Suppose at height h, the value of g reduces by 36 % i.e., it becomes 64 % of that at the surface. Then

\({ g }_{ h }=64\cdot /\cdot \quad of\quad g=\frac { 64 }{ 100 } g\) 
\(But\ { g }_{ h }=g{ \left( \frac { R }{ R+h } \right) }^{ 2 }\) 
\(\therefore \frac { 64 }{ 100 } g\quad =g{ \left( \frac { R }{ R+h } \right) }^{ 2 }\quad or\quad \frac { 8 }{ 10 } =\frac { R }{ R+h } \) 
\(or\ h=\frac { R }{ 4 } =\frac { 6400 }{ 4 } =1600\quad km.\) 

Two forces act on an object immersed in water. One is the gravitational force, which pulls the object downwards, and the other is the buoyant force, which pushes the object upwards. If the upward buoyant force is greater than the downward gravitational force, then the object comes up to the surface of the water as soon as it is released within water. Due to this reason, a block of plastic released under water comes up to the surface of the water.

Applications of Archimedes' principle

(i) Archimedes' principle is used in designing ships and submarines.

(ii) Lactometers based on Archimedes' principle are used to measure purity of a sample of milk.

(iii) Hydrometers used to measure density off liquids are based on Archimedes' principle.

Weight of an object on the moon. The weight of an object on the moon is the force with which it is attracted towards the centre of the moon.

The mass and radius of the moon is less than that of the earth. Due to this, the moon exerts lesser force of attraction on the object. Hence, the weight of an object on the moon is less than that on the earth. The gravitational force of the moon is about one-sixth of that on the earth.

A heavy coin and a piece of paper were placed inside it. The ends of the tube were closed. Air of the tube was removed by a vacuum pump. When the tube was quickly inverted, it was observed that both the coin and the paper hit the bottom at the same time. When the experiment was repeated with air inside the tube, it was observed that the piece of paper falls slowly while the coin hits the bottom immediately. This proves the Galileo assertion that in vacuum all bodies irrespective of their mass fall towards the earth with the same acceleration.

Hendry Cavendish first determined the value of G experimentally in the year 1779, by using a sensitive balance.
The presently accepted value of G=6.673x10^-11 Nm² kg^-2

(i) We tie our seat belts to remain intact on the seat, so that our body does not fall forward. 
(ii) When an object falls towards the earth under the gravitational force, then we say that the object in free fall.
(iii) Hemlata's friend showed the value of concerned, sympathetic responsible and caring friend
 

For the upward motion
v =u - gt₁ > 0 = u - gt_1, \({ t }_{ 1 }=\cfrac { u }{ g } \) and the downward motion,
v = u + gt₂, v = 0 + gt₂
The body falls back to the earth with the same speed as it was thrown vertically upwards.
\(\therefore\) v=u, u=0+gt2\(\Rightarrow \) \({ t }_{ 2 }=\cfrac { u }{ g } \)
From Eqs. (i) and (ii), we get
tl = t2 => Time of ascent = Time of descent

Let g = acceleration due to gravity at the earth surface
g_h = acceleration due to gravity at height h = 0.64g
\(\therefore\) \(g=\cfrac { GM }{ { R }_{ e }^{ 2 } } \)
 Similarly, \({ g }_{ h }\cfrac { g{ R }_{ e }^{ 2 } }{ \left( { R }_{ e }+h \right) ^{ 2 } } \)
From Eqs. (i) and (ii), we get​​​​​​​
\(\therefore\) \({ g }_{ h }=\cfrac { g{ R }_{ e }^{ 2 } }{ \left( { R }_{ e }+h \right) ^{ 2 } } \Rightarrow 0.64\quad g=\cfrac { g{ R }_{ e }^{ 2 } }{ \left( { R }_{ e }+h \right) ^{ 2 } } \)
\(\Rightarrow \)0.64 (R_e + h)² = R_e² \(\Rightarrow \) 0.8 (R_e + h) = R_e
\(\Rightarrow \) 0.8h = R_e - 0.8 R_e = 0.2 R_e
h=\(\cfrac { 2\times 6400 }{ 8 } =1600km\) [\(\therefore\) R, = 6400 km]
Thus, the height above the surface of the earth is 1600 km.​​​​​​​

No, it is not necessary that their weights in air should also be the same. This is because the two bodies have undergone the same loss in weight on completely immersing in water due to their equal volumes, not due to their equal weights. So, they may have different weights in air.

The average density of a fresh egg is more than of pure water but less than that of water in which salt is, dissolved. So a fresh egg sinks in pure water while it floats in salty water.

Archimedes' principle. This principle states that when a body is immersed fully or partially in a fluid, it experiences an upward thrust equal to the weight of the fluid displaced by it.
Applications of Archimedes' principle
(i) Archimedes' principle is used in designing ships and submarines.
(ii) Lactometers based on Archimedes' principle are used to measure purity of a sample of milk.
(iii) Hydrometers used to measure density off liquids are based on Archimedes' principle.

Difference between mass and weight:

Force of gravitation, F = \(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \)
(i) When mass of one body (m1 or m2) is doubled, the force gets doubled.
\(F'=G\frac { { (2m }_{ 1 }){ m }_{ 2 } }{ { r }^{ 2 } } =2G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } =2F\)
(ii) when the distance between the bodies is doubled, 
\(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ (2{ r }^{ 2 }) } =\frac { 1 }{ 4 } G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \frac { 1 }{ 4 } F\)i.e. the force becomes one-fourth of the original force.
(iii) When the masses of both bodies are doubled,
\(F'=G\frac { { (2m }_{ 1 }){ (2m }_{ 2 }) }{ { r }^{ 2 } } =4G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } =4F\)
i.e., the force becomes four times the original force.
(iii) When the distance between the two bodies is tripled,
\(F'=G\frac { { m }_{ 1 }{ m }_{ 2 } }{ (3{ r }^{ 2 }) } =\frac { 1 }{ 9 } G\frac { { m }_{ 1 }{ m }_{ 2 } }{ { r }^{ 2 } } \frac { 1 }{ 9 } F\)
i.e., the force becomes one-ninth of the originals force.