For students of class 11, to perform well for their class 11 exams as well as to prepare for upcoming competitive exams, they need to work on the NCERT Solutions for Class 11 Physics Chapter 8 Gravitation is a very popular subject for class 11 students as every higher studies consists of more advanced topics on this phenomenon. We should be knowing the difference between gravitation and gravity beforehand in order to understand more complex subjects mentioned above. This chapter consists of questions belonging to various important concepts such as how can a body be shielded from gravitational influences of any nearby matter. We can find out here, whose gravitational force if greater on earth, the sun’s or moon’s?.

We can come across various complex but easy to understand topics in grativation such as acceleration due to gravity, finding potential energy difference between two points which are at a certain distances from the earth’s center. We can even know that does the acceleration due to gravity increases/decreases when the depth increases or altitude decreases. We will be finding questions on interplanetary motion such as a planet is revolving the sun with a speed double than that of earth, this question will show you the use of kepler’s laws of motion. We will be seeing questions on jupiter’s satellites radius of revolutions.

This chapter talks about the distance and speed of light among stars which are far away in a different galaxy and the time taken for the stars to complete one revolution. We will see here questions related to a space shuttles escape velocity and it dependence on factors like an object’s mass, location and direction of projection and the gravitational influences on it. We will be finding the angular speed, kinetic energy, total energy, potential energy, linear speed and angular momentum of an asteroid revolving around a star.

We will be going through the medical problems faced by astronauts in space and the reason behind it. We will be seeing problems on gravitational intensity within a hemisphere. We will be knowing the mass of the sun in one of the questions asked below. Do you know the distance between Uranus and earth and the time taken to reach there ?. You will get to know it below.

We will see here the maximum height attained by a missile when it is fired vertically upwards before falling down on the earth’s surface. We will see the international space station’s motion around the earth and the influence of the earth’s gravitational field upon it. Do you know what will happen when two planets collide with each other, what will be their speed?, we will see it here. How a star turns into a black hole and the energy required to launch a space station in mars are the type of questions you will find below.

**Q.1: (a). By putting an electric charge inside a hollow conductor you can shield it from the influences of electrical forces. Can a body be shielded from the gravitational influences of nearby matter in a similar fashion? **

**(b). An astronaut in a shuttle orbiting Mars cannot feel gravity. If he was traveling in a very large space ship would he have detected gravity?**

**(c). The Sunâ€™s gravitational force on Earth is greater than the moonâ€™s. However, the Moon affects the tidal waves much more than the Sun. Why?Â Â Â **

**Sol:**

**(a).** **No**, as of now, no method has been devised to shield a body from gravity because gravity is independent of medium and it is the virtue of each and every matter. So the shield would exert the gravitational forces.

**(b).** **Yes**, if the spaceship is large enough then the astronaut will definitely detect the Mars gravity.

**(c).** **Gravitational force is inversely proportional to the square of the distance** whereas, **Tidal effects** are **inversely** **proportional** to the **cube** of the **distance**. So as the distance between the earth and moon is smaller than the distance between earth and sun, the moon will have a greater influence on the earthâ€™s **tidal waves**.

**Q2.Â Pick the right choice:
(a). Acceleration due to gravity does not depend upon the mass of the [body/earth]**

**(b).Â The formula,Â mg (r _{2 }â€“Â r_{1})Â is [more/less] accurate than the formula \(\frac{GMm}{\left ( \frac{1}{r_{2}}-\frac{1}{r_{2}} \right )}\) for the difference of potential energy between two pointsÂ at a distance of r_{1 }and r_{2 }Â from the earthâ€™s center.**

**(c)Â Acceleration due to gravity [decreases/increases] with increasing depth. (Consider earth has a uniform density). **

**(d) Acceleration due to gravity [decreases/increases] with decreasing altitude**

**Sol:**

**(a). body.**

**(b). more.**

**(c). decreases.**

**(d). increases.**

**Q.3: Imagine a planet that revolves around the sun at a double speed of the earth. Find the size of its orbit as compared to the orbit of the earth.**

**Sol:**

**Time taken** by the earth for one complete **revolution, ****T**_{E }**= 1 Year**

**Radius of Earthâ€™s orbit, ****R**_{E }**= 1 AU **

Thus, the time taken by the planet to complete one complete revolution:

T_{P }= \(\frac{1}{2}\)**T**_{E }**Â = \(\frac{1}{2}\) year**

**Let, the orbital radius of this planet = ****R**_{P}

Now, according to the **Keplerâ€™s third law of planetary motion**:

\(\left ( \frac{R_{P}}{R_{E}} \right )^{3}=\left ( \frac{T_{P}}{T_{E}} \right )^{2}\\\)

\(\frac{R_{P}}{R_{E}}=\left ( \frac{T_{P}}{T_{E}} \right )^{\frac{2}{3}}\\\)

\(\frac{R_{P}}{R_{E}}=\left ( \frac{T_{P}}{T_{E}} \right )^{\frac{2}{3}}=\left ( \frac{\frac{1}{2}\;T_{E}}{T_{E}} \right )^{\frac{2}{3}}=\left ( \frac{1}{2} \right )^{\frac{2}{3}}=0.63\)

**Therefore, radius of orbit of this planet is 0.63 times smaller than the radius of orbit of the Earth.**

**Â **

**Q.4: One of Jupiterâ€™s satellites, Io has an orbital radius of 4.22 x 10 ^{8} m and its revolution time around Jupiter is 1.769 days. Prove that the mass of the Sun is a thousand times that of Jupiter.**

**Â **

**Sol:**

**Given,**

**Orbital period ofÂ Io,Â T _{I0}Â =** 1.769 daysÂ

**=Â 1.769Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 s**

**Orbital radius ofÂ Io,Â R _{I0}Â =**

**4.22Â Ã—Â 10**

^{8}Â mWe know mass of Jupiter:

**M**_{J}**Â = 4Ï€ ^{2}R_{I0}^{3}Â / GT_{I0}^{2}Â Â Â . . . . . . . . . . . . . . . (1)**

Where;

M_{J}Â = Mass of Jupiter

G = Universal gravitational constant

Also,

Orbital period of the earth,

**T _{EÂ }= 365.25 days = 365.25Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 s**

**Orbital radius of the Earth, R _{EÂ }= 1 AU = 1.496Â Ã—Â 10^{11Â }m**

We know that the mass of sun is:

**M**_{S}**Â =** \(\frac{4\;\pi ^{2}\;R_{E}^{3}}{G\;T_{E}^{2}}\)**. . . . . . . . . . . . .Â (2)**

Therefore, Â \(\\\frac{M_{S}}{M_{J}}\) = \(\frac{4\;\pi ^{2}\;R_{E}^{3}}{G\;T_{E}^{2}}\;\times \;\frac{G\;T_{10}^{2}}{4\;\pi ^{2}\;R_{10}^{3}}\\\) = \(\\\frac{R_{E}^{3}}{T_{E}^{2}}\;\times \;\frac{T_{10}^{2}}{R_{10}^{3}}\)

**Now, on substituting the values, we will get:**

= \(\left [ \frac{1.769\times 24\times 60\times 60}{365.25\times 24\times 60\times 60} \right ]^{2}\times \left [ \frac{1.496\times 10^{11}}{4.22\times 10^{8}} \right ]^{3}\) = 1045.04

Therefore, \(\frac{M_{S}}{M_{J}}\) ~ 1000

**M**_{SÂ }**~ 1000Â Ã—Â M _{JÂ Â Â Â Â }**

**[Which proves that, the Sunâ€™s mass is 1000 times that of Jupiterâ€™s)**

**Q.5: Consider that the Sombrero galaxy has 3 x ****10 ^{11}Â stars, each of one solar mass. If the diameter of the galaxy is 10^{5} light years, calculate the time of a star, at a distance of 5000 light years from the galactic center, takes to complete one revolution.**

**Â **

**Sol:**

**Mass **of Sombrero Galaxy,Â MÂ **= 3 Ã— 10 ^{11}Â solar mass**

**Solar mass**= Mass of Sun =

**2.0 Ã— 10**

^{36}Â kg**Mass**of the galaxy,Â

**MÂ =**3 Ã— 10

^{11}Â Ã— 2 Ã— 10

^{36}Â

**= 6 Ã— 10**

^{41}Â kg**Diameter**of Sombrero Galaxy,Â

**dÂ = 5 x 10**

^{4}Â ly**Radius**of Sombrero Galaxy,Â

**rÂ = 2.5 Ã— 10**

^{4}Â lyWe know that:

1 light year = 9.46 Ã— 10^{15}Â m

Therefore, rÂ = 2.5 Ã— 10^{4}Â Ã— 9.46 Ã— 10^{15} **= 2.365 x 10 ^{20} m**

As this star revolves around the massive black hole in center of the Sombrero galaxy, its time period can be found with the relation:

TÂ = \(\left [ \frac{4\;\pi ^{2}\;r^{3}}{GM} \right ]^{\frac{1}{2}}\\\)

=Â \(\\\left [ \frac{4\times 3.14^{2}\times (2.365\times 10^{20})^{3}}{(6.67\times 10^{-11})\times (6\times 10^{41})} \right ]^{\frac{1}{2}}\\\) **= Â 4.246 x 10 ^{15 }Â s**

Now we know, 1 year = 365Â Ã—Â 24Â Ã—Â 60Â Ã—Â 60 s

1s = \(\frac{1}{365\times 24\times 60\times 60}\\\)Â years

Therefore, 4.246 Ã—Â 10^{15}s = \(\frac{4.246\times 10^{15}}{365\times 24\times 60\times 60}\) =Â 1.34 Ã—Â 10^{8}Â years.

**Q.6: Pick the right alternative:**

**(a).** The energy spent on sending a rocket vertically upwards out of the earthâ€™s gravitational influence is** more /Â less** than the energy spent on sending an orbiting satellite to the same height out of the earthâ€™s influence.

**(b).** If potential energy is zero at infinity, the total energy of an orbiting comet is negative of its **potential / kinetic** energy.

**Sol:**

**(a). Â more**

**(b). Â kinetic.**

**Â **

**Â **

**Q7. Choose the factors upon, which the escape velocity of an object from earth depends upon:**

**(a) bodyâ€™s mass.**

**(b) location of projection.**

**(c) direction of projection. **

**Â **

**Sol: (b)**

**Explanation:**

Escape velocity is independent of the direction of projection and the mass of the body. It depends upon the gravitational potential at the place from where the body is projected. **Gravitational potential depends slightly on the altitude and the latitude **of the place, thus **escape velocity depends slightly upon the location from where it is projected.**

**Â **

**Â **

**Q.8****: An asteroid orbits a star in an elliptical orbit.Â Does the asteroid have a constant (a) angular speed, (b) kinetic energy, (c) total energy (d), linear speed, (e) potential energy and (f) angular momentum throughout its revolution? **

**Â **

**Sol:**

An asteroid orbiting a star will have constant angular momentum and the constant value of total energy throughout its orbit.

**Â **

**Â **

**Q9. Which of the following problems is likely to affect an astronaut in space? (a) Swollen feet, (b) bone loss, (c) orientational problem, (d) headache.**

**Â **

**Sol:**

**(a).** In zero gravity the blood flow to the feet isnâ€™t increased so the astronaut does not get **swollen feet.**

**Â (b).** Due to zero gravity, the weight the bones have to bear is greatly reduced, this causes **bone loss** in astronauts spending greater amounts of time in space.

**Â (c).** Space has different orientations, so **orientational problems** can affect an astronaut.

**(d).** Due to increased blood supply to their faces, astronauts can be affected by **headaches****.**

**Q.10: Pick up the right answers from the given ones:**

**The direction of gravitational intensity inside a hemispherical sphere of a uniform mass is indicated by the arrow: **

**(i) c **

**(ii) x**

**(iii) z **

**(iv) o**

**Â **

**Sol: (i) c**

**Reason:**

**Inside a hollow sphere**, gravitational forces on any particle at any point is symmetrically placed. However, in this case, the upper half of the sphere is removed. Since gravitational intensity is gravitational force per unit mass it will act in direction point **downwards along â€˜câ€™.**

**Q.11: For the above problem the gravitational intensity at P will be directed towards: **

**(i) h**

**(ii) f**

**(iii) m**

**(iv) v**

**Sol: (iii) m**

**Reason:** Making use of the logic/explanation from the above answer we can conclude that the **gravitational intensity at P is directed downwards along m.**

**Â **

**Â **

**Q12. A rocket is launched off from the earthâ€™s surface for a massive comet nearing the earth. At what distance from the earthâ€™s center will the shuttle of the rocket be free from the earthâ€™s gravitational force. Mass of the comet = 7.4 x 10 ^{22}, mass of Earth = 6 x 10^{24} and distance between earth and the comet= 3.84 x 10^{10} m.**

**Sol:**

Given:

**Mass of the comet,Â M _{C}Â = 7.4 Ã— 10^{24}Â kg **

**Mass of the Earth,Â M _{E}Â = 6 Ã— 10Â ^{24}Â kg**

Orbital radius,Â rÂ = 3.84 Ã— 10^{10}Â m

Mass of the shuttle =Â m kg

LetÂ **â€˜xâ€™**Â be the **distance** from the center of the **Earth** where the **gravitational** **force** acting on the Shuttle **â€˜Sâ€™** becomes **zero**.

According to **Newtonâ€™s law of gravitation**, we have:

\(\\\Rightarrow \frac{G\;m\;M_{A}}{(r-x)^{2}}=\frac{G\;m\;M_{E}}{x^{2}}\\\)

\(\\\Rightarrow \)Â Â \(\frac{M_{A}}{M_{E}}=\left ( \frac{r-x}{x} \right )^{2}\)

\(\Rightarrow\)Â Â \(\frac{r}{x}-1=\left ( \frac{7.4\times 10^{24}}{6\times 10^{24}} \right )^{\frac{1}{2}}\\\)

\(\Rightarrow\) Â \(\frac{r}{x}-1=1.1106\)

\(\\\Rightarrow\)Â Â \(x=\frac{r}{2.1106}=\frac{3.84\times 10^{10}}{2.1106}=1.819\times 10^{10}\;m\)

**Q.13: If the earthâ€™s orbit around the sun is 1.5x 10 ^{8} km long, estimate the mass of the Sun.**

**Sol:**

**Given:**

** Earthâ€™s orbit,Â rÂ = 1.5 Ã— 10 ^{11}Â m**

**Time taken**by the Earth for one complete revolution,

**T**Â = 1 year =

**365.25 days**

i.e.

**T = (365.25 Ã— 24 Ã— 60 Ã— 60) seconds**

Since, Universal gravitational constant, G = 6.67 Ã— 10^{â€“11}Â Nm^{2}Â kg^{â€“2}

Therefore, **mass of the Sun, MÂ =** \(\frac{4\;\pi ^{2}r^{3}}{G\;T^{2}}\\\)

\(\\\Rightarrow M =\frac{4\times (3.14)^{2}\times (1.5\times 10^{11})^{3}}{(6.67\times 10^{-11})\times (365.25\times 24\times 60\times 60)^{2}}\\\)

\(\\\Rightarrow M =\frac{4\times (3.14)^{2}\times (1.5\times 10^{11})^{3}}{(6.67\times 10^{-11})\times (365.25\times 24\times 60\times 60)^{2}}\\\)

\(\\\Rightarrow\)Â Â \(M=\frac{1.331\times 10^{35}}{66.425\times 10^{3}}=2.004\times 10^{30}\;kg\\\)

**Therefore, the estimated mass of the Sun is 2.004 ****Ã—**** 10 ^{30} Kg**

**Q14. Uranus is 84 times the earth year. If the Earth is 15 x 10 ^{7} km away from the Sun, what is the distance between the Sun and Uranus?**

**Sol:**

Given:

**Distance** between Earth and the Sun,Â **r _{e}Â = **15 Ã— 10

^{7}Â km

**= 1.5 Ã— 10**

^{11}Â m**Time**

**period**of the Earth

**=Â TÂ**

_{e}**Time period**of Uranus,Â

**T**

_{u}Â = 84Â T_{e}Let, the **distance** **between** the **Sun** and the **Uranus** be **r _{uÂ }**

Now, according to the **Keplerâ€™s third law of planetary motion**:

TÂ =\(\left ( \frac{4\pi ^{2}r^{3}}{GM} \right )^{\frac{1}{2}}\)

For **Uranus** and **Sun**, we can write:

\(\Rightarrow \frac{r_{u}^{3}}{r_{e}^{3}}=\frac{T_{u}^{2}}{T_{e}^{2}}\\\\\\ \Rightarrow r_{u}=r_{e}\left [ \frac{T_{u}}{T_{e}} \right ]^{\frac{2}{3}}\\\\\\ \Rightarrow r_{u}=1.5\times 10^{11}\left [ \frac{84\;T_{e}}{T_{e}} \right ]^{\frac{2}{3}}=\boldsymbol{1.5\times 10^{11}\times (84)^{\frac{2}{3}}}\)

\(\Rightarrow r_{u}=28.77\times 10^{12}\\\)m

**Therefore, Uranus isÂ 28.77Â Ã—Â 10 ^{12}Â m away from the Sun.**

**Â **

**Q.15: A man weighing 60 N on the earthâ€™s surface is taken to the height that is equal to half of the earthâ€™s radius. What is the amount of gravitational force which the earth will exert on this man now? **

**Sol:**

Given:

**Weight of the man,Â WÂ = 60 N**

We know that acceleration due to **gravity** at **heightÂ â€˜hâ€™**Â from the **Earthâ€™s** **surface** is:

**g****‘ =Â \(\frac{g}{\left [ 1+\left ( \frac{h}{R_{e}} \right ) \right ]^{2}}\)**

Where, **gÂ = Acceleration due to gravity on the Earthâ€™s surface**

And, **R**_{e} = Radius of the Earth

ForÂ hÂ =Â \(\frac{R_{e}}{2}\\\)

g’ =Â \(\\\frac{g}{\left [ 1+\left ( \frac{R_{e}}{2R_{e}} \right ) \right ]^{2}}\\\)

\(\\\Rightarrow\) **g****‘ =Â \(\frac{g}{\left [ 1+\left (\frac{1}{2} \right ) \right ]^{2}}=\frac{4}{9}\;g\)**

Also , the **weight of a body** of mass **â€˜mâ€™ kg** at a height of **â€˜hâ€™ meters** can be represented as :

**W’ =Â mg**

=Â mÂ Ã—Â \(\frac{4}{9}\)gÂ =Â \(\frac{4}{9}\\\)mg

= \(\\\frac{4}{9}\\\)W

= \(\\\frac{4}{9}\)Â Ã—Â 60Â **=Â 26.66 N.**

**Â **

**Q.16:Â Considering the earth to be a sphere of uniform mass density. Find the weight of a body at one-third of the way down to the center of the earth if it weighed 200 N on the surface?**

**Sol:**

Given,

The weight of a body having **massÂ â€˜mâ€™**Â at the surface of earth,Â **WÂ =**Â mgÂ = **200 N**

Body of **mass**Â **â€˜****mâ€™**Â is located at **depth**,Â **dÂ = \(\frac{1}{3}\) R _{e}**

Where,Â

**R**

_{e}Â = Radius of the EarthÂNow, acceleration due to

**gravity gâ€™**at

**depthÂ (d)**is given by the relation:

**g’ = \(1-\left ( \frac{d}{R_{e}} \right )g\\\)**

\(\Rightarrow g’ = 1-\left ( \frac{R_{e}}{3R_{e}} \right )g=\frac{2}{3}\;g\)

**Now, weight of the body at depthÂ d,**

**W’ =Â mg’**

W’ =Â mÂ Ã—Â \(\frac{2}{3}\)gÂ = \(\frac{2}{3}\) mg

W’ = \(\frac{2}{3}\) W

\(\Rightarrow\) W =Â \(\frac{2}{3}\) Ã—Â 200Â **=Â 133.33 N**

**Therefore, the weight of a body at one-third of the way down to the center of the earth = ****133.33 N**

**Â **

**Q.17: A missile is fired vertically upwards from the surface at 4 km m/s. What is the greatest height the missile can attain before falling back to the earth? Also find the final distance of the missile from the center of the earth. [Mass of earth = 6 x 10 ^{24 }kg, mean radius of earth = 6.4 x 10^{6}m and take G =6.67 x 10^{-11} N m^{2} kg^{-2}] **

**Sol:**

Given:

**Velocity** of the **missile**, Â **vÂ =** 4 km/s **= 4 Ã— 10 ^{3}Â m/s**

**Mass of the Earth,**Â Â Â Â

**M**

_{E}Â = 6Â Ã—Â 10^{24}Â kg**Radius**of the

**Earth**,Â Â Â

**R**

_{E}Â = 6.4Â Ã—Â 10^{6}Â mLet, the **height** reached by the **missile **be **â€˜hâ€™** and the mass of the **missile Â beÂ **‘m**â€™.**

Now, at the **surface of the Earth**:

**Total energy** of the rocket at the surface of the Earth = Kinetic energy + Potential energy

**T _{E1 }= \(\\\frac{1}{2}mv^{2}+\frac{-GM_{E}\;m}{R_{E}}\)**

Now, at **highest** **point**Â **â€˜****hâ€™:**

Kinetic Energy = 0Â Â [Since, vÂ = 0]

And, Potential energy = \(\\\frac{-GM_{E}\;m}{R_{E}+h}\)

**Therefore, total energy of the ****missile ****at highest pointÂ â€˜hâ€™:**

T_{E2} = 0 + \(\frac{-GM_{E}\;m}{R_{E}+h}\)

\(\\\Rightarrow\) **T _{E2} = \(\frac{-GM_{E}\;m}{R_{E}+h}\)**

According to the **law of conservation of energy**, we have :

**Total** **energy** of the rocket at the **Earthâ€™s** **surface** **T _{E1}** =

**Total**

**energy**at

**height**Â

**â€˜**

**hâ€™**

**T**

_{E2}**:**

\(\\\Rightarrow \frac{1}{2}mv^{2}+\frac{-GM_{E}\;m}{R_{E}}=\frac{-GM_{E}\;m}{R_{E}+h}\\\\\\ \Rightarrow \frac{1}{2}v^{2}+\frac{-GM_{E}}{R_{E}}=\frac{-GM_{E}}{R_{E}+h}\\\\\\ \Rightarrow v^{2} = 2GM_{E}\times \left [ \frac{1}{R_{E}}-\frac{1}{(R_{E}+h)} \right ]=2GM_{E}\left [ \frac{h}{(R_{E}+h)R_{E}} \right ]\\\)

\(\\\Rightarrow v^{2}=\frac{g\;R_{E}\;h}{R_{E}+h}\\\)

Where,Â \(\\g=\frac{GM}{R_{E}^{2}}\\\)Â

**= 9.8 ms**

^{-2}Therefore, v

^{2}Â (R

_{E}Â +Â h) = 2gR

_{E}h

\(\\\Rightarrow\) v

^{2}R

_{E}Â =Â h (2gR

_{E}Â –Â v

^{2})

\(\\\Rightarrow\) \(h=\frac{R_{E}\;v^{2}}{2gR_{E}-v^{2}}=\frac{(6.4\times 10^{6})\times (4\times 10^{3})^{2}}{2\times 9.8\times 6.4\times 10^{6}-(4\times 10^{3})^{2}}\\\)

**\(\\\Rightarrow\) hÂ = 9.356Â Ã—Â 10 ^{5 }Â m**

Therefore, the

**missile**reaches at height of 9.36Â Ã—Â 10

^{5 }m from the surface.

Now, the **distance** from the **center** of the **earth** **= h + R _{E }**= 9.36Â Ã—Â 10

^{5 }+ 6.4 x 10

^{6 }

**= 7.33 x10**

^{6}m**Therefore, ****the greatest height the missile can attain before falling back to the earth isÂ Â 9.356Â Ã—Â 10 ^{5 }m and the final distance of the missile from the center of the earth is **

**7.33 x10**

^{6}m**Â **

**Q.18: On the surface of the earth escape velocity is 11.2 km/s. If a body (rocket) is projected upward with twice this speed, find the speed of this rocket at a distance far away from the surface of the earth. Neglect the presence of other heavenly bodies like the sun and other planets.**

**Sol:**

Given,

**Escape velocity** of the Earthâ€™s surface,Â **v _{escÂ }= 11.2 km/s**

Projection

Projection

**velocity**of the rocket,Â

**v**

_{P}Â = 2v_{esp}Let, **mass** of the body **=Â m kg**

And, the **velocity** of the **rocket** at a **distance** very **far away** from the surface of earth **=Â v _{F}**

**Total** **energy** of the rocket on the surface **= \(\frac{1}{2}\) mv _{p}^{2}Â – \(\frac{1}{2}\)mv^{2}_{esp}**

**Total** **energy** of the rocket at a distance very far away from the Earth **= \(\frac{1}{2}\) mv _{F}^{2}**

Now, **according** to the **law of conservation of energy**, we have:

**\(\frac{1}{2}\)mv _{P}^{2}Â – \(\frac{1}{2}\)mv_{ESC}^{2}Â =Â \(\frac{1}{2}\)mv_{F}^{2}**

mv_{ESC}^{2}Â =Â \(\frac{1}{2}\) mv_{F}^{2}

v_{F}Â = \(\\\sqrt{(v_{p})^{2}-(v_{esc})^{2}}\\\)Â Â Â **[Since, v _{P}Â = 2v_{esp }]**

v

_{F}Â = \(\\\sqrt{(2v_{esp})^{2}-(v_{esc})^{2}}\\\)

v

_{F}Â =\(\\\sqrt{3}\) v

_{esc}

\(\Rightarrow\)Â v_{F} =\(\sqrt{3}\)Â Ã—Â 11.2 **=Â 19.39 km/s**

**Therefore, t****he speed of rocket at a distance far away from the surface of earth is ****19.39 km/s.**

**Â **

**Q.19: International Space Station (ISS) orbits the earth at a height of 380 km and has a mass of ****420000 kg. Find the amount of energy required to take it out from the gravitational influence of earth. [Mass of earth = 6.0 x 10 ^{24} kg, radius of earth = 6.4 x10 ^{6}m and take G = 6.67x 10 ^{-11} N m^{2}Â kg^{â€“2}]**

**Sol:**

Given:

**Mass** of the **satellite,Â mÂ = 420000 kg**

**Radius** of the **Earth,Â R _{E}Â = 6.4 Ã— 10^{6}Â m**

**Mass** of the **Earth,Â M _{E}Â = 6.0 Ã— 10^{24}Â kg**

**Universal gravitational constant,**

**G = 6.67 Ã— 10**

^{â€“11}Â Nm^{2 }kg^{â€“2}**Height** of the **satellite**,Â **h**Â = 380 km **= 0.38 Ã—10 ^{6}Â m**

We know that the

**Total**

**energy**of the satellite at

**heightÂ â€˜hâ€™**Â

**\(=\frac{1}{2}mv^{2}+\left [ \frac{-GM_{E}\;m}{R_{E}+h} \right ]\)**

Also, **orbital velocity** of the satellite,Â **v \(=\left [ \frac{GM_{E}}{R_{E}+h} \right ]^{\frac{1}{2}}\)**

**Total** **energy** at **height,Â h**\(=\frac{1}{2}\left [ \frac{GM_{E}}{R_{E}+h} \right ]-\left [ \frac{GM_{E}\;m}{R_{E}+h} \right ]\)

**Therefore, T _{E}**

**\(=\frac{-1}{2}\left [ \frac{GM_{E}\;m}{R_{E}+h} \right ]\)**

This negative sign means that the ISS is bounded to Earth. This is the

**bound**

**energy**of the satellite.

Now,

**Total Energy**required to send the satellite out of its orbit =

**â€“ (Bound energy)**

**T**

_{E}\(\\=\frac{1}{2}\left [ \frac{GM_{E}\;m}{R_{E}+h} \right ]\\\)

T

_{E}\(\\=\frac{1}{2}\left [ \frac{(6.67\times 10^{-11})\times (6\times 10^{24})\times (4.2\times 10^{5})}{(6.4\times 10^{6})+(0.380\times 10^{6})} \right ]=\frac{1}{2}\times \left [ \frac{1.681\times 10^{20}}{6.78\times 10^{6}} \right ]\\\)

\(\\\Rightarrow\)Â

**T**

_{E }= 1.2396 x 10^{13}JTherefore, **the amount of energy required to take ISS out from the gravitational influence of earth is ****1.2396 x 10 ^{13} J**

**Â **

**Q.20: Two planets each of mass 2 x 10 ^{31} kg are on a path of collision with each other. If they have negligible speeds at a distance of 10^{10} km. Find the speed at which they collide if each planet has a radius of 10^{3 }km. [Take G = 6.67 x 10^{-11}]**

**Sol:**

Given:

**Radius **of each planet,Â **R**Â = 10^{3}Â km **= 10 ^{6}Â m**

**Distance **between the planet,Â **r**Â = 10^{10}Â km **= 10 ^{13 }m**

**Mass** of each planet,Â **MÂ = 2 Ã— 10 ^{31Â }kg**

For negligible speeds,Â **vÂ = 0 **

So the **total energy** of two planets separated by a **distanceÂ â€˜râ€™**:

**T _{E} =\(\frac{-GMM}{r}+\frac{1}{2}mv^{2}\) **

Since, v = 0;

Therefore, **T _{E} =\(\frac{-GMM}{r}\)Â . . . . . . . . . . . . . . . . (1)**

Now, when the planets are just about to collide:

Let, the **velocity** of the planets **=Â v**

The centers of the two planets are at a **distance** of **= 2R**

**Total kinetic energy** of the two planets = \(\frac{1}{2}\) Mv^{2}Â + \(\frac{1}{2}\) Mv^{2}Â **=Â Mv ^{2}**

**Total potential energy**of the two planets

**= \(\frac{-GMM}{2R}\)**

Therefore,

**Total**

**energy**of the two stars =Â

**Mv**

^{2}Â -\(\\\frac{GMM}{2R}\)Â .Â .Â . . . . . . . . . . (2)Now, according to **the law of** **conservation** **of** **energy** :

Mv^{2}Â â€“ \(\\\frac{GMM}{2R}\)Â = \(\frac{-GMM}{r}\\\)

v^{2}Â = \(\\\frac{-GM}{r}\)Â + \(\frac{GM}{2R}\\\)

v^{2} = GM Ã— \(\\\left [ \frac{-1}{r}+\frac{1}{2R} \right ]\)

\(\Rightarrow\) v^{2} = 6.67Â Ã—Â 10^{-11}Â Ã—Â 2Â Ã—Â 10^{31}Â Ã— \(\left [ \frac{-1}{(10^{13})}+\frac{1}{(2\times 10^{6})} \right ]\)

\(\Rightarrow\)v^{2 }= 6.67Â Ã—Â 10^{14}

**Therefore, v =Â ****(6.67Â Ã—Â 10 ^{14})^{1/2}Â =Â 2.583Â **

**Ã—**

**10**

^{7}**m/s.**

**Hence, the speed at which they will collide = 2.583Â ****Ã—**** 10 ^{7} **

**m/s.**

**Â **

**Q.21:Â Two spheres eachÂ having a radius of 0.15 m and mass 1000 kg are placed 1.0 m apart from each other on a horizontal plane. Calculate the gravitational potential and force atÂ the midpoint of the line connecting the centers of the two spheres. If anÂ object is kept at that point will it be in equilibrium? If yes, is this equilibrium**** position is unstable or stable?**

**Sol: **

Given:

**Radius** of spheres, **R = 0.15 m**

**Distance** between two spheres, **r = 1.0 m**

**Mass** of each sphere, **M = 1000 kg**

From the above figure, â€˜Aâ€™ is the mid-point and since each sphere will exert the gravitational force in opposite direction. Therefore, **the gravitational force at this point will be zero.**

Gravitational potential at the midpoint (A) is;

**U****=** \(\left [ \frac{-GM}{\frac{r}{2}}+\frac{-GM}{\frac{r}{2}} \right ]\)

**U=** \(\left [ \frac{-4GM}{r} \right ]\)

**U****=** \(\left [ \frac{-4\times (6.67\times 10^{-11})\times (1000)}{1.0} \right ]\)

**\(\Rightarrow\) U****= -2.668 x 10 ^{-7} J /kg**

**Therefore, the gravitational potential and force atÂ the mid-point of the line connecting the centers of the two spheres is** **= -2.668 x 10 ^{-7} J /kg**

The net force on an object, placed at the mid-point is **zero**. However, if the object is displaced even a little towards any of the two bodies it will not return to its equilibrium position. **Thus, the body is in unstable equilibrium.**

**Â **

**Q.22: Find the gravitational potential due to Earthâ€™s gravity on a geo-stationary satellite orbiting earth at 36000 km above the surface. [Mass of earth = 6 x 10 ^{24 }kg, radius = 6400 km and Assume that the potential energy is zero at infinity]**

**Sol:**

Given:

**Radius **of the Earth, **R =** 6400 km **= 0.64 ****Ã—**** 10 ^{7 }m**

**Mass** of Earth, **M = 6 x 10 ^{24 }kg**

**Height of the geo-stationary satellite** from earthâ€™s surface, **h =** 36000 km = **3.6 x 10 ^{7} m**

Therefore, **gravitational potential at height â€˜hâ€™** on the geo-stationary satellite due to the earthâ€™s gravity:

**G _{P }= \(\frac{-GM}{R+h}\\\)**

\(\\\Rightarrow\)

**G**\(\frac{-(6.67\times 10^{-11})\times (6\times 10^{24})}{(0.64\times 10^{7})+(3.6\times 10^{7})}\\\)

_{P}=\(\\\Rightarrow\)

**G**\(\frac{-40.02\times 10^{13}}{4.24\times 10^{7}}\)

_{P }=**= -9.439**

**Ã—**

**10**

^{6}J/Kg**Therefore, the gravitational potential due to Earthâ€™s gravity on a geo-stationary satellite orbiting earth is -9.439 ****Ã—**** 10 ^{6} J/Kg**

**Â **

**Q.23: A star 10 times the size of the sun collapses on itself and gets reduced to a black hole of radius 10km, rotating at a speed of 10 revolutions per second. Prove that an object placed on its equator will remain stuck on the equator due to its gravity. [Mass of the sun = 2 x 10 ^{3} kg]Â Â **

**Sol.**

Any matter/ object will remain stuck to the surface if the **outward centrifugal force** is **lesser** than the **inward** **gravitational** **pull**.

**Gravitational force,Â f _{G}Â = \(\frac{GM\;m}{R^{2}}\)** [Neglecting negative sign]

Here,

**M**Â = Mass of the star = 10 Ã— 2 Ã— 10^{30}Â = **2 Ã— 10 ^{31Â }kg**

**mÂ**= Mass of the object

**R**=Â Radius of the star = 10 km =

**1 Ã—10**

^{4}Â mTherefore,

**f**

_{G}**Â**

**= \(\\\frac{(6.67\times 10^{-11})\times (2\times 10^{31})\;m}{(1\times 10^{4})^{2}}=1.334\times 10^{13}m\;N\)**

Now, **Centrifugal force,Â f _{C}^{Â }=Â m r Ï‰^{2}**Here, Ï‰Â = Angular speed = 2Ï€Î½

Î½ = Angular frequency = 10 rev s

^{â€“1}

f

_{c}Â =Â m RÂ (2Ï€Î½)

^{2}

**f**** _{c}** =Â mÂ Ã— (10

^{4}) Ã— 4 Ã— (3.14)

^{2}Â Ã— (10)

^{2}Â

**= (3.9 Ã—10**

^{7}m) N**AsÂ f _{G}Â >Â f_{C}, the object will remain stuck to the surface of black hole.**

**Q.24: Find the amount of energy required to launch a spaceship stationed on Mars to out of the solar system. **

(Mass of the Sun = 2 Ã— 10^{30}Â kg; mass of the space ship = 2000 kg, mass of mars = 6.4 Ã— 10^{23}Â kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 Ã— 10^{8 }km; G= 6.67 Ã— 10^{â€“11}Â m^{2}kg^{â€“2})

**Sol: **

Given,

**Mass of the Sun,Â MÂ = 2 Ã— 10 ^{30}Â kg**

**Mass of the spaceship**,Â **m _{SÂ }= 2000 kg**

**Radius of Mars,Â r**Â = 3395 km **= 3.395 Ã— 10 ^{6}Â m**

**Mass of Mars,Â M _{m}Â = 6.4 Ã— 10Â ^{23}Â kg**

**Orbital radius of Mars,Â RÂ =** 2.28 Ã— 10^{8Â }km = **2.28 Ã— 10 ^{11 }m**

Universal gravitational constant, G = 6.67 Ã— 10^{â€“11}Â m^{2 }kg^{â€“ 2}

Now,

**Potential energy** of the spaceship due to the **gravity of Sun** = **\(\frac{-GMm_{s}}{R}\)**

**Potential energy **of the spaceship due to **gravity of Mars** = **\(\frac{-GM_{M}\;m_{s}}{r}\)**

As the spaceship is stationed on Mars, its **velocity** is **â€˜zeroâ€™** and thus, its **kinetic energy is also â€˜zeroâ€™.****
**Thus,

**Total energy of the spaceship**

**\(\\=\frac{-GM_{M}\;m_{s}}{r}-\frac{GMm_{s}}{R}\)**

**T**\(\\=-Gm_{s}\left [ \frac{M}{R}+\frac{M_{M}}{r} \right ]\)

_{E}The **negative** **sign** means that the system is in **bound state**.

Therefore, energy required to launch the spaceship out of the solar system:

**T _{E} = â€“ (bound energy)**

**T**\(\\=Gm_{s}\left [ \frac{M}{R}+\frac{M_{M}}{r} \right ]\)

_{E}Now, on substituting the values of **G, M, R, m _{s}, M_{M} and r** we will get:

**T _{E}** \(=(6.67\times 10^{-11})\times (2\times 10^{3})\times \left [ \frac{2\times 10^{30}}{2.28\times 10^{11}}+\frac{6.4\times 10^{23}}{3.395\times 10^{6}} \right ]\)

\(\\\Rightarrow\)Â Â \(13.34\times 10^{-8}\times \left [ 87.7\times 10^{17} \;+\;1.885\times 10^{17}\right ]\boldsymbol{=1.195\times 10^{12}J}\)

**Therefore, the total amount of energy required to launch a spaceship stationed on Mars to out of the solar system = 1.195 ****Ã—**** 10 ^{12} J **

**Â **

**Q.25: A missile is shot vertically upward from the surface of Venus; if 25% of its initial energy is lost in overcoming Venusâ€™s atmospheric resistance, find the maximum height the missile will achieve before returning back to the surface. Take Initial velocity of the missile = 3 km s ^{â€“ 1}, Mass of Venus = 4.8 Ã— 10^{24} kg and the radius of Venus= 6052 km. [G = 6.67Ã— 10^{-11}Â N m^{2}Â kg^{â€“2 }]**

**Sol.**

Given:

**Mass** of Venus,Â **MÂ = 4.8 Ã— 10 ^{24}Â kg**

**Initial** **velocity** of the missile,Â **vÂ = 3 km/s = 3 Ã— 10 ^{3}Â m/s**

**Radius** of Venus,Â **RÂ **= 6052 km **= 6.05 x 10 ^{6}Â m**

**Universal gravitational constant, G = 6.67Ã— 10 ^{â€“11}Â N m^{2}Â kg^{â€“2}**

Let the **mass** of the **missile** =Â **m kg**

Since, the Initial kinetic energy of the missile = \(\frac{1}{2}\) mv^{2}

and the Initial potential energy of the missile = \(\frac{-GMm}{R}\)

Thus, total initial energy \(=\frac{1}{2}mv^{2}+\left ( \frac{-GMm}{R} \right )\)

It is given that **25 % of initial kinetic energy is lost **in overcoming the atmospheric resistance of Venus; this means that only **75% of the total kinetic** **energy** is **available** for **propelling** it upwards.

Hence, the total available initial energy \(=\left [ \frac{75}{100}\times \frac{1}{2}mv^{2} \right ]-\frac{GMm}{R}=0.375mv^{2}-\frac{GMm}{R}\)

Let **â€˜hâ€™ be the maximum height attained by the missile.**

Now, at **height â€˜hâ€™ **the **final velocity = 0** and hence, the **kinetic energy = 0**

Therefore, the **total energy of the missile** at heightÂ â€˜hâ€™ = \(\frac{-GMm}{R+h}\)

Now, according to the law of conservation of energy:

\(\\\Rightarrow\;\;\; 0.375mv^{2}-\frac{GM}{R}=\frac{-GMm}{R+h}\)

\(\\\Rightarrow\) Â Â \(\;\;\; 0.375\;v^{2}=GMm\left ( \frac{1}{R}-\frac{1}{R+h} \right )\)

\(\\\Rightarrow 0.375 v^{2}=GM\times \left [ \frac{h}{R\;(R+h)} \right ]\\\)

\(\\\Rightarrow\) Â \(\frac{R+h}{h}=\frac{GM}{0.375\;v^{2}R}\\\)

\(\\\Rightarrow \)Â Â \(\frac{R}{h}=\frac{GM}{0.375\;v^{2}R}-1\\\)

\(\\\Rightarrow\) Â \(\frac{R}{h}=\frac{GM-0.375\;v^{2}R}{0.375\;v^{2}R}\\\)

\(\\\Rightarrow\)Â Â \(\frac{1}{h}=\frac{GM-0.375\;v^{2}R}{0.375\;v^{2}R^{2}}\\\)

\(\\\Rightarrow\) Â \(h=\frac{0.375\;v^{2}R^{2}}{GM-0.375\;v^{2}R}\)

Now, on substituting the values of **G, M, v and R** we will get:

\(\Rightarrow h=\frac{0.375\;\times \;(3\;\times \;10^{3})^{2}\;\times \;(6.05\;\times \;10^{6})^{2}}{\left [ (6.67\;\times \;10^{-11})\;\times \;(4.8\;\times \;10^{24}) \right ]-[0.375\;\times \;(3\;\times \;10^{3})^{2}\;\times \;6.05\;\times \;10^{6}]}\\\)

\(\\\Rightarrow h=\frac{0.375\;\times \;(3\;\times \;10^{3})^{2}\;\times \;(6.05\;\times \;10^{6})^{2}}{\left [ (6.67\;\times \;10^{-11})\;\times \;(4.8\;\times \;10^{24}) \right ]-[0.375\;\times \;(3\;\times \;10^{3})^{2}\;\times \;6.05\;\times \;10^{6}]}\\\)

\(\\\Rightarrow\)Â Â \(\frac{1.235\;\times \;10^{20}}{3.2016\;\times \;10^{14}-2.042\;\times \;10^{13}}\boldsymbol{=412023.75\;m=4.12\;km}\)

**Therefore, the maximum height that is achieved by the missile before returning back to the surface = 4.12 km.**

In order to help you to cover this topic in a detailed and comprehensive way by solving problems, we have provided NCERT Solutions for Class 11 Physics Chapter 8 pdf to help students to learn better.