Gravitation

The chapter “Gravitation” introduces the universal force of attraction between objects – gravity. It explains Newton’s law of gravitation, free fall, weight, mass, buoyancy, and Archimedes’ principle.

🔷 Key Concepts Covered

Gravitation: Universal force
Newton’s Law of Gravitation
Acceleration due to gravity (g)
Mass and Weight
Free fall and equations of motion under gravity
Thrust and Pressure
Archimedes’ Principle and Buoyancy
Relative density

🔹 1. Gravitation – Universal Force

✅ What is Gravitation?

Gravitation is a force of attraction acting between any two objects in the universe.

Example: The Earth pulling objects toward itself; Moon revolving around Earth.

✅ Newton’s Universal Law of Gravitation

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

📘 Formula:

$\boxed{F = G \frac{m_1 m_2}{r^2}}$

$F$ = Gravitational force
$m_1, m_2$ = masses of two objects
$r$ = distance between their centers
$G$ = Universal Gravitational Constant
Value of $G = 6.674 \times 10^{-11} Nm^2/kg^2$

1. State the universal law of gravitation.

Universal Law of Gravitation:
Every object in the universe attracts every other object with a force which is:

directly proportional to the product of their masses
inversely proportional to the square of the distance between their centers

This force acts along the line joining the centers of the two objects.

2. Write the formula to find the magnitude of the gravitational force between the Earth and an object on the surface of the Earth.

The formula is:
$F = G \frac{M m}{R^2}$

Where:
– $F$ = gravitational force
– $G$ = universal gravitational constant = $6.674 \times 10^{-11} \, Nm^2/kg^2$
– $M$ = mass of the Earth
– $m$ = mass of the object
– $R$ = radius of the Earth

🔹 2. Free Fall and Acceleration Due to Gravity (g)

When only gravity acts on an object, it is in free fall.
The acceleration due to gravity is denoted by g.

📘 Value of g on Earth:

$\boxed{g = 9.8 \, m/s^2}$

1. What do you mean by free fall?

Free fall is the motion of an object when it is falling solely under the influence of gravity, with no other forces (like air resistance) acting on it. During free fall, the only force acting on the object is the gravitational force.

2. What do you mean by acceleration due to gravity?

Acceleration due to gravity is the rate at which the velocity of a freely falling object increases. It is denoted by $g$ and on Earth, its average value is approximately 9.8 m/s². It acts in the downward direction toward the center of the Earth.

✅ Equations of Motion under Gravity

Substitute $a$ with $g$ in equations of motion:

$v = u + gt$
$s = ut + \frac{1}{2}gt^2$
$v^2 = u^2 + 2gs$

For objects falling: $g = +9.8 \, m/s^2$
For objects thrown upward: $g = -9.8 \, m/s^2$

🔹 3. Mass and Weight

Quantity	Mass	Weight
Definition	Quantity of matter in a body	Force with which Earth attracts the body
Symbol	$m$	$W$
SI Unit	kg	Newton (N)
Formula	—	$W = mg$
Nature	Scalar	Vector

Weight changes with location, mass remains constant.

1. What are the differences between the mass of an object and its weight?

Mass	Weight
Mass is the amount of matter in a body.	Weight is the force with which the Earth attracts a body.
Mass is constant everywhere.	Weight varies with gravity.
SI unit: Kilogram (kg)	SI unit: Newton (N)
It is a scalar quantity.	It is a vector quantity.

2. Why is the weight of an object on the moon ¹⁄₆ of its weight on the Earth?

The weight of an object depends on the acceleration due to gravity.
The gravity on the Moon is about one-sixth that on Earth, i.e.

$g_{\text{moon}} = \frac{1}{6} g_{\text{earth}}$

Therefore, the weight of an object on the Moon is:

$W_{\text{moon}} = mg_{\text{moon}} = m \times \frac{1}{6}g = \frac{1}{6} W_{\text{earth}}$

Hence, the weight of an object on the Moon is one-sixth of its weight on the Earth.

🔹 4. Thrust and Pressure

✅ Thrust:

Force acting perpendicular to a surface.

✅ Pressure:

Thrust per unit area.

📘 Formula:

$\boxed{\text{Pressure} = \frac{\text{Thrust}}{\text{Area}}} \quad \text{(Unit: Pascal or N/m²)}$

1. Why is it difficult to hold a school bag having a strap made of a thin and strong string?

A thin strap has a smaller area in contact with the shoulder. Since pressure is inversely proportional to area ( $P = \frac{F}{A}$ ), the smaller area causes greater pressure on the shoulder, making it painful and difficult to carry. A broader strap distributes the pressure, making it easier to hold.

2. What do you mean by buoyancy?

Buoyancy is the upward force exerted by a fluid (like water or air) on an object placed in it. This force acts in the opposite direction of gravity and makes objects feel lighter in a fluid.

3. Why does an object float or sink when placed on the surface of water?

An object floats if the buoyant force acting on it is greater than or equal to its weight. It sinks if its weight is greater than the buoyant force.

If density of object < density of water → it floats
If density of object > density of water → it sinks

🔹 5. Pressure in Fluids and Buoyancy

Fluids (liquids and gases) exert pressure in all directions.
Buoyancy: Upward force exerted by a fluid on a body submerged in it.
An object floats if buoyant force ≥ weight; sinks if weight > buoyant force.

🔹 6. Archimedes’ Principle

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

✅ Applications:

Design of ships and submarines
Measuring volume/density of irregular objects
Hydrometers

1. You find your mass to be 42 kg on a weighing machine. Is your mass more or less than 42 kg?

The weighing machine actually measures the $\bf apparent\ weight$ , which includes the effect of buoyant force due to air.

Since the air exerts a small upward force (buoyancy), your $\bf actual\ mass\ is\ slightly\ more\ than\ 42\ kg$ . However, the difference is very small.

2. You have a bag of cotton and an iron bar, each indicating a mass of 100 kg on a weighing machine. In reality, one is heavier than the other. Can you say which one is heavier and why?

Both show 100 kg on the scale, but $\bf cotton\ displaces\ more\ air$ because it has a larger volume.

The $\bf buoyant\ force$ acting on cotton is more than on iron. As a result, the actual mass of cotton is slightly more than that of iron (because the weighing machine shows a lesser reading for cotton due to more upward force by air).

🔹 7. Relative Density

$\text{Relative Density} = \frac{\text{Density of substance}}{\text{Density of water}}$

It has no unit because it’s a ratio.

📊 Formula Summary Table

Concept	Formula	Unit
Newton’s Gravitation	$F = G \frac{m_1 m_2}{r^2}$	Newton (N)
Acceleration due to gravity	$g = \frac{F}{m}$	m/s²
Weight	$W = mg$	Newton (N)
Pressure	$P = \frac{F}{A}$	Pascal (Pa)
Buoyant Force	$=$ weight of displaced fluid	Newton (N)
Relative Density	$= \frac{\text{Density of body}}{\text{Density of water}}$	—

🧠 Key Takeaways

Gravitation is a universal force acting between any two masses.
Weight is a force, hence it varies with gravity; mass does not.
Free fall means motion under gravity alone.
Buoyancy and Archimedes’ Principle explain floating and sinking.

Exercise

1. How does the force of gravitation between two objects change when the distance between them is reduced to half?

According to Newton’s law of gravitation:

$F \propto \frac{1}{r^2}$

If the distance is reduced to half,

$F' = \frac{1}{(r/2)^2} = \frac{4}{r^2}$

So, the force becomes **4 times** greater.

2. Gravitational force acts on all objects in proportion to their masses. Why then, a heavy object does not fall faster than a light object?

Though gravitational force is more on heavier objects, their greater mass also makes them harder to accelerate. Both effects cancel out, so all objects fall with the same acceleration due to gravity (g), neglecting air resistance.

3. What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface?

$F = G \frac{M m}{R^2} = \frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 1}{(6.4 \times 10^6)^2} \approx 9.8 \, N$

So, the gravitational force is approximately **9.8 N**.

4. The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Why?

The gravitational forces are **equal in magnitude** and **opposite in direction**, as per Newton’s Third Law of Motion. Each exerts the same force on the other.

5. If the moon attracts the earth, why does the earth not move towards the moon?

The Earth does move slightly, but due to its large mass, the acceleration is extremely small and not noticeable.

6. What happens to the force between two objects if:

(i) Mass of one object is doubled:
Force is directly proportional to mass. → Force is doubled.

(ii) Distance is doubled:

$F \propto \frac{1}{r^2} \Rightarrow F \text{ becomes } \frac{1}{4}$

(iii) Both masses are doubled:
Force becomes 4 times.

7. What is the importance of the universal law of gravitation?

The universal law of gravitation helps to:

Explain planetary motion
Understand tides due to the moon
Describe motion of satellites
Understand the fall of objects on Earth

8. What is the acceleration of free fall?

The acceleration experienced by a body when falling freely under the influence of gravity alone is called acceleration due to gravity. Its average value on Earth is **9.8 m/s²**.

9. What do we call the gravitational force between the earth and an object?

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

10. Amit buys few grams of gold at the poles and gives it to a friend at the equator. Will the friend agree with the weight? Why?

No, the friend will find the gold slightly lighter. The value of acceleration due to gravity **g is lower at the equator** than at the poles. Hence, weight measured at the equator will be less.

11. Why will a sheet of paper fall slower than one that is crumpled into a ball?

A sheet has a larger surface area and faces more **air resistance**. The crumpled paper has less surface area, less air drag, and thus falls faster.

12. What is the weight in newtons of a 10 kg object on the moon and on the earth?

On Earth:

$W = mg = 10 \times 9.8 = 98 \, N$

On Moon (1/6th gravity):

$W = 10 \times \frac{9.8}{6} \approx 16.3 \, N$

Answer: Earth = 98 N, Moon ≈ 16.3 N

13. A ball is thrown vertically upwards with a velocity of 49 m/s. Calculate: (i) the maximum height it rises to, (ii) the total time it takes to return to the surface.

Given: u = 49 m/s, g = 9.8 m/s²

(i) $h = \frac{u^2}{2g} = \frac{49^2}{2 \times 9.8} = \frac{2401}{19.6} = 122.5 \, \text{m}$
(ii) Time to reach max height: $t = \frac{u}{g} = \frac{49}{9.8} = 5 \, s$
Total time = 2 × 5 = **10 seconds**

14. A stone is released from the top of a tower of height 19.6 m. Calculate its final velocity just before touching the ground.

Given: h = 19.6 m, u = 0, g = 9.8 m/s²

$v^2 = u^2 + 2gh = 0 + 2 \times 9.8 \times 19.6 = 384.16 \Rightarrow v = \sqrt{384.16} = 19.6 \, \text{m/s}$

Answer: 19.6 m/s

15. A stone is thrown vertically upward at 40 m/s. Find: (i) Maximum height (ii) Net displacement and total distance

(i)

$h = \frac{u^2}{2g} = \frac{40^2}{2 \times 10} = \frac{1600}{20} = 80 \, \text{m}$

(ii) Net displacement = 0 (it returns to starting point), Total distance = 2 × 80 = **160 m**

16. Calculate the gravitational force between the earth and the sun.

$F = G \frac{M m}{r^2} = \frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 2 \times 10^{30}}{(1.5 \times 10^{11})^2} \approx 3.56 \times 10^{22} \, \text{N}$

17. A stone is dropped from a tower 100 m high. Another stone is projected upward at 25 m/s. When and where will they meet?

Let time to meet = t seconds.
Downward stone: $s_1 = \frac{1}{2}gt^2$
Upward stone: $s_2 = 25t - \frac{1}{2}gt^2$
Total distance = 100 m = $s_1 + s_2$

$100 = \frac{1}{2} \cdot 10 \cdot t^2 + 25t - \frac{1}{2} \cdot 10 \cdot t^2 \Rightarrow 100 = 25t \Rightarrow t = 4 \, \text{seconds}$

$\text{Height from top} = \frac{1}{2} \cdot 10 \cdot 4^2 = 80 \, \text{m}$

So, they meet **20 m above ground**.

18. A ball returns to the thrower after 6 s. Find: (a) the velocity with which it was thrown up (b) the max height (c) its position after 4 s

Time to reach top = 3 s (half of 6 s), g = 10 m/s²
(a) $u = gt = 10 \times 3 = 30 \, \text{m/s}$
(b) $h = \frac{u^2}{2g} = \frac{900}{20} = 45 \, \text{m}$
(c) In 4 seconds, it’s falling down:
Displacement in next 1 s = $\frac{1}{2}gt^2 = 5 \times 1 = 5 \, \text{m below top}$
Position = 45 – 5 = **40 m above ground**

19. In what direction does the buoyant force on an object immersed in a liquid act?

Buoyant force always acts in the **upward direction**, opposite to gravity.

20. Why does a block of plastic released under water come up to the surface?

The density of plastic is less than water, so the **buoyant force** is greater than its weight. Hence, it experiences a net upward force and floats to the surface.

21. The volume of 50 g of a substance is 20 cm³. Will it float in water?

Density = mass/volume = 50 / 20 = 2.5 g/cm³
Since 2.5 > 1 (density of water), the substance is **denser** and will **sink**.

22. A sealed packet of 500 g has volume 350 cm³. Will it float in water? What is the mass of displaced water?

Density of packet = 500 / 350 ≈ 1.43 g/cm³ > 1 → It will **sink**.
Mass of displaced water = Volume × density = 350 × 1 = **350 g**

📘 Chapter Overview

🔷 Key Concepts Covered