Gravitation – CBSE NCERT Study Resources

Gravitational force is the attraction between two objects with mass. We studied that it depends on their masses and distance.

According to Newton's law, force increases with mass but decreases with distance squared (F = Gm₁m₂/r²). Our textbook shows Earth's gravity pulling objects downward.

Example: Moon orbiting Earth (NCERT). Real-world: Tides caused by the Moon's gravity.

Free fall is motion under only gravity. We learned all objects accelerate equally at g (9.8 m/s²).

From Newton's second law and gravitation: g = GM/R² (M=Earth's mass, R=radius). NCERT demonstrates this with a feather and coin in vacuum.

On the Moon, g is 1/6th of Earth's (Apollo astronauts' jumps).

Mass is matter quantity (constant), while weight is gravitational force (W=mg).

Our textbook shows a spring balance measuring weight. Weight changes with g (W ∝ g). On Jupiter, higher g increases weight.

Astronauts weigh less on Moon (NCERT example). Spacecraft calculate fuel based on planetary g.

The law states F=Gm₁m₂/r², where G is the gravitational constant (6.67×10⁻¹¹ Nm²/kg²).

Earth's large mass creates strong pull (NCERT: apple falling). Gravity decreases with altitude as r increases.

Satellites use this law for orbital motion. Geostationary satellites maintain fixed positions.

Thrust is force perpendicular to surface, pressure is thrust per unit area (P=F/A).

Our textbook shows a nail tip creating high pressure. Underwater, gravity increases pressure with depth (P = hρg).

Submarines withstand high pressure (NCERT). Divers feel ear pain due to pressure changes.

Gravitational force is the attractive force between two masses. Newton's law states it is directly proportional to the product of masses and inversely proportional to the square of the distance between them.

• Formula: F = G(m₁m₂)/r²
• G is the gravitational constant (6.67×10⁻¹¹ Nm²/kg²)

Our textbook shows Earth's gravity pulling objects downward. Similarly, the moon's gravity causes tides.

Acceleration due to gravity (g) is the force per unit mass exerted by a planet. It depends on the planet's mass and radius.

• Earth's g ≈ 9.8 m/s²
• Moon's g ≈ 1.63 m/s² (1/6th of Earth's)

Our textbook explains astronauts weigh less on the moon. A 60 kg person weighs only 98 N there vs 588 N on Earth.

Free fall is motion under only gravity's influence. All objects accelerate equally in vacuum because g is independent of mass.

• Galileo demonstrated this at the Leaning Tower of Pisa
• Air resistance causes variations in real-world falls

Our textbook shows a feather and coin falling equally in a vacuum tube. Skydivers reach terminal velocity due to air resistance.

Mass is constant matter quantity (kg), while weight is gravitational force (N) that varies with location.

• Mass measured by balance; weight by spring scale
• Weight = mass × g
• Mars' g ≈ 3.7 m/s² (about 38% of Earth's)

Our textbook mentions a 50 kg person would weigh 185 N on Mars vs 490 N on Earth.

A simple pendulum's time period relates to g through the formula T=2π√(L/g).

• Measure length (L) and time for 20 oscillations
• Calculate T (time/20) and substitute in formula

Our textbook shows this experiment. Seismometers use similar principles to detect ground motion.

Gravitational force is the attractive force between two masses. It keeps planets in orbit around the Sun.

Our textbook shows that the Earth's gravity pulls objects toward its center, causing tides due to the Moon's pull.

Satellites use gravitational force to stay in orbit, like communication satellites.

Newton's law states every mass attracts another with force proportional to their masses and inversely to distance squared.

We studied F = G(m₁m₂)/r². Weight is F when one mass is Earth.

This law helps calculate satellite orbits, like in ISRO missions.

'g' decreases with height as distance from Earth's center increases, and decreases with depth as mass below reduces.

Our textbook shows 'g' is 9.8 m/s² at surface but less on mountains.

This variation affects spacecraft launch calculations.

Objects float if buoyant force equals weight, else sink. Archimedes' Principle quantifies this.

We studied iron nails sink (density > water) while ships float (displaced water weight = ship weight).

Hot-air balloons rise as heated air becomes less dense.

Mass is constant matter quantity, weight is gravitational force on that mass.

Our textbook's spring balance activity shows weight = mass × g. Moon's 'g' is 1/6th Earth's.

Astronauts weigh less on Moon but mass remains same.

Free fall is the motion of an object under the influence of gravitational force alone, without any other external forces like air resistance acting on it. During free fall, all objects, regardless of their masses, experience the same acceleration towards the Earth, known as the acceleration due to gravity (g).

To derive the expression for g, we use Newton's law of gravitation, which states that the gravitational force (F) between two masses (M and m) separated by a distance (r) is given by:

Where G is the universal gravitational constant. For an object of mass m near the Earth's surface, the gravitational force is also equal to its weight:

Equating the two expressions:

Canceling m from both sides, we get:

Here, M is the mass of the Earth, and r is the distance from the center of the Earth to the object (approximately the Earth's radius near the surface).

The value of g varies with altitude and depth as follows:

• With altitude: As we move above the Earth's surface, r increases, causing g to decrease because g is inversely proportional to r².
• With depth: Inside the Earth, only the mass enclosed within the radius from the center contributes to g. As depth increases, the effective mass decreases, reducing g linearly until it becomes zero at the Earth's center.

The term free fall refers to the motion of an object under the sole influence of gravity, without any other forces (like air resistance) acting on it. During free fall, all objects, regardless of their mass, experience the same acceleration towards the Earth, known as the acceleration due to gravity (g).

To derive the expression for g, we use Newton's Law of Universal Gravitation and Second Law of Motion:

1. Gravitational force (F) between Earth (mass M) and an object (mass m) is given by: F = G * (M * m) / R², where G is the gravitational constant and R is Earth's radius.

2. According to Newton's Second Law: F = m * a. Here, a = g (acceleration due to gravity).

3. Equating both expressions: m * g = G * (M * m) / R².

4. Simplifying, we get: g = G * M / R².

This shows that g depends on Earth's mass (M) and radius (R).

Variation of g with altitude and depth:

• Altitude: As we go higher above Earth's surface, g decreases because the distance (R + h) from Earth's center increases. The formula becomes: g' = G * M / (R + h)².
• Depth: Inside Earth, g decreases linearly with depth (d) because only the mass enclosed by the smaller radius (R - d) contributes. The formula is: g' = g * (1 - d/R).

Thus, g is maximum at Earth's surface and reduces both above and below it.

The term free fall refers to the motion of an object under the sole influence of gravity, without any other forces (like air resistance) acting on it. During free fall, all objects, regardless of their mass, experience the same acceleration towards the Earth, known as the acceleration due to gravity (g).

To derive the expression for g, we use Newton's law of gravitation, which states that the force of attraction (F) between two masses (M and m) separated by a distance (r) is given by:
F = G * (M * m) / r²
Here, G is the universal gravitational constant.

According to Newton's second law of motion, F = m * a. For an object of mass m near the Earth's surface, this force is the weight of the object, and the acceleration is g. Thus:
m * g = G * (M * m) / r²
Canceling m from both sides, we get:
g = G * M / r²
Here, M is the mass of the Earth, and r is the distance from the object to the Earth's center (approximately the Earth's radius near the surface).

The value of g varies with altitude and depth as follows:

• With altitude: As we move higher above the Earth's surface, r increases, causing g to decrease because g ∝ 1/r².
• With depth: Below the Earth's surface, only the mass enclosed by the radius at that depth contributes to g. Since the effective mass decreases, g also decreases linearly with depth.

This understanding is crucial for applications like satellite motion, geophysical studies, and even everyday phenomena like weight variation at different locations.

Free fall is the motion of an object under the influence of gravitational force alone, without any other forces (like air resistance) acting on it. During free fall, all objects, regardless of their mass, experience the same acceleration towards the Earth.

To derive the expression for g, we start with Newton's law of gravitation:

F = G * (M * m) / r²

Where:
F = Gravitational force
G = Universal gravitational constant
M = Mass of the Earth
m = Mass of the object
r = Distance between the object and Earth's center

From Newton's second law of motion, F = m * a. Here, a = g (acceleration due to gravity).

Equating both expressions:
m * g = G * (M * m) / r²
Canceling m from both sides:
g = G * M / r²

This is the expression for acceleration due to gravity.

Factors affecting g:

• Altitude: As height increases, g decreases because r (distance from Earth's center) increases.
• Shape of the Earth: Earth is not perfectly spherical, so g varies slightly at different latitudes.

The gravitational force is the attractive force between two objects with mass. According to Newton's law of gravitation, the force (F) between two masses (m₁ and m₂) separated by a distance (r) is given by:

F = G (m₁m₂)/r²

where G is the universal gravitational constant.

To derive the acceleration due to gravity (g) on Earth's surface, consider Earth's mass (M) and radius (R). The force on an object of mass m is:

F = G (Mm)/R²

From Newton's second law (F = ma), the acceleration (g) is:

g = F/m = G (M)/R²

This is the expression for g on Earth's surface.

Variation of g:

• With altitude: As height (h) increases, g decreases because the distance from Earth's center increases. The modified formula is:
  g' = g (R²)/(R + h)²
• With depth: Inside Earth, g decreases linearly with depth (d) due to reduced effective mass. The formula becomes:
  g' = g (1 - d/R)

Understanding these variations helps in applications like satellite motion and geophysical studies.

The concept of free fall refers to the motion of an object under the sole influence of gravity, without any other forces (like air resistance) acting on it. During free fall, all objects, regardless of their mass, experience the same acceleration towards the Earth, known as the acceleration due to gravity (g).

To derive the expression for g, we start with Newton's law of gravitation:
F = G * (M * m) / r²
where:
- F is the gravitational force
- G is the universal gravitational constant
- M is the mass of the Earth
- m is the mass of the object
- r is the distance between the centers of the Earth and the object.

According to Newton's second law of motion:
F = m * a
For free fall, the acceleration (a) is g, so:
m * g = G * (M * m) / r²
Canceling 'm' from both sides:
g = G * M / r²

This shows that g depends on the mass of the Earth (M) and the distance from its center (r).

Variation of g:

• With altitude: As altitude increases, r increases, causing g to decrease (since g ∝ 1/r²).
• With depth: Inside the Earth, g decreases linearly with depth because only the mass enclosed by the smaller radius contributes to the gravitational force.

Free fall is the motion of an object under the influence of gravitational force alone, without any other forces (like air resistance) acting on it. During free fall, all objects, regardless of their masses, experience the same acceleration called the acceleration due to gravity (g).

To derive g using Newton's law of gravitation:
1. Newton's law states: F = G * (M * m) / r², where F is the gravitational force, G is the gravitational constant, M is Earth's mass, m is the object's mass, and r is the distance between their centers.
2. From Newton's second law: F = m * a. Here, a = g (acceleration due to gravity).
3. Equating both expressions: m * g = G * (M * m) / r².
4. Simplifying: g = G * M / r². This gives the acceleration due to gravity.

Variation of g:

• With altitude: As height (h) increases, r (distance from Earth's center) increases, reducing g as per g' = g * (R² / (R + h)²), where R is Earth's radius.
• With depth: Inside Earth, g decreases linearly with depth (d) as g' = g * (1 - d/R), since only the inner mass contributes to gravity.

Note: g is maximum at Earth's surface and zero at its center.

Free fall refers to the motion of an object under the sole influence of gravity, without any other forces (like air resistance) acting on it. During free fall, all objects experience the same acceleration, known as acceleration due to gravity (g), regardless of their mass.

To derive g using Newton's law of gravitation:

Variation of g:

• With altitude: As height increases, r increases, reducing g (since g ∝ 1/r²).
• With depth: Inside Earth, only the mass enclosed by the radius contributes to g, so g decreases linearly with depth.

Gravitational potential energy (U) is the energy possessed by an object due to its position in a gravitational field. It represents the work done to move the object from a reference point (usually Earth's surface) to its current position against gravity.

Derivation for height h:

Change at infinity:

• As h → ∞, U → 0 (since gravitational force diminishes to zero).
• The energy required to move the object to infinity is equal to the binding energy of the object-Earth system.

Free fall refers to the motion of an object under the sole influence of gravity, without any other forces (like air resistance) acting on it. During free fall, all objects experience the same acceleration, known as acceleration due to gravity (g), regardless of their masses.

To derive g using Newton's law of gravitation:

1. Newton's law states: F = G (M_em)/r², where:
- F = Gravitational force
- G = Universal gravitational constant
- M_e = Mass of Earth
- m = Mass of object
- r = Distance between centers (Earth's radius for surface)

2. From Newton's second law: F = ma, where a = g for free fall.
Thus, mg = G (M_em)/r².

3. Canceling m from both sides: g = G M_e/r².

Variation with altitude: As altitude increases, r increases, causing g to decrease (inverse square law).

Variation with depth: Inside Earth, only the mass enclosed by the radius contributes to g, so g decreases linearly with depth, reaching zero at Earth's center.

The universal law of gravitation states that every object in the universe attracts every other object with a force (F) that is:
1. Directly proportional to the product of their masses (m₁ and m₂).
2. Inversely proportional to the square of the distance (r) between their centers.

Mathematically: F = G (m₁m₂)/r², where G is the gravitational constant.

Significance:
- Explains planetary motion and tides.
- Binds the solar system and galaxies.
- Determines weight of objects on Earth.

Calculation:
Given: m₁ = 50 kg, m₂ = 100 kg, r = 2 m, G = 6.67 × 10^-11 Nm²/kg².

1. Substitute values into the formula:
F = (6.67 × 10^-11 × 50 × 100)/2².

2. Simplify numerator:
6.67 × 10^-11 × 5000 = 3.335 × 10^-7.

3. Divide by denominator:
3.335 × 10^-7/4 = 8.3375 × 10^-8 N.

Thus, the gravitational force is 8.34 × 10^-8 N (very small due to weak gravitational force between everyday objects).