NCERT Solutions for Class 11 Physics Chapter 8 Gravitation: Download Topic-wise Questions & Answers PDF

ncert solutions physics class 11th 2023

Updated on Nov 29, 2024 12:38 IST

Gravitation Class 11 NCERT Solutions helps students in improving their overall academic performance. Gravitation Class 11 Important Questions with Answers helps students to develop a deeper understanding of the subject, which translates into better grades and academic success. NCERT Class 11 Solutions Physics help students to build confidence in their abilities.

The NCERT Solutions for Class 11 Physics Chapter 8 Gravitation provides comprehensive explanations and step-by-step solutions to the exercise questions, helping students develop a clear understanding of the concepts. These Class 11 Physics Chapter 8 Exercise Solutions are an essential resource for mastering topics such as Kepler's laws and gravitational potential energy. By studying the NCERT Gravitation Class 11 solutions, students can effectively prepare for exams and gain confidence in applying the principles of gravitation to solve real-world problems.

Class 11 Physics Gravitation is the force of attraction between any two objects in the universe. While all objects in the universe exert some level of gravitational pull on each other, the distance between them is often so vast that the force is too weak to be detected. Additionally, the effects of gravity weaken as objects move further apart, although its range is theoretically infinite. Sir Isaac Newton was the first to observe and describe this force of attraction.

NCERT Physic\s Class 11th Gravitation: Topics Covered

  • Introduction to gravitation
  • Kepler’s laws
  • Universal law of gravitation
  • The gravitational constant
  • Acceleration due to gravity of the earth
  • Acceleration due to gravity below and above the surface of earth
  • Gravitational potential energy
  • Escape speed
  • Earth satellites
  • Energy of an orbiting satellite
  • Geostationary and polar satellites
  • Weightlessness

NCERT Physics Class11 th Solution PDF For Gravitation

NCERT Solution for Class XI Physics Chapter Gravitation PDF download is available here for the students' reference. Physics NCERT Class 11 PDF downloads for the Gravitation chapter will allow students to save and print the solutions, making it easier for them to study and revise.

Download Here: NCERT Solution for Class XI Physics Chapter Gravitation PDF

Explore exams which ask questions on ncert solutions physics class 11th

Select your preferred stream

NCERT Physics Class11th Gravitation Solutions and FAQs

Q.8.1 Answer the following:

(a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?

(b) An astronaut inside a small space ship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?

(c) If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that the Sun’s pull is greater than the moon’s pull.

(you can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the moon’s pull is greater than the tidal effect of sun. Why?

Ans.8.1 Unlike electrical forces, gravitational force is independent of the status of the objects.

  • Yes, the size of the space station is large enough and the astronaut will detect the change in Earth’s gravity.
  • Tidal effect depends inversely upon the cube of the distance while gravitational force depends inversely on the square of the distance. Since the distance between the Moon and the Earth is smaller than the distance between the Sun and the Earth, the tidal effect of the Moon’s pull is greater than the tidal effect of the Sun’s pull.

 

Q.8.2 Choose the correct alternative:

(a) Acceleration due to gravity increases/decreases with increasing altitude.

(b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).

(c) Acceleration due to gravity is independent of mass of the earth/mass of the body.

(d) The formula –G Mm(1/r2 – 1/r1) is more/less accurate than the formula mg(r2 r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the earth.

Ans.8.2 Decreases - Acceleration due to gravity at depth h is given by

g h  = (1 –  2 h R e  )g, where  R e = R a d i u s o f t h e e a r t h ,  g = acceleration due to gravity on the surface of the Earth. From this equation, it is clear that acceleration due to gravity decreases with increase in height

Decreases – Acceleration due to gravity at depth d is given by  g d  = (1-  d R e  )g. So the acceleration due to gravity decreases with increase in depth.

mass of the body – Acceleration due to gravity of body mass m is given by the relation g =  G M R 2  , where G = Universal gravitation constant, M = mass of the Earth and R = radius of the Earth. Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body.

More – Gravitational potential energy of two points  r 1  and  r 2  distance away from the centre of the Earth is respectively given by:

V(  r 1  ) =  -  -  G m M r 1  and V(  r 2  ) =  -  -  G m M r 2

Difference in potential energy, V = V(  r 2  )  -  V(  r 1  ) =  - G m M ( 1 r 2   - 1 r 1  )

Hence this formula is more accurate than mg(  r 2  -  r 1  )

 

Q.8.3 Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth?

Ans.8.3 Time taken by the Earth to complete one revolution around the Sun,  T e  = 1 year

Orbital radius of the Earth in its orbit,  R e  = 1 AU

Time taken by the planet to complete one revolution around the Sun,  T p  =  1 2 T e  =  1 2  year

Orbital radius of the planet =  R p

From Kepler’s 3rd law of planetary motion, we can write:

R p R e ) 3  = (  T p T e ) 2

R p R e  = (  T p T e ) 2 / 3  =(  1 2 ) 2 / 3  = 0.63

Hence, the orbital radius of the planet will be 0.63 times smaller than that of the Earth.

 

Q.8.4 Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Ans.8.4 The rotation period of the satellite Io,  T j u  = 1.769 days = 1.769 x 24 x 60 x 60s

Radius of the Orbit is given by the relation,  R j u  = 4.22  × 10 8  m

Mass is given by the relation:  M j  =  4 π 2 R j u 3 G T j u 2  ……(i)

Where  M j  = mass of Jupiter and G = Universal gravitational constant

Orbital period of the Earth,  T e  = 365.25 days = 365.25 x 24 x 60 x 60 s

Orbital radius of the Earth,  R e  = 1 AU = 1.496 x  10 11  m

Mass of Sun is given as :  M s  =  4 π 2 R e 3 G T e 2  …(ii)

M s M j  =  4 π 2 R e 3 G T e 2   ×   G T j u 2 4 π 2 R j u 3  =  R e 3 T e 2   ×   T j u 2 R j u 3  = (  1.496 x 10 11 4.22 × 10 8 ) 3   × ( 1.769 x 24 x 60 x 60 365.25 x 24 x 60 x 60 ) 2  = 1045.04

Hence it can be inferred that the mass of Jupiter is about one – thousandth that of the Sun.

 

Q.8.5 Let us assume that our galaxy consists of 2.5 × 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milk Way to be 105 ly.

Ans.8.5 Mass of our galaxy Milky Way, M = 2.5  × 10 11  solar mass

Solar mass = Mass of the Sun = 2.0  × 10 30  kg

Mass of our galaxy = 2.5  × 10 11 ×  2.0  × 10 30  = 5  × 10 41  kg

Diameter of the Milky Way, d = 105 ly

Radius of the Milky Way = 5  × 10 4  ly

1 ly = 9.46  × 10 15  m

r = 5  × 10 4 ×  9.46  × 10 15  m = 4.73  × 10 20  mAs a star revolves around the galactic centre of the Milky Way, its time period is given by the relation:  τ  = (  4 π 2 r 3 G M ) 1 2  = (  4 × 3.14 2 × 4.73 3 × 10 60 6.67 × 10 - 11 × 5 × 10 41 ) 1 2  = 1.12  × 10 16 s  = 3.55 x  10 8  years

 

Q.8.6 Choose the correct alternative:

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

(b) The energy required to launch an orbiting satellite out of earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth’s influence.

Ans.8.6 Total mechanical energy of a satellite is the sum of its kinetic energy (always positive) and potential energy (can be either positive or negative). At infinity, the gravitational potential energy of the satellite is zero. As the Earth-Satellite system is a bound system, the total energy of the satellite is negative. Thus the total energy of an orbiting satellite at infinity is equal to the negative of its kinetic energy.

  • An orbiting satellite acquires a certain amount of energy that enables it to revolve around the Earth. This energy is provided by its orbit. It requires relatively lesser energy to move out of the influence of the Earth’s gravitational field than a stationary object on the Earth’s surface that initially contains no energy.

 

Q.8.7 Does the escape speed of a body from the earth depend on

(a) the mass of the body,

(b)the location from where it is projected,

(c) the direction of projection,

(d) the height of the location from where the body is launched?

Ans.8.7 Escape velocity of a body from the Earth is given by the relation:

v e s c  =  2 g R  ….(1)

Where g = acceleration due to gravity and R = radius of the Earth

So escape velocity is independent of the mass of the body.

  • It does not depend on the location from where it is projected.
  • Does not depend on the direction of projection
  • Depends on the height of the location from where the body is launched.

 

Q.8.8 A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant

(a) linear speed,

(b) angular speed,

(c) angular momentum,

(d) kinetic energy,

(e) potential energy,

(f) total energy throughout its orbit?

Neglect any mass loss of the comet when it comes very close to the Sun.

Ans.8.8 Angular momentum and total energy at all points of the orbit of a comet moving in a highly elliptical orbit around the Sun are constant. Is linear speed, angular speed, kinetic and potential energy varies from point to point in the orbit.

  • No
  • No
  • Yes
  • No
  • No
  • Yes

 

Q.8.9 Which of the following symptoms is likely to afflict an astronaut in space

(a) swollen feet,

(b) swollen face,

(c) headache,

(d) orientational problem.

Ans.8.9 Legs hold the entire mass of a body in standing position due to gravitational pull. In space, an astronaut feels weightlessness because of the absence of gravity. Therefore swollen feet of an astronaut do not affect him/her in space.

  • A swollen face is caused generally because of apparent weightlessness in space. Sense organs such as eyes, ears, nose and mouth constitute a person’s face. These symptoms can affect astronaut in space.
  • Headaches are caused because of mental strain. It can affect the working of an astronaut in space.
  • Space has different orientations. Therefore, orientational problem can affect an astronaut in space.

 

Q.8.10 In the following two exercises, choose the correct answer from among the given ones:

The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12) (i) a, (ii) b, (iii) c, (iv) 0.

Ans.8.10 Gravitational potential (V) is constant at all points in a spherical shell. Hence the gravitational gradient (  d V d r )  is zero everywhere inside the spherical shell. The gravitational potential gradient is equal to the negative of gravitational intensity. Hence intensity is also zero at all points inside the spherical shell. This indicates that gravitational forces acting at a point in a spherical shell are symmetric.

If the upper half of a spherical shell is cut out then the net gravitational force acting on a particle located at the centre O will be in the downward direction.

Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction. Thus the gravitational intensity at centre O of the given hemispherical shell has the direction as indicated by arrow c.

 

Q.8.11 For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow (i) d, (ii) e, (iii) f, (iv) g.

Ans.8.11 If the upper half of the spherical shell is cut out, then the net gravitational force acting on a particle at an arbitrary point P will be in the downward direction. Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction. Thus, the gravitational intensity at an arbitrary point P of the hemispherical shell has the direction as indicated by arrow e.

 

Q.8.12 A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero? Mass of the sun = 2×1030 kg, mass of the earth = 6×1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 × 1011 m).

Ans.8.1 Mass of the Sun,  M s  =  2 × 10 30  kg, Mass of the Earth,  M e  =  6 × 10 24  kg

Orbital radius, r =  1.5 × 10 11  m

Let the mass of the rocket be, m

Let x be the distance from the centre of the Earth where the gravitational force acting on satellite P becomes zero.

From Newton’s law of gravitation, we can equate gravitational forces acting on the satellite P under the influence of the Sun and the Earth as:

G m M s ( r - x ) 2  =  G m M e x 2  or(  r - x x ) 2  =  M s M e

r - x x ) 2  =  2 × 10 30 6 × 10 24  ,  r - x x  = 577.35 , r = 578.35x , x =  1.5 × 10 11 578.35  = 2.59  × 10 8  m

 

Q.8.13 How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 108 km.

Ans.8.13 Orbital radius of the Earth around the Sun, r = 1.5 × 108 km = 1.5 × 1011 m

Time taken by the Earth to complete 1 revolution around the Sun , T = 1 year = 365.25 days = 365.25  ×  24  × 24 × 60 × 60 s

Universal gravitational constant, G = 6.67  × 10 - 11  N  m 2 / k g 2

Thus, mass of the Sun can be calculated as,

M =  4 π 2 r 3 G T 2  =  4 × ( 3.1416 ) 2 × ( 1.5 × 10 11 ) 3 6.67 × 10 - 11 × ( 365.25 × 24 × 60 × 60 ) 2  = 2.0  × 10 30 kg

 

Q.8.14 A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50 × 108 km away from the Sun?

Ans.8.14 Distance of the Earth from the Sun,  r e  = 1.50 × 108 km = 1.5 × 1011 m

Time period of the Earth =  T e  , Time period of the Saturn  T s  =  29.5 T e

Distance of Saturn from the Sun =  r s

From Kepler’s 3rd law of planetary motion, we have T =(  4 π 2 r 3 G M ) 1 / 2

For Saturn and Sun, we can write,  r s 3 r e 3  =  T s 2 T e 3

r s  =  r e ( 29.5 T e T e ) 2 / 3  = 1.5 × 1011  × ( 29.5 1 ) 2 / 3  = 1.4 X  10 12  m

 

Q.8.15 A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth?

Ans.8.15 Weight of the body, W = 63 N

Acceleration due to gravity at h from the Earth’s surface is given by

g’ =  g ( 1 + h R e ) 2  where g = acceleration due to gravity on the Earth’s surface,  R e  = Radius of the Earth. h =  R e 2

g’ =  g ( 1 + R e / 2 R e ) 2  =  g ( 1 + h 2 h ) 2  = (4/9)g

Weight of the body of mass m at a height h is given by

W’ = m X g’ = (4/9) mg = (4/9) x w = (4/9) x 63 N = 28 N

 

Q.8.16 Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?

Ans.8.16 Weight of a body of mass m at Earth’s surface, W = mg = 250 N

Body of mass m is located at depth, d =  1 2 R e  , where  R e  is radius of the Earth

Acceleration due to gravity at depth g (d) is given by : g’ = (1 -  d R e  )g = (1/2)g

Weight of the body at depth d

W’ = mg’ = (1/2) mg = (1/2)W = (1/2) x 250 N = 125 N

 

Q.8.17 A rocket is fired vertically with a speed of 5 km s-1 from the earth’s surface. How far from the earth does the rocket go before returning to the earth? Mass of the earth = 6.0 × 1024 kg; mean radius of the earth = 6.4 × 106 m; G = 6.67 × 10–11 N m2 kg2.

Ans.8.17 Velocity of the rocket, v = 5 km/s = 5  × 10 3  m/s

Mass of the Earth,  M e  = 6.0 × 1024 kg

Radius of the Earth,  R e  = 6.4 × 106 m

Height reached by rocket mass, m = h

At the Earth’s surface

Total energy of the rocket = Kinetic energy + Potential energy =  1 2  m  v 2  + (  - G m M e R e  )

At highest point h, v = 0, and potential energy =  - G m M e R e + h

Total energy of the rocket at height h =  - G m M e R e + h

From the law of conservation of energy, we have,

Total energy of the rocket at Earth surface = Total energy at height h

1 2  m  v 2  + (  - G m M e R e  ) =  - G m M e R e + h  or  1 2 v 2  =  G M e ( 1 R e - 1 R e + h )

1 2 v 2  =  G M e ( 1 R e - 1 R e + h )  =  G M e h R e ( R e + h )  =  G M e R e 2 × R e h ( R e + h )  =  g R e h ( R e + h )  where g =  G M e R e 2  = 9.8  m s 2

v 2 ( R e + h )  = 2  g R e h   v 2 R e + v 2 h  = 2  g R e h  h(2  g R e - v 2 )  =  v 2 R e

h =  v 2 R e ( 2 g R e - v 2 )  = 1.59 x  10 6  m

 

Q.8.18 The escape speed of a projectile on the earth’s surface is 11.2 km s–1. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.

Ans.8.18 Escape velocity of the projectile on the Earth’s surface,  v e s c  = 11.2 km/s = 11.2  × 10 3  m/s

Projection velocity of the projectile = 3  v e s c

Mass of the projectile = m

Velocity of the projectile far away from the Earth =  v f

Total energy of the projectile on the Earth =  1 2  m  v p 2   -   1 2  m  v e s c 2

Gravitational potential energy of the projectile far away from the Earth is zero.

Total energy of the projectile far away from the Earth =  1 2  m  v f 2

From the law of conservation of energy

1 2  m  v p 2     1 2  m  v e s c 2  =  1 2  m  v f 2

v p 2 - v e s c 2 = v f 2

v f  =  v p 2 - v e s c 2  =  ( 3 v e s c ) 2 - v e s c 2  =  8 ×  11.2= 31.68 km/s

 

Q.8.19 A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0×1024 kg; radius of the earth = 6.4 × 106 m; G = 6.67 × 10–11 N m2 kg2.

Ans.8.19 Mass of the Earth, M = 6.0×1024 kg

Mass of the satellite, m = 200 kg

 

Radius of the Earth,  R e = 6.4 ×106 m

Universal gravitational constant, G = 6.67 × 10–11 N m2 kg2

Height of the satellite, h = 400 km = 0.4  × 10 6  m

Total energy of the satellite at height h =  1 2 m v 2  + (  - G M e m R e + h  )

Orbital velocity of the satellite, v =  G M e R e + h

Total energy of the satellite at height h =  m 2  (  G M e R e + h )  - (  G M e m R e + h  ) =  -   1 2 ( G M e m R e + h )

The negative sign indicates that the satellite is bound to the Earth. This is called bound energy of the satellite  T y p e e q u a t i o n h e r e .

Energy require to send the satellite out of its orbit = - (bound energy) =  1 2 ( G M e m R e + h )

1 2 ( 6.67 X 10 - 11 × 6 × 10 24 × 200 6.4 × 10 6 + 0.4 × 10 6 )  = 5.88  × 10 9  J

 

Q.8.20 Two stars each of one solar mass (= 2×1030 kg) are approaching each other for a head on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

Ans.8.20 Mass of each star, M = 2×1030 kg, Radius of each star, R = 104 km = 10 7  m

Distance between stars, r =  10 9  km =  10 12  m

For negligible speed, v = 0

The total energy of two stars separated at a distance r

- G M M r + 1 2 m v 2  =  - G M M r  ……..(i)

Now, consider the case when the stars are about to collide. Velocity of the stars = V, distance between the centres of the stars = 2R

Total kinetic energy of both stars =  1 2  M  V 2  +  1 2  M  V 2  = M  V 2

Total potential energy of both stars =  - G M M 2 R

Total energy of two stars = M  V 2 - G M M 2 R  …….(ii)

Using the law of conservation of energy, we can write

V 2 - G M M 2 R  =  - G M M r

V 2 = G M M 2 R   - G M M r

V 2  = GM (  1 2 R   -   1 r )  = 6.67  × 10 - 11 × 2 × 10 30  (  1 2 × 10 7 - 1 10 - 12  )

= 1.334  × 10 20 ×  (  1 2 × 10 7 - 1 10 - 12  )

V = 2.6  × 10 6  m/s

 

Q. 8.21 Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the midpoint of the line joining the centers of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?

Ans.8.21

Mass of each sphere, M = 100 kg

Separation between the spheres, r = 1 m

X is the midpoint between the spheres.

Gravitational force at point x will be zero. This is because gravitational force exerted by each spheres will act in opposite directions.

Gravitational potential at point x:

- G M r 2   - G M r 2  =  -  4  G M r  =  - 4 × 6.67 × 10 - 11 × 100 1  =  - 2.668 × 10 - 8  J/kg

Any object placed at point x will be in equilibrium state, but the equilibrium is unstable. This is because any change in the position of the object will change the effective force in that direction.

 

Q.8.22 As you have learnt in the text, a geostationary satellite orbits the earth at a height of nearly 36,000 km from the surface of the earth. What is the potential due to earth’s gravity at the site of this satellite? (Take the potential energy at infinity to be zero). Mass of the earth = 6.0×1024 kg, radius = 6400 km.

Ans.8.22 Mass of the Earth, M = 6.0  × 10 24  kg

Radius of the Earth, R = 6400 km = 6.4  × 10 6  m

Height of geostationary satellite from the surface of the Earth, h = 36000 km = 3.6  × 10 7  m

Gravitational potential energy due to Earth’s gravity at height h:

- G M ( R + h )

- 6.67 × 10 - 11 × 6.0 × 10 24 3.6 × 10 7 + 6.4 × 10 6  =  - 6.67 × 6.0 × 10 13 4.24 × 10 7  =  -  9.44  × 10 6  J/kg

 

Q.8.23 A star 2.5 times the mass of the sun and collapsed to a size of 12 km rotates with a speed of 1.2 rev. per second. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (mass of the sun = 2×1030 kg).

Ans.8.23 Yes, a body gets stuck to the surface of a star if the inward gravitational force is greater than the outward centrifugal force caused by the rotation of the star.

Gravitational force,  f g  =  G M m R 2  , where M = mass of the star = 2.5  × 2 × 10 30  = 5  × 10 30  kg

M = mass of the body, R = radius of the star = 12km = 1.2  × 10 4 m

f g  =  6.67 × 10 - 11 × 5 × 10 30 × m ( 1.2 × 10 4 ) 2 =  2.31  × 10 12 m N

Centrifugal force  f c  = mr  ω 2  where  ω =  angular speed = 2  π γ  and angular frequency  γ  = 1.2 rev/s

f c =  mR(  2 π γ ) 2  = m  ×  (1.2  × 10 4 ) × 2 × 3.14 × 1.2 × ( 2 × 3.14 × 1.2 )  = 6.81  × 10 5 m N

Since  f g > f c  , the body will remain stuck to the surface of the star.

 

Q.8.24 A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the space ship = 1000 kg; mass of the sun = 2×1030 kg; mass of mars = 6.4×1023 kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 ×108 km; G = 6.67×10-11 N m2 kg–2.

Ans.8.24 Mass of the spaceship,  m s  = 1000 kg

Mass of the Sun, M = 2×1030 kg

Mass of Mars,  m m = 6.4×1023 kg

Orbital radius of Mars, R = 2.28 ×108 km = 2.28 ×1011 m

Radius of Mars, r = 3395 km = 3.395  × 10 6 m

Universal Gravitational constant, G = 6.67×10-11 N m2 kg–2

Potential energy of the spaceship due to the gravitational attraction of the Sun =  - G M m s R

Potential energy of the spaceship due to the gravitational attraction of Mars=  - G m s m m r

Since the spaceship is stationed on Mars, its velocity and hence its kinetic energy will be zero

Total energy of the spaceship =  - G M m s R   - - G m s m m r  =  - G m s ( M R  +  m m r  )

The negative sign indicates that the system is in bound state.

Energy required for launching the spaceship out of the solar system = - (Total energy of the spaceship)

G m s ( M R  +  m m r  )

= 6.67  × 10 - 11 × 10 3 × ( 2 × 10 30 2.28 × 10 11  +  6.4 × 10 23 3.395 × 10 6 )

= 6.67  × 10 - 8 × ( 8.77 × 10 18  + 1.88  × 10 17  )

= 6.67  × 10 - 8 × 8.958 × 10 18

= 5.97  × 10 11  J

 

Q.8.25 A rocket is fired ‘vertically’ from the surface of mars with a speed of 2 km s–1. If 20% of its initial energy is lost due to Martian atmospheric resistance, how far will the rocket go from the surface of mars before returning to it? Mass of mars = 6.4×1023 kg; radius of mars = 3395 km; G = 6.67×10-11 N m2 kg–2.

Ans.8.25 Initial kinetic energy of the rocket =  1 2 m v 2

Initial potential energy of the rocket =  - G M m R

Total initial energy =  1 2 m v 2 - G M m R

If 20% of initial kinetic energy is lost due to Martian atmosphere resistance, then only 80% of its kinetic energy helps in reaching a height

Total initial energy available = 0.8  × 1 2 m v 2   - G M m R

Maximum height reached by the rocket = h

At this height, the velocity and hence the kinetic energy of the rocket becomes zero.

Total energy of the rocket at height h =  - G M m R + h

Applying the law of conservation of energy for the rocket, we can write:

0.4  × m v 2   - G M m R  =  - G M m R + h

0.4  v 2  = GM(  1 R   - 1 R + h )

0.4  v 2  = GM(  R + h - R R ( R + h )  ) =  G M h R ( R + h )

R + h h  =  G M 0.4 R v 2

R h  + 1 =  G M 0.4 R v 2  =  6.4 × 10 23 × 6.67 × 10 - 11 0.4 × 3395 × 10 3 × 2000 2

h = 495 km

 

 

qna

ncert solutions physics class 11th Exam

Student Forum

chatAnything you would want to ask experts?
Write here...