Understanding gravity, universal law of gravitation, and motion of objects
Gravitation is the force of attraction between any two objects in the universe. Every object attracts every other object with a force called gravitational force.
Imagine every object in the universe has invisible rubber bands connecting it to every other object, always pulling them together! The bigger the objects and the closer they are, the stronger the pull. This invisible pull is gravity - it's everywhere, always working, holding the universe together!
Gravitation: Universal force of attraction
between any two objects in the universe
Gravity: Specifically the force of attraction
exerted by Earth (or any planet/celestial body) on objects near
its surface
In simple words: Gravity is Earth's gravitation!
Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
F = G × (m₁ × m₂) / r²
Where:
F = Gravitational force between two objects (Newton, N)
G = Universal gravitational constant = 6.67 × 10⁻¹¹ N⋅m²/kg²
m₁ = Mass of first object (kg)
m₂ = Mass of second object (kg)
r = Distance between centers of the two objects (m)
Two objects of masses 100 kg and 200 kg are separated by 2 m.
Calculate the gravitational force between them.
(G = 6.67 × 10⁻¹¹ N⋅m²/kg²)
Solution:
m₁ = 100 kg
m₂ = 200 kg
r = 2 m
G = 6.67 × 10⁻¹¹ N⋅m²/kg²
F = G × (m₁ × m₂) / r²
F = (6.67 × 10⁻¹¹) × (100 × 200) / (2)²
F = (6.67 × 10⁻¹¹) × 20000 / 4
F = (6.67 × 10⁻¹¹) × 5000
F = 3.335 × 10⁻⁷ N
The gravitational force is very small (0.0000003335 N) - that's
why we don't feel attraction between everyday objects!
We ARE attracted to each other gravitationally, but the force is
extremely tiny! The universal gravitational constant G is so small
(0.0000000000667) that the gravitational force between everyday
objects is negligible.
We only notice gravity when at least one object is VERY massive
(like Earth, Moon, or Sun). That's why we feel pulled toward Earth
but not toward a building or car!
Free fall is the motion of an object under the influence of gravity alone, without any other force acting on it (no air resistance, no applied force).
Galileo dropped a heavy iron ball and a light wooden ball from the
Leaning Tower of Pisa. People expected the heavy ball to hit
ground first, but both hit at the SAME time!
This proved that all objects fall with the same acceleration,
regardless of their mass. The acceleration depends only on the
gravitational pull of Earth, not on the object's mass!
Why does a feather fall slower than a stone?
It's not because of mass! It's because of air resistance. The
feather has a large surface area relative to its mass, so air
pushes against it more.
In vacuum (no air), a feather and a stone dropped together would
hit the ground at exactly the same time! This was famously
demonstrated by astronauts on the Moon.
Acceleration due to gravity (g) is the acceleration produced in an object when it falls freely under the influence of Earth's gravity alone.
g = 9.8 m/s² (approximately 10 m/s²)
This means:
• Velocity of freely falling object increases by 9.8 m/s every
second
• After 1 second: velocity = 9.8 m/s
• After 2 seconds: velocity = 19.6 m/s
• After 3 seconds: velocity = 29.4 m/s
And so on...
From Newton's Law of Gravitation:
F = G × (M × m) / R²
From Newton's Second Law:
F = m × g
Equating both:
m × g = G × (M × m) / R²
g = G × M / R²
Where:
g = Acceleration due to gravity
G = Universal gravitational constant
M = Mass of Earth (6 × 10²⁴ kg)
R = Radius of Earth (6.4 × 10⁶ m)
Notice that 'm' (mass of object) cancels out in the equation!
This mathematically proves that acceleration due to gravity is
INDEPENDENT of the mass of the falling object. Whether you drop a
pebble or a boulder, both accelerate at 9.8 m/s²!
For objects moving under gravity, we use equations of motion with
a = g:
1. v = u + gt
(Final velocity = Initial velocity + g × time)
2. s = ut + ½gt²
(Height = Initial velocity × time + ½ × g × time²)
3. v² = u² + 2gs
(Final velocity² = Initial velocity² + 2 × g × height)
Important:
• For upward motion: Use g = -9.8 m/s² (negative, opposes
motion)
• For downward motion: Use g = +9.8 m/s² (positive, aids motion)
A stone is dropped from a height of 80 m. Find:
(a) Time taken to reach ground
(b) Final velocity when it hits ground
(Take g = 10 m/s²)
Solution:
Initial velocity (u) = 0 (dropped, not thrown)
Height (s) = 80 m
g = 10 m/s²
(a) Finding time:
Using s = ut + ½gt²
80 = 0 × t + ½ × 10 × t²
80 = 5t²
t² = 16
t = 4 seconds
(b) Finding final velocity:
Using v = u + gt
v = 0 + 10 × 4
v = 40 m/s
Or using v² = u² + 2gs
v² = 0 + 2 × 10 × 80
v² = 1600
v = 40 m/s
A ball is thrown vertically upward with velocity 30 m/s. Find:
(a) Maximum height reached
(b) Time to reach maximum height
(c) Total time to return to ground
(Take g = 10 m/s²)
Solution:
Initial velocity (u) = 30 m/s (upward)
g = -10 m/s² (negative because upward motion)
At maximum height, final velocity (v) = 0
(a) Maximum height:
Using v² = u² + 2gs
0 = (30)² + 2 × (-10) × s
0 = 900 - 20s
20s = 900
s = 45 m
(b) Time to reach maximum height:
Using v = u + gt
0 = 30 + (-10) × t
10t = 30
t = 3 seconds
(c) Total time:
Time to go up = Time to come down = 3 seconds
Total time = 3 + 3 = 6 seconds
Mass is the quantity of matter contained in an object. It is a measure of inertia of the object.
Weight is the force with which Earth (or any celestial body) attracts an object toward its center. Weight = Mass × Acceleration due to gravity.
Weight (W) = Mass (m) × Acceleration due to gravity (g)
W = m × g
On Earth: W = m × 9.8
On Moon: W = m × 1.6 (Moon's gravity is about 1/6th of Earth's)
SI unit: Newton (N)
| Feature | Mass | Weight |
|---|---|---|
| Definition | Amount of matter in object | Force of gravity on object |
| Type | Scalar quantity | Vector quantity |
| SI Unit | Kilogram (kg) | Newton (N) |
| Measuring Device | Physical balance | Spring balance |
| Variation | Constant everywhere | Changes with location |
| Can be Zero? | No (never zero) | Yes (in space) |
| Formula | - | W = m × g |
A person has mass of 60 kg. Calculate weight on:
(a) Earth (g = 9.8 m/s²)
(b) Moon (g = 1.6 m/s²)
(c) Jupiter (g = 25 m/s²)
Solution:
Mass = 60 kg (same everywhere)
(a) Weight on Earth:
W = m × g = 60 × 9.8 = 588 N
(b) Weight on Moon:
W = m × g = 60 × 1.6 = 96 N
(c) Weight on Jupiter:
W = m × g = 60 × 25 = 1500 N
Notice: Mass stays 60 kg everywhere, but weight changes
dramatically!
Imagine your body is like a box full of marbles (that's your mass
- the marbles never change). Now imagine Earth is like a giant
magnet pulling on your box.
On Earth, the magnet pulls hard, so the box feels heavy (588
N).
On Moon, it's a weaker magnet, so same box feels lighter (96
N).
On Jupiter, it's a super-strong magnet, so the box feels super
heavy (1500 N)!
The marbles inside (mass) never changed - only the pull (weight)
changed!
In everyday life, we incorrectly say "my weight is 60 kg" - but
technically, 60 kg is MASS, not weight!
Correct way: "My mass is 60 kg, and my weight on Earth is 588
N."
We use mass because it's constant everywhere, while weight
changes. When you step on a weighing scale, it actually measures
weight (force) but shows it as mass (in kg) by dividing by 9.8.
This works fine on Earth but would give wrong readings on Moon or
other planets!
Weightlessness is the condition in which an object appears to have no weight or feels no gravitational force. This happens when an object is in free fall or there is no reaction force acting on it.
Many people think astronauts float because there's no gravity in
space - WRONG! There IS gravity in space (Earth's gravity is still
strong at the space station's altitude).
Astronauts float because they are in continuous free fall! The
spacecraft is falling toward Earth, and astronauts inside are also
falling at the same rate. Since everything falls together, there's
no reaction force, creating the sensation of weightlessness.
Think of it like this: Imagine you're in an elevator, and the
cable breaks. As the elevator falls, you would float inside it
because both you and the elevator are falling together at the same
speed! That's exactly what's happening in the space station - it's
constantly "falling" around Earth (that's what an orbit is).
Weightlessness doesn't mean gravity is absent! It means there's no reaction force (normal force) acting on the body. The gravitational force is still there, but since everything is in free fall, we don't "feel" it.
Thrust is the total force acting perpendicular to a surface. When an object rests on a surface, it exerts a force perpendicular to that surface - this force is called thrust.
Thrust = Force perpendicular to surface = Weight (in many
cases)
Thrust = m × g
SI unit: Newton (N)
Direction: Perpendicular to the surface
Pressure is the thrust (force) acting per unit area. It tells us how concentrated the force is.
Pressure (P) = Thrust (F) / Area (A)
P = F / A
SI unit: Pascal (Pa) or Newton per square meter (N/m²)
1 Pascal = 1 N/m²
Larger unit: 1 atmosphere = 101,325 Pa ≈ 10⁵ Pa
A box weighing 500 N rests on a table. The area of contact is 0.5
m². Calculate the pressure exerted by the box on the table.
Solution:
Thrust (Force) = 500 N
Area = 0.5 m²
Pressure = Thrust / Area
P = 500 / 0.5
P = 1000 Pa or 1000 N/m²
The box exerts a pressure of 1000 Pascal on the table.
A sharp knife has a very small edge area. When you apply force,
all that force is concentrated on a tiny area, creating HUGE
pressure! That's why it cuts through things easily.
Same principle:
• Nails have pointed ends - small area, high pressure, easy to
pierce
• Needles are sharp - concentrate force on tiny point
• Your feet hurt when you step on a pebble - the pebble's small
contact area creates high pressure on your foot
On the other hand, a blunt knife has larger contact area, so same
force creates less pressure - that's why it doesn't cut well!
1. Why Do Tractors Have Wide Tires?
Wide tires have larger contact area, so pressure on ground is
less. This prevents them from sinking into soft soil.
2. Why Do Skiers Wear Broad Skis?
Broad skis spread weight over larger area, reducing pressure on
snow. This prevents skiers from sinking into snow.
3. Why Do We Carry Heavy Bags with Wide Straps?
Wide straps spread the weight over larger area of shoulder,
reducing pressure and making it more comfortable.
4. Why Are Railway Tracks Laid on Large-Sized Wooden or Iron
Sleepers?
Sleepers spread the weight of train over large area, reducing
pressure on ground and preventing tracks from sinking.
5. Why Do Cutting Tools Have Sharp Edges?
Sharp edges have very small area, so even moderate force creates
very high pressure, making cutting easy.
Buoyancy (or buoyant force) is the upward force exerted by a fluid (liquid or gas) on an object immersed in it. This is why objects feel lighter when submerged in water.
When you enter a swimming pool, you feel lighter because water
pushes you upward! This upward push is the buoyant force.
Imagine water molecules as tiny hands pushing against every part
of your body that's underwater. The push from below is stronger
than the push from above (because pressure increases with depth),
creating a net upward force. That's buoyancy!
This is why it's easier to lift someone in water than on land. The
water is helping you by pushing them upward!
When an object is immersed wholly or partially in a fluid, it experiences an upward force (buoyant force) equal to the weight of the fluid displaced by it.
Buoyant Force = Weight of fluid displaced
Fbuoyant = V × ρfluid × g
Where:
V = Volume of object immersed (or volume of fluid displaced)
ρfluid = Density of fluid
g = Acceleration due to gravity
Alternative form:
Fbuoyant = mdisplaced fluid × g
A metal cube of side 5 cm is completely immersed in water.
Calculate the buoyant force acting on it.
(Density of water = 1000 kg/m³, g = 10 m/s²)
Solution:
Side of cube = 5 cm = 0.05 m
Volume of cube = (0.05)³ = 0.000125 m³
Density of water = 1000 kg/m³
g = 10 m/s²
Buoyant force = V × ρ × g
Fbuoyant = 0.000125 × 1000 × 10
Fbuoyant = 1.25 N
The buoyant force is 1.25 N acting upward on the cube.
Legend says King Hiero of Syracuse asked Archimedes to check if
his crown was pure gold or mixed with silver. Archimedes was
puzzled - how to find this without melting the crown?
One day, while taking a bath, he noticed water overflowing when he
got in. He realized that his body displaced water equal to his
volume! Excited, he shouted "Eureka!" (I found it!) and ran out
(reportedly naked!) to solve the problem.
He compared the crown with pure gold of equal weight. Both
displaced different amounts of water, proving the crown was
impure! This discovery led to Archimedes' Principle - one of the
most important principles in physics!
Object Floats when:
Buoyant force ≥ Weight of object
Or: Density of object < Density of fluid
Object Sinks when:
Buoyant force < Weight of object
Or: Density of object > Density of fluid
Object Remains Suspended when:
Buoyant force = Weight of object
Or: Density of object = Density of fluid
A ship is made of steel, which is denser than water. So why
doesn't it sink?
Answer:
The ship is hollow inside, filled with air! When you consider the
entire volume of the ship (steel + air space), the average density
is LESS than water.
The hollow shape displaces a huge volume of water, creating enough
buoyant force to support the ship's weight. That's why ships float
even though they're made of steel!
If you make a solid steel ball of the same mass, it would sink
because it displaces much less water.
Water has a unique property: it expands when it freezes! This
means ice is LESS dense than liquid water (density of ice ≈ 920
kg/m³, density of water = 1000 kg/m³).
Since ice is less dense, it floats on water. This is unusual - for
most substances, the solid form is denser and sinks in its liquid
form.
Why is this important?
When lakes and ponds freeze, ice forms on top while water remains
liquid below. This allows fish and aquatic life to survive in the
liquid water underneath during winter!
Imagine objects in water are playing tug-of-war:
Team Gravity: Pulls object downward with force =
weight
Team Buoyancy: Pushes object upward with force =
weight of displaced water
• If Team Gravity wins → Object sinks (stone, iron)
• If Team Buoyancy wins → Object floats (wood, plastic bottle)
• If it's a tie → Object stays suspended (fish, submarine with
right ballast)
Fish are masters at controlling this! They have an air bladder
that they can inflate or deflate to adjust buoyancy and stay at
any depth they want.
Relative density (or specific gravity) of a substance is the ratio of the density of the substance to the density of water at 4°C.
Relative Density = Density of substance / Density of water
RD = ρsubstance / ρwater
Since density of water = 1000 kg/m³ or 1 g/cm³
RD = ρsubstance / 1000 (if ρ in kg/m³)
RD = ρsubstance / 1 (if ρ in g/cm³)
Note: Relative density has no unit (it's a ratio)
The density of iron is 7800 kg/m³. Calculate its relative
density.
Solution:
Density of iron = 7800 kg/m³
Density of water = 1000 kg/m³
Relative Density = Density of iron / Density of water
RD = 7800 / 1000
RD = 7.8
This means iron is 7.8 times denser than water. Since RD > 1, iron
sinks in water.
| Substance | Density (kg/m³) | Relative Density | Floats or Sinks? |
|---|---|---|---|
| Cork | 250 | 0.25 | Floats |
| Wood (Pine) | 500 | 0.5 | Floats |
| Ice | 920 | 0.92 | Floats |
| Water | 1000 | 1.0 | Reference |
| Aluminum | 2700 | 2.7 | Sinks |
| Iron | 7800 | 7.8 | Sinks |
| Gold | 19300 | 19.3 | Sinks |
| Mercury | 13600 | 13.6 | Sinks |
Mercury is a liquid metal with very high density (13.6 times
denser than water). It's so dense that even iron will FLOAT on
mercury!
Since iron's density (7800 kg/m³) is less than mercury's density
(13600 kg/m³), iron floats on mercury. You can literally float an
iron ball on liquid mercury!
Two objects of mass 50 kg each are separated by 1 m. Calculate the
gravitational force between them.
(G = 6.67 × 10⁻¹¹ N⋅m²/kg²)
Solution:
m₁ = 50 kg
m₂ = 50 kg
r = 1 m
G = 6.67 × 10⁻¹¹ N⋅m²/kg²
F = G(m₁m₂)/r²
F = (6.67 × 10⁻¹¹) × (50 × 50) / (1)²
F = (6.67 × 10⁻¹¹) × 2500
F = 16.675 × 10⁻⁸ N
F = 1.67 × 10⁻⁷ N
Answer: 1.67 × 10⁻⁷ N (very small force!)
A ball is dropped from a building of height 45 m. Calculate:
(a) Time taken to reach ground
(b) Velocity when it hits the ground
(Take g = 10 m/s²)
Solution:
Initial velocity (u) = 0 (dropped)
Height (s) = 45 m
g = 10 m/s²
(a) Time:
Using s = ut + ½gt²
45 = 0 + ½ × 10 × t²
45 = 5t²
t² = 9
t = 3 seconds
(b) Final velocity:
Using v = u + gt
v = 0 + 10 × 3
v = 30 m/s
Answers: (a) 3 seconds, (b) 30 m/s
An astronaut has mass 75 kg. Calculate weight on:
(a) Earth (g = 10 m/s²)
(b) Moon (g = 1.6 m/s²)
(c) Mars (g = 3.7 m/s²)
Solution:
Mass = 75 kg (same everywhere)
(a) Weight on Earth:
W = mg = 75 × 10 = 750 N
(b) Weight on Moon:
W = mg = 75 × 1.6 = 120 N
(c) Weight on Mars:
W = mg = 75 × 3.7 = 277.5 N
Answers: (a) 750 N, (b) 120 N, (c) 277.5 N
A rectangular block of dimensions 4 m × 2 m × 1 m and weight 1600
N is placed on ground. Calculate the minimum and maximum pressure
it can exert.
Solution:
Weight (Thrust) = 1600 N
Possible contact areas:
• 4 m × 2 m = 8 m²
• 4 m × 1 m = 4 m²
• 2 m × 1 m = 2 m²
Minimum pressure (largest area):
P = F/A = 1600/8 = 200 Pa
Maximum pressure (smallest area):
P = F/A = 1600/2 = 800 Pa
Answers: Minimum = 200 Pa, Maximum = 800 Pa
A wooden block of volume 0.002 m³ is completely immersed in water.
Calculate the buoyant force acting on it.
(Density of water = 1000 kg/m³, g = 10 m/s²)
Solution:
Volume immersed (V) = 0.002 m³
Density of water (ρ) = 1000 kg/m³
g = 10 m/s²
Buoyant force = V × ρ × g
F = 0.002 × 1000 × 10
F = 20 N
Answer: 20 N (upward)
A substance has density 2600 kg/m³. Calculate its relative
density. Will it float or sink in water?
Solution:
Density of substance = 2600 kg/m³
Density of water = 1000 kg/m³
Relative Density = Density of substance / Density of water
RD = 2600 / 1000
RD = 2.6
Since RD > 1, the substance is denser than water and will SINK.
Answer: Relative Density = 2.6, Will sink in
water
For Force and Motion Problems:
• Always write given data first
• Identify whether motion is upward or downward
• For upward motion: use g = -9.8 m/s²
• For downward motion: use g = +9.8 m/s²
• Choose appropriate equation of motion
For Weight Problems:
• Remember: W = mg
• Mass remains constant, weight changes with g
• Always write units (kg for mass, N for weight)
For Pressure Problems:
• Identify thrust (usually weight)
• Find area of contact
• Use P = F/A
• For minimum pressure: use maximum area
• For maximum pressure: use minimum area
For Buoyancy Problems:
• Find volume of object immersed
• Identify density of fluid
• Use Buoyant force = V × ρ × g
• Compare with weight to determine floating/sinking
Common Mistakes to Avoid:
• Confusing mass (kg) with weight (N)
• Using wrong sign for g in upward motion
• Forgetting to convert units (cm to m, etc.)
• Not writing units in final answer
• Mixing up pressure and thrust