NCERT Solutions for Class 9th Science Chapter 9: GRAVITATION

According to the law of gravitation, the force of attraction between two objects is inversely proportional to the square of the distance between them.
When the distance is reduced to half, the new force becomes four times greater than before.
So, the gravitational force increases byfour times.

Although the gravitational force on a heavy object is greater because of its larger mass, the acceleration due to gravity is the same for all objects.
The acceleration due to gravity (g = 9.8 m/s²) does not depend on the mass of the object.
Therefore, both heavy and light objects fall at the same rate when only gravity acts on them (if air resistance is ignored).

The formula for gravitational force is:
F = G × (m₁ × m₂) / r²

Here,
G = 6.67 × 10⁻¹¹ N·m²/kg²
m₁ = 6 × 10²⁴ kg
m₂ = 1 kg
r = 6.4 × 10⁶ m

F = (6.67 × 10⁻¹¹ × 6 × 10²⁴ × 1) / (6.4 × 10⁶)²
F = (4.002 × 10¹⁴) / (4.096 × 10¹³)
F = 9.77 N

Therefore, the gravitational force between the Earth and the object is approximately 9.8 N.

The Earth attracts the Moon with a force equal in magnitude to the force with which the Moon attracts the Earth.
According to Newton’s Third Law of Motion, every action has an equal and opposite reaction.
Therefore, the gravitational force of the Earth on the Moon and the gravitational force of the Moon on the Earth are equal in size but opposite in direction.

Both the Earth and the Moon attract each other with equal gravitational force, but the Earth does not move noticeably towards the Moon because the mass of the Earth is much greater than the mass of the Moon.
Due to its very large mass, the acceleration produced in the Earth is extremely small, while the lighter Moon moves more easily around the Earth.

The universal law of gravitation explains many natural phenomena. Its importance is as follows:

1. It helps us understand how all objects in the universe attract each other.
2. It explains the motion of the Moon around the Earth and the motion of planets around the Sun.
3. It helps us understand the cause of tides due to the gravitational pull of the Moon and the Sun.
4. It explains why objects fall towards the Earth when dropped.

Thus, the universal law of gravitation helps us understand and explain the structure and behavior of the universe.

The acceleration produced in a body when it falls freely under the influence of gravity alone is called the acceleration due to gravity or acceleration of free fall.
It is denoted by g and its value on Earth is approximately 9.8 m/s².
This means that the velocity of a freely falling object increases by 9.8 m/s every second.

The gravitational force between the Earth and an object is called the weight of the object.

This is a specific type of gravitational force, which is the universal attractive force that exists between any two objects with mass.

No, the friend will not agree with the weight of the gold.

This is because the value of acceleration due to gravity (g) is greater at the poles and smaller at the equator.

Since weight = mass × gravity (W = m × g), the weight of the same gold will be less at the equator than at the poles.

Therefore, the gold will weigh slightly less at the equator, even though its mass remains the same.

A sheet of paper falls slower than a crumpled paper ball because of air resistance.
The flat sheet has a larger surface area, so it experiences more air resistance while falling.
When the paper is crumpled into a ball, its surface area facing the air becomes smaller, reducing air resistance.

Therefore, the crumpled paper ball falls faster than the flat sheet.

We know that Weight = Mass × Acceleration due to gravity (W = m × g)

On Earth:
m = 10 kg
g = 9.8 m/s²
Wₑ = 10 × 9.8 = 98 N

On Moon:
Gravity on Moon = (1/6) × gravity on Earth
gₘ = 9.8 / 6 = 1.63 m/s²
Wₘ = 10 × 1.63 = 16.3 N

Therefore,
Weight on Earth = 98 N
Weight on Moon = 16.3 N

Height (h) = 19.6 m
Initial velocity (u) = 0 (since the stone is released from rest)
Acceleration due to gravity (g) = 9.8 m/s²

Using the equation of motion:
v² = u² + 2gh

Substitute the values:
v² = 0 + 2 × 9.8 × 19.6
v² = 384.16
v = √384.16
v = 19.6 m/s

Therefore, the final velocity of the stone just before touching the ground is 19.6 m/s.

Initial velocity (u) = 40 m/s
Acceleration due to gravity (g) = –10 m/s² (negative because gravity acts downward)
At maximum height, final velocity (v) = 0

Using the equation of motion:
v² = u² + 2as

0 = (40)² + 2(–10)s
0 = 1600 – 20s
20s = 1600
s = 80 m

Maximum height = 80 m

When the stone returns to the ground:

Net displacement = 0 (since it comes back to the starting point)
Total distance covered = Distance up + Distance down = 80 + 80 = 160 m

Final Answers:
Maximum height = 80 m
Net displacement = 0
Total distance covered = 160 m

Mass of Earth, M₁ = 6 × 10²⁴ kg
Mass of Sun, M₂ = 2 × 10³⁰ kg
Distance between Earth and Sun, r = 1.5 × 10¹¹ m
Gravitational constant, G = 6.67 × 10⁻¹¹ N·m²/kg²

Formula:
F = G × M₁ × M₂ / r²

Substituting values:
F = (6.67 × 10⁻¹¹ × 6 × 10²⁴ × 2 × 10³⁰) / (1.5 × 10¹¹)²
F = (8.004 × 10⁴⁴) / (2.25 × 10²²)
F = 3.56 × 10²² N

Therefore, the gravitational force between the Earth and the Sun is 3.56 × 10²² N.

Height of tower = 100 m
Acceleration due to gravity, g = 10 m/s²

Let the stones meet after t seconds.

For the first stone (falling from the top):
Initial velocity, u₁ = 0
Distance fallen, s₁ = ½ g t² = 5t²

For the second stone (projected upwards):
Initial velocity, u₂ = 25 m/s
Distance covered, s₂ = u₂t – ½ g t² = 25t – 5t²

They meet when the total distance covered by both is equal to the height of the tower.
s₁ + s₂ = 100
5t² + (25t – 5t²) = 100
25t = 100
t = 4 seconds

Distance fallen by the first stone, s₁ = 5t² = 5 × 4² = 80 m
Therefore, they meet 20 m above the ground (100 – 80 = 20).

Final Answer:
Time when they meet = 4 seconds
Height above the ground where they meet = 20 meters

(a) Time to rise to top = half the total time = 6/2 = 3 s.
Initial velocity u = g × time = 10 × 3 = 30 m/s.

(b) Maximum height H = u² / (2g) = (30)² / (2 × 10) = 900 / 20 = 45 m.

(c) Position after t = 4 s (take upward as positive, origin at thrower):
y = ut − ½ g t² = 30×4 − ½×10×4² = 120 − 80 = 40 m above the thrower.

Final answers:
(a) 30 m/s upward
(b) 45 m
(c) 40 m above the thrower (after 4 s)

The buoyant force on an object immersed in a liquid acts vertically upward, that is, opposite to the direction of gravity.

A block of plastic comes up to the surface of water because the buoyant force acting on it is greater than its weight. Since the density of plastic is less than the density of water, the upward buoyant force pushes it up until it reaches the surface.

Density of the substance = Mass / Volume = 50 g / 20 cm³ = 2.5 g/cm³

Since the density of the substance (2.5 g/cm³) is greater than the density of water (1 g/cm³), the substance will sink in water.

Density of the packet = Mass / Volume = 500 g / 350 cm³ = 1.43 g/cm³

Since the density of the packet (1.43 g/cm³) is greater than the density of water (1 g/cm³), the packet will sink in water.

Mass of water displaced = Volume of packet × Density of water
= 350 cm³ × 1 g/cm³
= 350 g

Final Answer:
The packet will sink in water, and the mass of water displaced is 350 g.