\(\frac{GMm}{R^2}=mR\omega^2\ or\ \frac{G\frac{4}{3}\pi R^3\rho m}{R^2}=mR\frac{4\pi^2}{T^2}\ or\ \rho T^2=\frac{3\pi}{G}=a\ constant.\)

Percentage increase in speed \(=\frac{v_e-v_0}{v_0}\times100=(\frac{v_e}{v_0}-1)\times100\)
\(=(\sqrt 2-1)\times100=41.4\%\)

About 100 minutes.

No, artificial satellite is like a freely falling body and the.weightof. the body inside the satellite is zero.

\(\frac{v_1}{v_2}=\frac{T_1}{T_2}=1\) because both satellites are revolving in same orbit and for a given orbit the orbital velocity, as well as time period, is independent of the mass of satellite.

They are used for (i) ground water survey, (ii) detecting the areas under forest.

9.789\({ m }/s^{ 2 }\)

9.77\({ m }/s^{ 2 }\)

The change in gravitational potential energy of a body between two given points depends only upon the position of the given points and is independent of the path followed.is due to the fact that the gravitational force is a conservative force and work done by a conservative force depends only on the position of initial and final poits and is independent of path followed.

\({ v }_{ p }\) (escape speed on a planet) =\(\sqrt { \frac { { GM }_{ p } }{ { R }_{ p } } } \)
\({ v }_{ e }\) (escape speed on the earth)  =\(\sqrt { \frac { { GM }_{ e } }{ { R }_{ e } } } \)
Clearly ,   \(\frac { { v }_{ p } }{ { v }_{ e } } =\sqrt { \frac { { M }_{ p } }{ M_{ e } } \times \frac { { R }_{ e } }{ { R }_{ p } } } =\sqrt { 8\times \frac { 1 }{ 2 } } =2\)
\( { v }_{ p }={ 2v }_{ e }=22.4km/s\)       

Any change in the distribution of the earth's mass will not affect the variation of acceleration due to gravity with height. This is because for a point outside the earth, the whole mass of the earth is effective and the earth behaves as a homogenous sphere.

The safety of a person depends upon the momentum with which the person hits the planet. Since, the mass of the person is constant, therefore the maximum velocity v is the limiting factor.

6.67 x 10^-7 N

​​Putting, T₁ = 365 days, R₁=R, R₂ =R/2
We get
(365/T₂)² = (R/(R/2))³
(365/T₂)² = 2³ = 8
​​​​​T₂³ = 365² / 8
​​​​​​T₂=129.04
Therefore, Number of days in one year will be about 129 days.

From the law of conservation of angular momentum
\(m{ r }_{ 1 }{ v }_{ 1 }=m{ r }_{ 2 }{ v }_{ 2 }\quad \Rightarrow \quad { r }_{ 1 }{ v }_{ 1 }={ r }_{ 2 }{ v }_{ 2 }\ \ or\quad \frac { { v }_{ 1 } }{ { v }_{ 2 } } =\frac { { r }_{ 2 } }{ { r }_{ 1 } } \)