Chapter 4 · Section 2Textbook pages 75–77

Universal Law of Gravitation സാർവിക ഗുരുത്വാകർഷണനിയമം

The Earth pulls a stone; the Moon pulls the Earth's oceans; the Sun pulls the Earth. Isaac Newton saw that these are all the same force and wrote one law for all of them. With it you can calculate the pull between any two objects – two planets, or you and your friend.

📖 What you need (pages 3 and 5 of this guide)

Direct proportion (നേർ അനുപാതം): one doubles → the other doubles. Inverse proportion (വിപരീത അനുപാതം): one doubles → the other halves.
Square (വർഗം): 2² = 4, (½)² = ¼. Powers of ten: 10³ × 10⁻¹¹ = 10⁻⁸.
Page 5: the pull grows with mass and shrinks with distance.

1. Newton's law

Sir Isaac Newton was the first to put the mutual attraction of objects into one complete law (textbook page 75).

⭐ Universal law of gravitation (textbook page 75)

പ്രപഞ്ചത്തിലുള്ള എല്ലാ വസ്തുക്കളും പരസ്പരം ആകർഷിക്കുന്നു. രണ്ടു വസ്തുക്കൾ തമ്മിലുള്ള ആകർഷണബലം (Gravitational force) അവയുടെ മാസുകളുടെ ഗുണനഫലത്തിന് നേർ അനുപാതത്തിലും അകലത്തിന്റെ വർഗത്തിന് വിപരീത അനുപാതത്തിലും ആണ്.

Every object in the universe attracts every other object. The attractive force between two objects is directly proportional to the product of their masses (ഗുണനഫലം) and inversely proportional to the square of the distance between them (അകലത്തിന്റെ വർഗം).

F = G m₁ m₂d²

🎬 Building the formula from Fig 4.4

⚠️ d is measured between the CENTRES

In Fig 4.4, d is the distance between the centres of the two objects, not the gap between their surfaces. For the Earth and an object on it, d is the Earth's radius (6400 km) – that is why the 100 kg object on page 5 was “6400 km away” even when sitting on the ground.

2. The gravitational constant G

G is the gravitational constant (ഗുരുത്വാകർഷണസ്ഥിരാങ്കം). Its value is

G = 6.67 × 10⁻¹¹ N m²/kg²

G has the same value everywhere in the universe – on Earth, on the Moon, between two galaxies. That is why the law is called universal.
Meaning: two objects of mass 1 kg each, kept 1 m apart, attract each other with a force of G newton: F = G × 1 × 1 ÷ 1² = G = 0.0000000000667 N. That is tiny!
Unit: from F = G m₁m₂/d², G = F d²/(m₁m₂), so its unit is N × m² ÷ kg² = N m²/kg².

🎬 How Henry Cavendish measured G (1798) – torsion balance, seen from above

3. Gravity force lab

The textbook suggests the PhET simulation “Gravity Force Lab” (link in the Explore-online box below). Here is an offline version. Change the masses and the distance and watch the two force arrows.

🧪 Two masses pulling each other

m₁ kg m₂ kg d m

m₁ m₂ d

The two arrows are always equal in length and opposite in direction, even when one ball is 50 times heavier. Arrow lengths use a stretched (logarithmic) scale so that both tiny and large forces fit; the number printed is exact. The balls are drawn larger than real lead balls of that mass.

4. Table 4.2 – complete it

Using Newton's law, the textbook asks you to complete Table 4.2 (page 76). Row 1 is done for you; step through the rest.

🎬 Filling Table 4.2 row by row

No.	m₁ (kg)	m₂ (kg)	d (m)	Mutual attractive force (N)
1	5	10	2	G × 10 × 5 ÷ 2² = 12.5 G
2	10	10	2	G × 100 ÷ 4 = 25 G
3	10	30	2	G × 300 ÷ 4 = 75 G
4	5	10	4	G × 50 ÷ 16 = 3.125 G
5	5	10	1	G × 50 ÷ 1 = 50 G
6	5	10	½	G × 50 ÷ ¼ = 200 G

Now answer the textbook's questions by comparing rows (page 76–77):

Q1. ഒരു നിശ്ചിത അകലത്തിലുള്ള രണ്ട് വസ്തുക്കളിൽ ഒന്നിന്റെ മാസ് രണ്ട് മടങ്ങാക്കിയാൽ പരസ്പരാകർഷണബലം എത്ര മടങ്ങാകും? – If the mass of one of two objects at a fixed distance is doubled, how many times does the force become?

2 times (doubles). Row 1 → row 2: m₁ 5 → 10 kg, d unchanged; 12.5 G → 25 G. Why: F is directly proportional to m₁ × m₂; doubling one factor doubles the product.

Q2. ഒരു വസ്തുവിന്റെ മാസ് രണ്ട് മടങ്ങും രണ്ടാമത്തെ വസ്തുവിന്റെ മാസ് മൂന്ന് മടങ്ങും ആക്കിയാലോ? – What if one mass is doubled and the other is tripled?

6 times. Row 1 → row 3: 5 → 10 (× 2) and 10 → 30 (× 3); 12.5 G → 75 G = 6 × 12.5 G. Why: the product m₁m₂ becomes 2 × 3 = 6 times bigger.

Q3. വസ്തുക്കൾ തമ്മിലുള്ള അകലം രണ്ട് മടങ്ങ് ആക്കിയാലോ? – What if the distance between the objects is doubled?

One quarter (¼). Row 1 → row 4: d 2 → 4 m; 12.5 G → 3.125 G = 12.5 G ÷ 4. Why: F ∝ 1/d²; d × 2 makes d² × 4, so F ÷ 4.

Q4. വസ്തുക്കൾ തമ്മിലുള്ള അകലം പകുതിയാക്കിയാലോ? – What if the distance is halved?

4 times. Row 1 → row 5: d 2 → 1 m; 12.5 G → 50 G. Why: d ÷ 2 makes d² ÷ 4, so F × 4.

Q5. വസ്തുക്കൾ തമ്മിലുള്ള അകലം നാലിലൊന്നാക്കിയാലോ? – What if the distance is made one quarter?

16 times. Row 1 → row 6: d 2 → ½ m (one quarter); 12.5 G → 200 G = 16 × 12.5 G. (Also row 4 → row 5: 4 m → 1 m, 3.125 G → 50 G, again × 16.) Why: d ÷ 4 makes d² ÷ 16, so F × 16.

🧠 Quick rule for ratio questions

New force = old force × (change in m₁) × (change in m₂) ÷ (change in d)². Example: m₁ × 3, d × 3 → F × 3 ÷ 9 = F ÷ 3. Try it in the force lab above.

5. Worked example – two children 2 m apart

Example (textbook page 77). 40 kg, 50 kg വീതം മാസുള്ള രണ്ട് കുട്ടികൾ 2 m അകലത്തിൽ ആയിരിക്കുമ്പോൾ അവർ തമ്മിലുള്ള ഗുരുത്വാകർഷണ ബലം കണക്കാക്കുക. – Two children of masses 40 kg and 50 kg are 2 m apart. Calculate the gravitational force between them.

Write m₁ = 40 kg, m₂ = 50 kg, d = 2 m. Multiply 40 × 50 first, divide by 2² = 4, and only then multiply by 6.67 × 10⁻¹¹.

🎬 Solution in exam layout

Answer: F = G m₁m₂/d² = 6.67 × 10⁻¹¹ × 40 × 50 ÷ 2² = 500 × 6.67 × 10⁻¹¹ = 3335 × 10⁻¹¹ = 3.335 × 10⁻⁸ N = 0.00000003335 N.

6. Then why don't we slide towards each other?

In Fig 4.5 one boy asks: നമ്മൾ തമ്മിൽ പരസ്പരാകർഷണബലമുണ്ടല്ലേ. പിന്നെന്താ ഈ മേശ മാറ്റിയാലും നാം തമ്മിൽ സ്വയം അടുക്കാത്തത്? – “We attract each other, don't we? Then why don't we move towards each other even if this table is removed?”

Because the force is extremely small. 3.335 × 10⁻⁸ N is the weight of a speck of dust of about 3.4 micrograms. The friction between your feet (or chair) and the floor is hundreds of newtons – billions of times larger – and so are magnetic and other everyday forces. So we never notice gravity between ordinary objects in daily life. It becomes important only when at least one mass is enormous, like the Earth.

In Fig 4.6 the big man asks: നീ എന്നെ ആകർഷിക്കുന്നതിനേക്കാൾ വളരെ കൂടുതൽ ആയിരിക്കുമല്ലോ ഞാൻ നിന്നെ ആകർഷിക്കുന്ന ബലം? – “Since I'm much heavier, I must be pulling you much more than you pull me?” The other replies: നമ്മൾ തമ്മിലുള്ളത് പരസ്പരാകർഷണമല്ലേ. ഒരേ ബലത്താലാണ് നാം പരസ്പരം ആകർഷിക്കുന്നത്. – “It's mutual attraction! We attract each other with the same force.” Test it:

⚖️ Who pulls harder?

mass of the heavy person kg

Q. ഭൂമി ചന്ദ്രനെ ആകർഷിക്കുന്ന ബലം F എങ്കിൽ ചന്ദ്രൻ ഭൂമിയെ ആകർഷിക്കുന്ന ബലം എത്രയായിരിക്കും? – If the Earth attracts the Moon with a force F, with what force does the Moon attract the Earth?

The same force F, in the opposite direction (towards the Moon). Why: the formula G m₁m₂/d² gives the same number whichever body you call m₁; it is also Newton's third law (action and reaction are equal and opposite). The size is about 2 × 10²⁰ N (with the Moon 3.84 × 10⁸ m away).

🌍 Real life

Every coconut on a tree in your compound pulls the Earth upwards with exactly the same force as the Earth pulls it down. The ocean tides at Kochi are the Earth's water being pulled by the Moon; and the Moon is pulled by the Earth with the same force. Space agencies like ISRO use Newton's law to plan every orbit – including Chandrayaan-3's path to the Moon.

One more thought from the textbook: തുടർച്ചയായി ബലം പ്രയോഗിക്കുന്നതു മൂലം വസ്തുക്കൾക്ക് ത്വരണമുണ്ടാകുമെന്നറിയാമല്ലോ. എങ്കിൽ ഭൂഗുരുത്വാകർഷണ ബലം മൂലം ത്വരണമുണ്ടാകുമോ? – a force acting continuously produces acceleration. Does the Earth's gravity produce acceleration too? That is the next page.

🌐 Explore online needs internet · opens in a new tab

🧪 Gravity Force Lab PhET
Change the two masses and the distance and watch the force arrows grow or shrink – see the inverse-square law yourself (the sim named in your textbook).
🧪 Gravity Force Lab: Basics PhET
A simpler version: drag two spheres apart or together and read the force – both feel equal and opposite forces.
🧪 Gravity and Orbits PhET
Switch gravity off and watch the Moon fly away in a straight line – gravity is what keeps it orbiting.
🧪 My Solar System PhET
Build your own sun-and-planet system and see how mass and distance shape the orbits.
▶️ Introduction to Newton's law of gravitation (Khan Academy) Khan Academy
Sal Khan explains F = G m1 m2 / r² step by step with simple examples.
▶️ Calculating Gravitational Attraction YouTube – Veritasium · 3:31
Just how tiny is the gravitational pull between two everyday objects? Derek works it out.
▶️ Watch gravity pull two metal balls together YouTube – Steve Mould · 12:47
A home-built Cavendish torsion balance: you actually see lead balls being pulled together by gravity, the way G was first measured.
📄 Cavendish experiment (Wikipedia) Wikipedia
Diagram and history of Henry Cavendish's 1798 torsion-balance experiment that measured G (and 'weighed the Earth').

7. Check yourself

🧭 Summary

F = G m₁m₂/d²: F ∝ product of the masses, F ∝ 1/(distance between centres)².
G = 6.67 × 10⁻¹¹ N m²/kg², the same everywhere; measured by Cavendish (1798).
Mass × 2 → F × 2; masses × 2 and × 3 → F × 6; d × 2 → F ÷ 4; d ÷ 2 → F × 4; d ÷ 4 → F × 16.
40 kg and 50 kg, 2 m apart: F = 3.335 × 10⁻⁸ N – too tiny to notice.
The two bodies pull each other with equal and opposite forces.

Next: Acceleration due to gravity g ▶