common mistakes
- `r` mein Earth ka radius bhoolna: Jab object Earth ke surface par hai, toh `r = 0` nahi hai! `r` = Earth ka radius (`6.4 × 10⁶ m`) hai, kyunki distance centre-to-centre hoti hai.
- Powers of 10 mein galti: `10⁻¹¹ × 10²⁴` = `10¹³`, nahi ki `10⁻²⁶⁴`. Exponents ADD hote hain multiplication mein. Isko carefully likho step by step.
- Units miss karna: Force ka answer `N` (Newton) mein hona chahiye. Agar masses `kg` mein aur distance `m` mein nahi daali, toh answer galat aayega.
Recall questions
- What is `r` in the formula `F = G m₁ m₂ / r²` when an object is on Earth's surface?
- Derive the expression for acceleration due to gravity `g` from the universal law.
- If both masses and the distance are tripled, what happens to the gravitational force?
Questions & answers
Calculate the force of gravitation between the Earth and the Sun, given that the mass of the Earth = `6 × 10²⁴ kg` and of the Sun = `2 × 10³⁰ kg`. The average distance between the two is `1.5 × 10¹¹ m`.
`F = (6.674 × 10⁻¹¹ × 6 × 10²⁴ × 2 × 10³⁰) / (1.5 × 10¹¹)² = (6.674 × 12 × 10⁴³) / (2.25 × 10²²) = 80.09 × 10⁴³ / 2.25 × 10²² = 35.6 × 10²¹ ≈ 3.56 × 10²² N`.
Approach: Substitute values into `F = G m₁ m₂ / r²`, handle exponents carefully.
How does the force of gravitation between two objects change when the distance between them is reduced to half?
`F' = G m₁ m₂ / (r/2)² = G m₁ m₂ / (r²/4) = 4 × G m₁ m₂ / r² = 4F`. The force becomes four times.
Approach: Substitute `r/2` for `r` in the formula and simplify the ratio `F' / F`.
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