1. The Mystery of the Falling Apple | Gravitation class 9 NCERT solutions

Hello students! Today we are going to explore the invisible glue that holds our universe together. Imagine you are standing in a park holding a cricket ball. If you let it go, what happens? It falls down. Always down. Never up, never sideways. Why?

This question bothered a very smart man named Isaac Newton back in the 1600s. The story says he saw an apple fall from a tree and asked a brilliant question: “If the Earth pulls the apple, does it also pull the Moon?”.

The answer is YES. The Earth is constantly pulling the Moon. But if the Earth is pulling the Moon, why doesn’t the Moon crash into us just like the apple? To understand this, we need to understand circular motion.

1.1 Centripetal Force: The “Center-Seeking” Force

Imagine tying a stone to a string and whirling it around your head. The stone moves in a circle. What keeps it in that circle? It is the tension in the string, which you are pulling with your hand.

This inward pull is called Centripetal Force. In space, there is no string. The invisible string that holds the Moon in orbit around the Earth is the Gravitational Force. Without this force, the Moon would fly off into deep space in a straight line.

2. The Universal Law of Gravitation

Newton realized that gravity isn’t just an “Earth thing.” It is a “Universe thing.” He stated the Universal Law of Gravitation:

“Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them.”

2.1 Decoding the Law

Let’s break this complex sentence into simple parts using a diagram.

Suppose we have two objects:

1. A big object with mass M.

2. A small object with mass m.

3. The distance between their centers is d.

Rule 1: Mass Increases Force.

The force (F) depends on how much matter is there.

F ∝ M × m

This means if you double the mass of one object, the force doubles. If you double both masses, the force becomes 4 times stronger.

Rule 2: Distance Kills Force (Inverse Square Law).

The force (F) gets weaker quickly as you move away.

F ∝ 1 / d²

This is the tricky part. It is “d squared”.

– If you double the distance (2d), the force drops to 1/4th.

– If you triple the distance (3d), the force drops to 1/9th.

2.2 The Formula

Combining these rules, we get the famous formula:

F = G × (M × m) / d²

Here, G is the Universal Gravitational Constant.

Its value is 6.673 × 10⁻¹¹ Nm²/kg².

Why is G so small? The negative exponent (-11) tells us gravity is a very weak force. That is why you don’t stick to your chair or your friend sitting next to you. The force exists, but it is too tiny to notice unless one of the objects is gigantic, like the Earth.

3. Free Fall: When Gravity is the Only Boss

Imagine you are on the top of the Burj Khalifa. If you drop a coin, it falls. While it is falling, no one is pushing it, no engine is driving it. Only one force is acting on it: The Earth’s Gravity. This state is called Free Fall.

3.1 Acceleration due to Gravity (g)

When the coin falls, does it fall at a constant speed? No! It starts at zero speed and gets faster and faster every second. Since velocity is changing, there must be acceleration.

We denote this specific acceleration with the small letter g.

Calculating ‘g’:

From Newton’s 2nd Law (Force & Motion), we know: Force = Mass × Acceleration (F = ma).

From the Law of Gravitation, we know: Force = G × (Mass of Earth × Mass of Object) / Distance².

If we equate them:

ma = G (M × m) / R²

(The ‘m’ of the object cancels out from both sides!)

g = GM / R²

The Surprise Result:

Notice that in the formula g = GM/R², there is no ‘m’ (mass of the object). This means ‘g’ does not depend on the mass of the falling object.

In a vacuum (where there is no air friction), a heavy hammer and a light feather will fall at the exact same speed and hit the ground at the same time. This is counter-intuitive because, on Earth, air resistance usually slows the feather down.

3.2 Value of ‘g’

On Earth, the average value of g is 9.8 m/s². This means for every second you fall, your speed increases by 9.8 m/s.

Variation of g: The Earth isn’t a perfect sphere. It is squashed at the top and bottom (poles) and bulges at the middle (equator).

– At Poles: The radius is smaller (you are closer to the center). So, gravity is stronger.

– At Equator: The radius is larger (you are farther from the center). So, gravity is weaker.

3.3 Equations of Motion for Free Fall

Since ‘g’ is essentially constant near the surface, we can use our standard motion equations. We just replace ‘a’ with ‘g’.

Tip: If you throw an object UP, g is negative (-9.8). If it falls DOWN, g is positive (+9.8).

4. Mass vs. Weight: Investigating the Difference

In everyday English, “Mass” and “Weight” mean the same thing. In Physics, they are totally different. Let’s create a clear distinction.

4.1 Mass (m)

Mass is the measure of the quantity of matter inside a body. It is also a measure of Inertia (resistance to motion).

– Does it change? No. Never. Whether you are on Earth, the Moon, or floating in deep space, you are made of the same atoms. Your mass is constant.

– Unit: Kilogram (kg).

4.2 Weight (W)

Weight is a Force. Specifically, it is the force with which the Earth (or any planet) pulls you down.

– Formula: Since Force = mass × acceleration, Weight = Mass × g. (W = m × g).

– Does it change? Yes! Since ‘g’ changes from place to place, your weight changes.

– Unit: Newton (N) (because it is a force).

4.3 Weight on the Moon

The Moon is smaller than Earth, so its gravity is weaker. Specifically, gravity on the Moon is 1/6th of Earth’s gravity.

So, if your mass is 60kg:

– On Earth: Weight = 60 × 9.8 ≈ 588 N.

– On Moon: Weight = 588 / 6 ≈ 98 N.

You would feel very light on the Moon, but you would still contain the same amount of matter.

5. Thrust and Pressure: The Science of Surfaces

Why does a camel walk easily on sand while you sink? Why do army tanks have caterpillar tracks? It is all about how force is distributed.

5.1 Thrust

Thrust is simply the total force applied perpendicular (at 90 degrees) to a surface. If you stand on sand, the thrust is your body weight.

5.2 Pressure

Pressure is the “intensity” of force. It depends on the area over which the force acts.

Formula: Pressure = Thrust / Area

Unit: Pascal (Pa) or N/m².

Everyday Applications:

• Sharp Knife: The cutting edge has a very tiny area. Even a small force creates huge pressure to cut vegetables.
• School Bags: Straps are made wide to increase the area. This reduces the pressure on your shoulders, making the bag feel lighter.
• Balloon Bed: You can pop a balloon with one pin (tiny area, huge pressure). But you can push it onto a bed of 100 pins without popping it because the force is spread out (large area, low pressure).

6. Buoyancy: Why Do Ships Float?

Have you ever pushed a plastic ball underwater? It fights back! It pops up the moment you release it. This upward force exerted by water (or any fluid) is called Buoyancy or Upthrust.

6.1 Why things sink or float

When an object is in water, there is a battle between two forces:

1. Gravity (Weight): Pulls it DOWN.

2. Buoyant Force: Pushes it UP.

• If Density of Object < Density of Liquid → It Floats (e.g., Cork, Ice).
• If Density of Object > Density of Liquid → It Sinks (e.g., Iron nail, Stone).

6.2 Archimedes’ Principle

This is the most important rule of fluids. The story goes that Archimedes discovered this in his bathtub and ran out shouting “Eureka!” (I found it!).

Principle: “When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.”

The Iron Ship Paradox:

An iron nail sinks because it is small and displaces very little water. The weight of that little water is not enough to push the nail up.

An iron ship is hollow and huge. It displaces a MASSIVE amount of water. The weight of that displaced water creates a huge upward force that is strong enough to hold the heavy ship up.

7. Extensive Practice Set (Detailed Solutions)

Now, let’s apply what we learned. I have provided very detailed step-by-step solutions so you can understand the logic.

Part A: Multiple-Choice Questions (MCQs)

1. According to the Universal Law of Gravitation, if the distance between two objects is doubled, the gravitational force between them becomes:
   a) Double
   b) Half
   c) Four times stronger
   d) Four times weaker

   Answer: d) Four times weaker.
   
   Detailed Explanation: The formula is F ∝ 1/d². If distance ‘d’ becomes ‘2d’, then d² becomes (2d)² = 4d². Since the 4 is in the denominator (bottom), the force is divided by 4.

2. The weight of an object is:
   a) The amount of matter it contains.
   b) A measure of its inertia.
   c) The force with which it is attracted to the Earth.
   d) The same on the Earth and the Moon.

   Answer: c) The force with which it is attracted to the Earth.
   
   Explanation: Option (a) and (b) describe Mass. Option (d) is false because weight changes. Weight is the gravitational pull (Force).

3. An object is in free fall. Which of the following quantities remains constant?
   a) Velocity
   b) Displacement
   c) Acceleration
   d) Momentum

   Answer: c) Acceleration.
   
   Explanation: Velocity keeps increasing. Momentum (mass × velocity) also increases. However, the rate of increase is ‘g’ (9.8 m/s²), which remains constant near Earth’s surface.

4. An object will float in a liquid if:
   a) Its density is greater than the density of the liquid.
   b) Its density is less than the density of the liquid.
   c) Its density is equal to the density of the liquid.
   d) It has a very large mass.

   Answer: b) Its density is less than the density of the liquid.
   
   Explanation: Lighter density materials (like wood) float on heavier density fluids (like water).

5. The SI unit of pressure is:
   a) Newton (N)
   b) Kilogram (kg)
   c) Newton-meter (Nm)
   d) Pascal (Pa)

   Answer: d) Pascal (Pa).
   
   Explanation: Pressure = Force/Area = N/m². This unit is named Pascal.

Part B: Short Answer Questions

6. State Newton’s Universal Law of Gravitation.

   Answer: It states that every object in the universe attracts every other object with a force. This force is:
   
   1. Directly proportional to the product of their masses (heavier = stronger pull).
   
   2. Inversely proportional to the square of the distance between them (farther = weaker pull).

7. Why is the value of ‘g’ (acceleration due to gravity) greater at the poles than at the equator?

   Answer: The Earth is not a perfect sphere; it is an oblate spheroid. This means it is flattened at the poles and bulging at the equator.
   
   – The distance from the center to the Poles is smaller.
   
   – Gravity is stronger when the distance is smaller.
   
   – Therefore, ‘g’ is greater at the poles than at the equator.

8. Differentiate between the mass of an object and its weight, giving at least two key differences.

   Answer:
   
   Difference 1 (Definition): Mass is the amount of matter in an object. Weight is the force of gravity acting on that object.
   
   Difference 2 (Variability): Mass is constant everywhere in the universe. Weight varies depending on the local gravity (e.g., zero in space, less on Moon).

9. Why does a sharp knife cut objects more effectively than a blunt knife? Explain using the concept of pressure.

   Answer:
   
   This works on the formula: Pressure = Force / Area.
   
   A sharp knife has a very thin edge, meaning the surface area is extremely small. When we apply force, it is concentrated on this tiny area. This creates extremely high pressure, allowing the knife to pierce through the object easily. A blunt knife has a larger area, resulting in low pressure for the same force.

10. State Archimedes’ Principle.

    Answer: Archimedes’ Principle states that when a body is immersed fully or partially in a fluid, it experiences an upward force (buoyant force) that is equal to the weight of the fluid displaced by it. This principle helps us understand why huge ships float.

Part C: Long Answer Questions (Numerical Solving)

11. An object weighs 120 N on the surface of the Earth. What would be its weight on the surface of the Moon? What is its mass on the Earth and on the Moon? (Take g on Earth = 10 m/s²).

    Answer:
    
    Given:
    
    Weight on Earth (We) = 120 N
    
    Gravity on Earth (g) = 10 m/s²

    Step 1: Calculate Mass.
    
    Formula: Weight = Mass × g
    
    120 = Mass × 10
    
    Mass = 120 / 10 = 12 kg.
    
    Note: Mass is constant. So, Mass on Earth = Mass on Moon = 12 kg.

    Step 2: Calculate Weight on Moon.
    
    Concept: Gravity on Moon is 1/6th of gravity on Earth.
    
    Weight on Moon = (1/6) × Weight on Earth
    
    Weight on Moon = (1/6) × 120
    
    Weight on Moon = 20 N.

    Final Answer: Mass is 12 kg (everywhere). Weight on Moon is 20 N.

12. A stone is dropped from the top of a 49 m high tower. (a) How long will it take to reach the ground? (b) With what velocity will it strike the ground? (Take g = 9.8 m/s²).

    Answer:
    
    Given:
    
    Initial velocity (u) = 0 m/s (dropped from rest)
    
    Distance/Height (h) = 49 m
    
    Acceleration (g) = +9.8 m/s² (falling down)

    (a) Find Time (t):
    
    Using the 2nd Equation of Motion: h = ut + ½gt²
    
    49 = (0 × t) + ½ × 9.8 × t²
    
    49 = 0 + 4.9t²
    
    t² = 49 / 4.9
    
    t² = 10
    
    t = √10 ≈ 3.16 seconds.

    (b) Find Final Velocity (v):
    
    Using the 3rd Equation of Motion: v² = u² + 2gh
    
    v² = 0² + 2 × 9.8 × 49
    
    v² = 19.6 × 49
    
    v² = 960.4
    
    v = √960.4 ≈ 31 m/s.

13. Explain why a heavy object does not fall faster than a light object when dropped from the same height in a vacuum. Which law or principle explains this?

    Answer:
    
    This phenomenon is explained by the derivation of ‘g’ (acceleration due to gravity).
    
    The formula is g = GM / R².
    
    Here:
    
    – G is a universal constant.
    
    – M is the mass of the Earth.
    
    – R is the radius of the Earth.
    
    Key Observation: The term ‘m’ (mass of the falling object) is missing from the formula! This proves that acceleration due to gravity does not depend on the mass of the object. Therefore, in a vacuum (where air resistance doesn’t interfere), a heavy hammer and a light feather accelerate at the exact same rate (9.8 m/s²) and hit the ground together.

14. A sealed packet has a mass of 500 g and a volume of 400 cm³. The density of water is 1 g/cm³. (a) Calculate the density of the packet. (b) Will it float or sink? (c) Mass of water displaced?

    Answer:
    
    (a) Calculate Density:
    
    Formula: Density = Mass / Volume
    
    Density = 500 g / 400 cm³
    
    Density = 1.25 g/cm³.

    (b) Float or Sink?
    
    Compare densities:
    
    – Density of Packet = 1.25 g/cm³
    
    – Density of Water = 1.0 g/cm³
    
    Since the packet is denser than water (1.25 > 1.0), it will Sink.

    (c) Mass of Water Displaced:
    
    Since the packet sinks completely, it displaces water equal to its own volume.
    
    Volume of water displaced = 400 cm³.
    
    Mass of water = Density of water × Volume
    
    Mass = 1 g/cm³ × 400 cm³ = 400 g.

15. What is buoyancy? Explain the conditions under which an object floats or sinks when placed on the surface of a liquid, using the concepts of buoyant force and the weight of the object.

    Answer:
    
    Buoyancy: It is the upward force exerted by a fluid that opposes the weight of an immersed object.
    
    When an object is placed in water, two vertical forces act on it:
    
    1. Weight (W): Acting downwards.
    
    2. Buoyant Force (Fb): Acting upwards.

    Conditions:
    
    – Object Floats: If Buoyant Force > Weight. (This happens when the object’s density is less than the liquid’s density).
    
    – Object Sinks: If Weight > Buoyant Force. (This happens when the object’s density is greater than the liquid’s density).
    
    – Object Suspended (Submerged but not sinking): If Weight = Buoyant Force.