Powers of Ten, Squares and Ratios വർഗം · അനുപാതം
Gravitation works with numbers that are huge (the Earth's mass is 6 000 000 000 000 000 000 000 000 kg) and numbers that are tiny (G = 0.000 000 000 066 7). Nobody writes those out in full. This page shows the short way, powers of ten, and how to multiply, divide and square them. It also covers the three kinds of proportion behind the universal law: direct, inverse and inverse square.
- Multiplying and dividing decimals; squares such as 3² = 9.
- Units in SI (page 2).
1. Powers of ten
10² means 10 × 10 = 100, and 10³ = 10 × 10 × 10 = 1000. The small raised number (the power or exponent) tells you how many tens are multiplied – and, for big numbers, how many zeros follow the 1. So 10⁶ = 1 000 000 = 10 lakh (10,00,000), and 10⁷ = 1 crore. Going the other way, each step divides by 10: 10⁰ = 1, 10−1 = 1/10 = 0.1, 10−2 = 0.01, 10−3 = 0.001. A negative power means a tiny number, not a negative one.
Each step to the right multiplies by 10. The coloured dots mark the numbers this chapter uses: G (10−11), the Earth's radius (10⁶ m), the Moon's mass (10²² kg) and the Earth's mass (10²⁴ kg).
Indian number names are powers of ten too: a thousand = 10³, a lakh = 10⁵, ten lakh = 10⁶, a crore = 10⁷. Kerala's population is about 3.5 crore = 3.5 × 10⁷ people. The textbook's Table 4.1 writes heights as 1,00,000 m (= 10⁵ m = 100 km) and 10,00,000 m (= 10⁶ m = 1000 km).
2. Scientific notation: a × 10ⁿ
A number in scientific notation is written as a × 10ⁿ, where a is at least 1 but less than 10 (one non-zero digit before the decimal point). To get there, move the decimal point and count the hops: each hop to the left adds 1 to the power, each hop to the right takes 1 away.
Earth's radius R = 6400 km = 6 400 000 m = 6.4 × 10⁶ m · Earth's mass M = 6 × 10²⁴ kg (6 followed by 24 zeros) · gravitational constant G = 6.67 × 10−11 N m²/kg².
3. Multiplying, dividing and squaring powers
Since 10³ × 10² = (10 × 10 × 10) × (10 × 10) = 10⁵, powers of ten follow three simple rules:
multiply → add the powers: 10−11 × 10²⁴ = 10¹³
divide → subtract the powers: 10¹³ ÷ 10¹² = 10¹
square → double the power: (10⁶)² = 10¹²
With a number in front, deal with the front numbers and the powers separately: (3 × 10⁴) × (2 × 10⁵) = (3 × 2) × 10⁴⁺⁵ = 6 × 10⁹. Squaring (വർഗം) squares both parts: (6.4 × 10⁶)² = 6.4² × (10⁶)² = 40.96 × 10¹². If the front number ends up 10 or more, hop once more: 40.96 × 10¹² = 4.096 × 10¹³.
Set 6.7 × 10−11 times 6 × 10²⁴ (close to G × M of the Earth). Then switch to “divide” and try 4 × 10¹⁴ ÷ 4.1 × 10¹³ to see how the front number gets fixed when it drops below 1.
Worked example 1 · The number that gives g
Q. On page 7 you will calculate (6.67 × 10−11 × 6 × 10²⁴) ÷ (6.4 × 10⁶)². Work it out now, using the power rules.
Top: multiply 6.67 × 6 and add the powers −11 + 24. Bottom: square 6.4 and double the power 6. Then divide the front numbers and subtract the powers.
4. Direct and inverse proportion
Two quantities are in direct proportion (നേർ അനുപാതം) if doubling one doubles the other, tripling one triples the other, and so on. We write y ∝ x (“y is proportional to x”). Bananas at ₹60 a kilo: 2 kg cost ₹120 and 3 kg cost ₹180.
They are in inverse proportion (വിപരീത അനുപാതം) if doubling one halves the other: y ∝ 1/x. A jackfruit shared among friends: twice as many friends, half as big a share each.
If y is proportional to a product (ഗുണനഫലം), y ∝ m₁ × m₂, the changes multiply. Doubling m₁ and tripling m₂ makes y 2 × 3 = 6 times bigger. You will see exactly this in the universal law, F ∝ m₁ × m₂.
All three start at 12 when x = 1 (dashed outline). At x = 2 the direct bar doubles to 24, the inverse bar halves to 6, and the inverse-square bar drops to a quarter, 3.
5. Inverse square: double the distance, one quarter the effect
The Earth's pull, light from a lamp and paint from a spray can all weaken with distance in the same special way. The effect is proportional to 1 ÷ (distance)², an inverse-square law. Why a square? The lamp's light spreads out in every direction. At twice the distance the same light covers a patch twice as wide and twice as tall, which is 2 × 2 = 4 times the area. Each square therefore gets only ¼ of the light. At 3 times the distance it covers 9 times the area and each square gets 1/9.
Spray-painting a wall for a temple festival: hold the can close and you get a small, dark patch; step back to twice the distance and the same paint spreads over 4 times the area, so the coat is only a quarter as thick. A street light looks bright under it and dim a few lamp-posts away for the same reason.
distance × 2 → effect ÷ 4 · distance × 3 → effect ÷ 9 · distance ÷ 2 (halved) → effect × 4 · distance ÷ 4 → effect × 16. In general, the new value = old value × (old distance ÷ new distance)².
Worked example 2 · An inverse-square force
Q. A force that obeys the inverse-square law is 36 N when the distance is 1 m. What is it when the distance becomes 3 m?
The distance became 3 times as big. Inverse square: divide by 3² = 9.