Is g the Same Everywhere? The Moon ചന്ദ്രനിലെ ഗുരുത്വാകർഷണത്വരണം
On page 7 we found g = GM/R² ≈ 9.8 m/s². But R – your distance from the Earth's centre – is not the same everywhere. So is g the same at the equator and at the North Pole? Deep down a mine? High above in an aeroplane? And on the Moon? Let's find out.
- g = GM/R²: bigger R → smaller g (inverse square).
- F = G m1m2/d², and the two bodies feel equal forces.
- a = F/m: the same force gives a big mass a small acceleration.
1. The Earth is not a perfect sphere (Fig 4.10)
The thinking girl asks: ഭൂഗുരുത്വാകർഷണത്വരണം ഭൂമിയിൽ എല്ലായിടത്തും ഒരുപോലെയാണോ? – Is g the same everywhere on the Earth? Look at Fig 4.10 (drawn below, with the flattening greatly exaggerated) and answer the textbook's questions.
The real Earth's equatorial radius is about 6378 km and its polar radius about 6357 km – only 21 km less, so a true-to-scale picture would look perfectly round. The values of g follow the textbook: about 9.78 m/s² at the equator and 9.83 m/s² at the poles.
Textbook questions (page 79)
- ഭൂമി യഥാർഥ ഗോളാകൃതിയിലാണോ? Is the Earth a perfect sphere?
- ഭൂകേന്ദ്രത്തിൽ നിന്നുള്ള അകലം ഏറ്റവും കൂടിയ ഭാഗം ഏതാണ്? (ധ്രുവപ്രദേശം / ഭൂമധ്യരേഖാപ്രദേശം) Where is the distance from the centre greatest? (polar / equatorial region)
- അകലം കുറഞ്ഞ ഭാഗമോ? And where is it least?
- ഭൂകേന്ദ്രത്തിൽ നിന്ന് ഉപരിതലത്തിലേക്കുള്ള അകലം വ്യത്യാസപ്പെടുമ്പോൾ g യുടെ മൂല്യത്തിന് എന്തു മാറ്റമുണ്ടാകും? How does g change when the distance from the centre to the surface changes? (Check with g = GM/R².)
- g യുടെ മൂല്യം ഏറ്റവും കൂടുതൽ എവിടെയാണ്? Where is g greatest? (equatorial / polar region)
2. In the equatorial region (ഭൂമധ്യരേഖാപ്രദേശം) – about 6378 km.
3. In the polar region (ധ്രുവപ്രദേശം) – about 6357 km.
4. g = GM/R²: G and M are fixed, so a smaller R gives a bigger g and a bigger R gives a smaller g.
5. In the polar region: R is least there, so g is greatest – about 9.83 m/s². At the equator g is least – about 9.78 m/s².
ഭൂമധ്യരേഖാ പ്രദേശത്ത് g യുടെ ഏകദേശ മൂല്യം = 9.78 m/s². ധ്രുവപ്രദേശത്ത് g യുടെ ഏകദേശ മൂല്യം = 9.83 m/s².
In the equatorial region g ≈ 9.78 m/s²; in the polar regions g ≈ 9.83 m/s². The average value used in calculations is 9.8 m/s².
There is a second, smaller reason why g is less at the equator: the Earth spins. A person at the equator is carried round a big circle once a day, and a little of the Earth's pull is used up keeping them on that circular path (you will meet this “centripetal force” on page 11). At the poles you just turn on the spot. Both effects – the bulge and the spin – make g smaller at the equator.
2. Down to the centre, up into the sky
Q. ഭൂകേന്ദ്രത്തിൽ വസ്തുവിനെ എല്ലാഭാഗത്തു നിന്നും ആകർഷിക്കുന്ന ബലങ്ങൾ തുല്യമായതിനാൽ അവിടെ g യുടെ മൂല്യം എത്രയായിരിക്കും? – At the centre of the Earth the object is pulled equally from all sides. What will be the value of g there?
Model used: inside a uniform Earth, g = 9.8 × (r ÷ R); outside, g = 9.8 × (R ÷ r)². (The real Earth is denser in its core, so g actually stays near 9.8–10 m/s² for the first half of the way down, but it is still zero at the centre.)
3. Why doesn't the Earth rise up to meet a falling aeroplane? (Fig 4.11)
ഗുരുത്വാകർഷണബലം എന്നത് പരസ്പരാകർഷണബലം ആണല്ലോ. Gravity is a mutual force. If an aeroplane's engines stop, the plane falls to the Earth – but the Earth does not rise up towards the plane, although it is pulled by the same force. Why? Let's calculate (textbook page 80).
Example. An aeroplane of mass 10000 kg is 10 km above the Earth. Find (a) the gravitational force between the plane and the Earth, (b) the acceleration of the plane, (c) the acceleration produced in the Earth. (M = 6 × 1024 kg, R = 6.4 × 106 m, G = 6.67 × 10−11 N m²/kg².)
The distance from the Earth's centre is R + h = 6.4 × 106 + 104 = 6.41 × 106 m. Both bodies feel the same F; use a = F/m for the plane and a = F/M for the Earth.
ഒരേ ബലം പ്രയോഗിക്കപ്പെടുമ്പോൾ മാസ് കൂടിയ വസ്തുക്കൾക്ക് ലഭിക്കുന്ന ത്വരണം കുറവായിരിക്കും.
When the same force acts, an object with a larger mass gets a smaller acceleration. The plane and the Earth pull each other equally, but the Earth's enormous mass makes its acceleration practically zero – so we never notice the Earth moving.
4. g on the Moon
The thinking boy asks: ഭൂമിയിലെയും ചന്ദ്രനിലെയും ഗുരുത്വാകർഷണ ത്വരണത്തിന്റെ മൂല്യങ്ങൾ ഒരുപോലെയാകുമോ? – Are the values of g on the Earth and on the Moon the same? The same formula g = GM/R² works for the Moon – we just use the Moon's mass and radius.
Textbook (page 81). Mass of the Moon M = 7.34 × 1022 kg, radius R = 1.74 × 106 m. Find g on the Moon and compare it with g on the Earth (9.8 m/s²).
Top: 6.67 × 7.34 and 10−11 × 1022. Bottom: 1.74² and (106)². Then divide 1.62 by 9.8.
Vikram, the lander of Chandrayaan-3, touched down near the Moon's south pole on 23 August 2023. As it came down, the Moon's gravity (g = 1.62 m/s²) kept speeding it up, so its engines had to fire downwards to slow it to a gentle “soft landing”. Because the Moon's g is only about 1/6 of the Earth's, a much smaller engine thrust is enough to hover there than on Earth.
- A black hole is a body whose gravity is so enormous that not even light can escape from it (പ്രകാശത്തിനുപോലും വിട്ടുപോകുവാൻ കഴിയാത്ത വിധത്തിൽ അതിശക്തമായ ഗുരുത്വാകർഷണം ഉള്ള പ്രപഞ്ചവസ്തുക്കൾ).
- Their possible existence was predicted in 1916, on the basis of Albert Einstein's theory of gravitation (general relativity).
- Clear evidence came from observations made in 2017: scientists produced an image of the shadow of the huge black hole at the centre of the distant galaxy M87 (the Event Horizon Telescope image, released in April 2019).
- There is also evidence of a black hole at the centre of our own galaxy, the Milky Way (ആകാശഗംഗ).
- Some stars many times heavier than the Sun may become black holes at the end of their lives; galaxies contain many such black holes.
- 📄 GOCE (Gravity field and steady-state Ocean Circulation Explorer) ESA
ESA's satellite that mapped how Earth's gravity differs from place to place. - ▶️ GOCE: Geoid YouTube – European Space Agency, ESA · 4:34
Animated globe showing the geoid – the shape Earth's oceans would take under gravity alone. - 📄 GRACE-FO mission NASA
Twin NASA/German satellites chase each other around Earth and measure tiny changes in gravity. - ▶️ Tracking Water from Space: The GRACE-FO Mission YouTube – NASA Jet Propulsion Laboratory · 2:15
Short animation of how two satellites measure Earth's gravity by tracking the distance between them. - 🖼️ Press Release (April 10, 2019): Astronomers Capture First Image of a Black Hole Event Horizon Telescope
The first ever picture of a black hole's shadow, at the centre of galaxy M87. - 🖼️ First Image of a Black Hole – NASA Science NASA
NASA's page with the M87 black hole image. - 📄 Black Holes – NASA Science NASA
NASA's guide to black holes – gravity so strong that not even light escapes. - ▶️ What is a Black Hole? We Asked a NASA Expert YouTube – NASA · 1:34
A NASA scientist explains black holes in 90 seconds. - ▶️ First Image of a Black Hole! YouTube – Veritasium · 5:29
What the 2019 black hole picture actually shows and why it looks like a ring.
5. Check yourself
- The Earth is flattened at the poles: R is least there, so g is greatest (≈ 9.83 m/s²); at the equator g ≈ 9.78 m/s².
- At the Earth's centre g = 0 (pulled equally from all sides); far above the Earth g gets smaller (g ∝ 1/r²).
- Plane 104 kg at 10 km: F ≈ 97400 N; plane's a = 9.74 m/s², Earth's a = 1.6 × 10−20 m/s². Same force, bigger mass → smaller acceleration.
- Moon: g = GM/R² = 1.62 m/s² ≈ 1/6 of the Earth's.
- Black holes: gravity so strong that not even light escapes (predicted 1916; M87 image from 2017 data).