Centripetal force and Kepler's laws
Tie a stone to a string and whirl it in a circle. Your hand feels the string pulling inwards all the time. Let go, and the stone does not keep circling: it shoots off in a straight line.
The force that pulls a body towards the centre of its circle is the centripetal force (अभिकेंद्र बल): F = mv²/r.
In the figure : In the figure a body in circular motion is pulled to the centre; if released it flies off along the tangent.
Rule 1: Centripetal force — A body moving in a circle of radius r with speed v needs a force F = mv²/r directed towards the centre. The velocity direction keeps changing, so the body is always accelerating. Released, it moves along the tangent (स्पर्शरेषा).
- Easy : A 0.5 kg stone whirled at 4 m/s on a 2 m string: F = 0.5 × 16 ÷ 2 = 4 N.
- Medium : The same stone at double speed, 8 m/s: F = 0.5 × 64 ÷ 2 = 16 N, which is 4 times more.
- Exam-type : A 1000 kg car turns a bend of radius 50 m at 10 m/s: F = 1000 × 100 ÷ 50 = 2000 N. Friction between tyres and road supplies this force.
Rule 2: Kepler's laws — (1) Every planet moves in an ellipse (लंबवर्तुळ) with the Sun at one focus. (2) The line joining the planet and the Sun sweeps equal areas in equal times. (3) T²/r³ is the same for all planets.
In the figure : In the figure Kepler's second law: the planet sweeps equal areas in equal times.
- Easy : First law: Earth's path is an ellipse, only slightly flattened.
- Medium : Second law: a planet moves faster when it is nearer the Sun and slower when it is far.
- Exam-type : Third law: a planet at 4 times Earth's distance (4 AU): T² = 4³ = 64, so T = 8 years.
F = mv²/r towards the centre. Kepler: ellipse; equal areas; T²/r³ constant.
Thinking the stone is pushed outwards while whirling. The real force on it is inwards; "flying off" is only inertia along the tangent.
Memory clue: "Pull in, fly off straight; near Sun, run fast".
Spot the difference
| Item | Statement |
|---|---|
| Centripetal force | mv²/r towards the centre |
| Kepler 1 | Elliptical orbit |
| Kepler 2 | Equal areas in equal times |
| Kepler 3 | T²/r³ constant |
Practice questions
- 1Write the formula of centripetal force.
- 2In which direction does a released whirling stone move?
- 3What is the shape of a planet's orbit?
- 4State Kepler's third law.
- 5Find the force on a 2 kg body moving at 3 m/s in a circle of radius 1.5 m.
- 6If the speed in circular motion is tripled, how does the force change?
- 7A planet is at 9 AU from the Sun. Find its period in years.
- 8Why does a planet move fastest when it is closest to the Sun?
Show answers (try them yourself first)
- 1F = mv²/r.
- 2Along the tangent.
- 3An ellipse with the Sun at one focus.
- 4T²/r³ is the same for all planets.
- 52 × 9 ÷ 1.5 = 12 N.
- 6It becomes 9 times, since F depends on v².
- 7T² = 9³ = 729, so T = 27 years.
- 8The line to the Sun must sweep equal areas in equal times; a short line needs more speed to cover the same area.
Newton's universal law of gravitation
Drop a pen and it falls to the floor. Earth pulls it. The pen pulls the earth too, but the earth is so huge that we never see it move. Newton said this pull exists between every two objects.
Universal law of gravitation (वैश्विक गुरुत्वाकर्षणाचा नियम) : F = G m₁ m₂ / d².
In the figure : In the figure the inverse square law: the force falls quickly as the distance grows.
Rule 1: Newton's law — Every object attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. G = 6.67 × 10⁻¹¹ N m² kg⁻² (measured by Cavendish).
- Easy : Two bodies of 10 kg and 20 kg at 2 m: F = 6.67 × 10⁻¹¹ × 200 ÷ 4 ≈ 3.3 × 10⁻⁹ N, a tiny force.
- Medium : Distance doubled: F becomes 1/4. Distance tripled: F becomes 1/9. One mass doubled: F doubles.
- Exam-type : Both masses are doubled and the distance is also doubled: F becomes (2 × 2) ÷ 4 = 1 time, so the force stays the same.
Rule 2: Why G is called universal — The same value of G works for the apple, the moon, planets and stars. Tides in the sea are caused by the pull of the moon on the earth's water.
- Easy : An apple falling and the moon circling the earth obey the same law.
- Medium : The sun and moon both pull the sea; the nearer moon pulls more strongly, so it has the larger tidal effect.
- Exam-type : Two students sitting 1 m apart also attract each other, but the force is far too small to feel.
Newton's universal law of gravitation
In the app you can move this figure with your finger.
Open in the appF = G m₁m₂/d². Distance × 2 means force ÷ 4.
Forgetting to square the distance. Always use d², not d.
Memory clue: "Masses multiply on top, distance squares below".
Spot the difference
| Change | Effect on F |
|---|---|
| d doubled | F ÷ 4 |
| d halved | F × 4 |
| One mass doubled | F × 2 |
| Both masses doubled | F × 4 |
Practice questions
- 1Write Newton's law of gravitation as a formula.
- 2What is the value of G?
- 3Who measured G first?
- 4What causes tides in the sea?
- 5The distance between two bodies is tripled. How does the force change?
- 6One mass is tripled and the distance is halved. Find the factor by which F changes.
- 7Find the force between 5 kg and 8 kg masses 1 m apart (G = 6.67 × 10⁻¹¹).
- 8Why do we not feel the pull between two people standing next to each other?
Show answers (try them yourself first)
- 1F = G m₁ m₂ / d².
- 26.67 × 10⁻¹¹ N m² kg⁻².
- 3Cavendish.
- 4The gravitational pull of the moon (and the sun).
- 5It becomes 1/9.
- 63 × 4 = 12 times.
- 7F = 6.67 × 10⁻¹¹ × 40 ÷ 1 ≈ 2.7 × 10⁻⁹ N.
- 8Their masses are small and G is tiny, so the force is far too small to notice.
Acceleration due to gravity
A feather and a coin dropped from the same height in a closed room seem to fall differently, but that is the air. In truth the earth gives every falling thing the same acceleration.
Acceleration due to gravity (गुरुत्वीय त्वरण) : g = GM/R² ≈ 9.8 m/s² for the earth.
In the figure : In the figure how g changes from place to place.
Rule 1: What g depends on — g = GM/R² depends on the mass M and radius R of the planet, not on the mass of the falling body. g is highest at the poles (about 9.83) and lowest at the equator (about 9.78), and it decreases with height and with depth.
- Easy : Dropping a 1 kg and a 5 kg stone from the same height: both have g = 9.8 m/s² and land together.
- Medium : A planet has the same mass as the earth but twice the radius: g = 9.8 ÷ 4 = 2.45 m/s².
- Exam-type : On the moon g is about 1/6 of the earth's value, nearly 1.6 m/s². A stone dropped from 2 m: t = √(2 × 2 ÷ 1.6) ≈ 1.6 s, while on earth it takes about 0.64 s.
g = GM/R². Same g for all masses. Poles more, equator less.
Thinking a heavier body falls faster. In vacuum all bodies fall together.
Memory clue: "g likes small planets with big mass, and it is bigger at the poles".
Spot the difference
| Place | g (m/s²) |
|---|---|
| Poles | 9.83 |
| Equator | 9.78 |
| Moon | 1.6 |
| High up or deep inside | Less |
Practice questions
- 1Write the formula of g.
- 2Does g depend on the mass of the falling body?
- 3Where is g the largest on the earth?
- 4What is g on the moon roughly?
- 5How does g change as we go up a tall mountain?
- 6A planet has twice the earth's radius and the same mass. Find g if the earth's is 10.
- 7Find the weight of a 30 kg child on the moon (g = 1.6).
- 8Why do a feather and a coin fall together on the moon?
Show answers (try them yourself first)
- 1g = GM/R².
- 2No.
- 3At the poles.
- 4About 1.6 m/s², one sixth of the earth's.
- 5It decreases slightly.
- 610 ÷ 4 = 2.5 m/s².
- 730 × 1.6 = 48 N.
- 8The moon has no air, so only gravity acts, and g is the same for both.
Mass, weight and free fall
Your mass at school and your mass on the moon are the same, but a weighing spring would show less on the moon. Mass is how much matter you have; weight is how hard gravity pulls it.
Weight (वजन) W = mg, in newton. Free fall (मुक्तपतन) is motion under gravity alone: v = gt, s = ½gt², v² = 2gs.
In the figure : In the figure speed and distance in free fall grow with time.
Rule 1: Mass and weight — Mass is a scalar in kg and is the same everywhere. Weight W = mg is a vector in newton and changes with g.
- Easy : A 40 kg student on earth: W = 40 × 9.8 = 392 N.
- Medium : The same student on the moon (g = 1.6): W = 40 × 1.6 = 64 N, but the mass is still 40 kg.
- Exam-type : A body weighs 150 N on the moon (g = 1.5). Its mass is 150 ÷ 1.5 = 100 kg, and its weight on earth (g = 10) is 1000 N.
Rule 2: Equations of free fall — Starting from rest: v = gt, s = ½ g t², v² = 2 g s. For an upward throw, a = −g. True free fall happens only in vacuum.
- Easy : A stone dropped from rest for 3 s (g = 10): v = 30 m/s, s = ½ × 10 × 9 = 45 m.
- Medium : A ball dropped from 80 m (g = 10): t = √(2 × 80 ÷ 10) = 4 s and v = 40 m/s.
- Exam-type : A ball is thrown up at 20 m/s (g = 10). At the top v = 0, so rise = 20² ÷ (2 × 10) = 20 m and time up = 20 ÷ 10 = 2 s.
W = mg. Free fall: v = gt, s = ½gt², v² = 2gs. Mass stays same, weight changes.
Writing weight in kg. Weight is a force, so its unit is the newton.
Memory clue: "Mass = matter (kg), weight = pull (N)".
Spot the difference
| Property | Mass | Weight |
|---|---|---|
| Unit | kg | N |
| Type | Scalar | Vector |
| Changes with place? | No | Yes |
Practice questions
- 1Write the formula of weight.
- 2What is the SI unit of weight?
- 3What is free fall?
- 4What is the acceleration of a ball thrown upward?
- 5Find the weight of a 25 kg body (g = 10).
- 6A stone is dropped from rest. Find its speed after 5 s (g = 10).
- 7A stone falls from a 45 m tower (g = 10). Find the time taken and the landing speed.
- 8A man weighs 600 N on earth. Why will his mass be the same on the moon but his weight less?
Show answers (try them yourself first)
- 1W = mg.
- 2Newton.
- 3Motion under gravity alone.
- 4−g (downward).
- 5250 N.
- 6v = 10 × 5 = 50 m/s.
- 7t = √(2 × 45 ÷ 10) = 3 s; v = 30 m/s.
- 8Mass is the amount of matter and does not change; weight is mg and the moon's g is smaller.
Potential energy and escape velocity
To throw a ball up, you need effort. A very fast throw goes very high. If you could throw hard enough, the ball would never come back. That special speed is the escape velocity.
Escape velocity (पलायन वेग) is the minimum speed needed to leave the earth for ever: about 11.2 km/s.
In the figure : In the figure orbital velocity keeps a satellite circling, while escape velocity takes it away.
Rule 1: Potential energy and escape — Gravitational potential energy at distance r from the centre is −GMm/r, taking zero at infinity. For small heights, mgh is enough. A body given the escape velocity has total energy zero and never returns.
- Easy : A 5 kg body lifted 8 m (g = 10): PE = 5 × 10 × 8 = 400 J.
- Medium : The escape velocity does not depend on the mass of the body: a stone and a rocket need the same 11.2 km/s from the earth's surface.
- Exam-type : The moon is smaller and has weaker gravity, so its escape velocity is only about 2.4 km/s, much less than the earth's 11.2 km/s.
Rule 2: Satellites and orbits — The moon and satellites are in free fall around the earth. Their sideways orbital velocity (कक्षीय वेग) keeps them missing the ground, about 7.9 km/s close to the earth. Einstein predicted gravitational waves in 1916; LIGO detected them a hundred years later.
- Easy : A satellite at 7.9 km/s close to the earth circles it in about 90 minutes.
- Medium : An astronaut inside the satellite floats because she and the craft fall together, not because gravity is absent.
- Exam-type : Orbital speed is lower than escape speed: 7.9 km/s keeps the satellite circling, but 11.2 km/s would send it away from the earth for good.
Escape velocity of earth 11.2 km/s; orbital velocity about 7.9 km/s. PE = −GMm/r; mgh for small heights.
Thinking that astronauts float because there is no gravity up there. Gravity is still strong; they are in free fall.
Memory clue: "7.9 circles, 11.2 leaves".
Spot the difference
| Body | Escape velocity (km/s) |
|---|---|
| Earth | 11.2 |
| Moon | 2.4 |
| Orbit near earth | 7.9 (orbital) |
Practice questions
- 1What is escape velocity?
- 2What is the escape velocity of the earth?
- 3Does escape velocity depend on the mass of the body?
- 4Who predicted gravitational waves, and when?
- 5Find the PE of a 2 kg body at 15 m height (g = 10).
- 6Is the potential energy −GMm/r positive or negative? Why?
- 7Why is the escape velocity of the moon less than that of the earth?
- 8Why does an astronaut float inside a satellite?
Show answers (try them yourself first)
- 1The minimum speed needed to leave the earth for ever.
- 2About 11.2 km/s.
- 3No.
- 4Einstein, in 1916.
- 52 × 10 × 15 = 300 J.
- 6Negative, because zero is taken at infinity and the body is bound to the earth.
- 7The moon has less mass and weaker gravity at its surface.
- 8The astronaut and the satellite are both in free fall together.
Quick revision
- F = G m1m2/d2; g = GM/R2; W = mg.
- Kepler: ellipse; equal areas; T2/r3 = constant.
- g is more at poles than at equator; decreases with height and depth.
- Escape velocity of earth: 11.2 km/s.