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Page 1 Solved Examples on Gravitation JEE Mains Single Correct Type Q1. Three identical point masses, each of mass ?? ???? lie in the ?? - ?? plane at points (?? ,?? ),(?? , ?? .?? ?? ) and (?? .?? ?? ,?? ) . The net gravitational force on the mass at the origin is (a) ?? .???? × ???? -?? (??ˆ + ??ˆ)?? (b) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? (c) ?? .???? × ???? -?? (??ˆ - ??ˆ)?? (d) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? Ans: (a) Let particle ?? lies at origin, particle ?? and ?? on ?? and ?? -axis respectively ?? ???? ?????? = ?? ?? ?? ?? ?? ?? ???? 2 ??ˆ = 6.67× 10 -11 × 1× 1 (0.2) 2 ??ˆ = 1.67× 10 -9 ??ˆ?? Similarly ?? ???? ?????? = 1.67× 10 -9 ??ˆ N ? Net force on particle ?? ?? = ?? ???? + ?? ???? = 1.67× 10 -9 (??ˆ + ??ˆ)?? Q2. Acceleration due to gravity on moon is ?? /?? of the acceleration due to gravity on earth. If the ratio of densities of earth (?? ?? ) and moon (?? ?? ) is ( ?? ?? ?? ?? ) = ?? ?? then radius of moon ?? ?? in terms of ?? ?? will be (a) ?? ???? ?? ?? (b) ?? ?? ?? ?? (c) ?? ???? ?? ?? (d) ?? ?? v?? ?? ?? Ans: (a) Acceleration due to gravity ?? = 4 3 ???????? ? ?? ? ???? or ?? ?? ?? ?? = ?? ?? ?? ?? · ?? ?? ?? ?? [ As ?? ?? ?? ?? = 1 6 and ?? ?? ?? ?? = 5 3 (given)] ? ?? ?? ?? ?? = ( ?? ?? ?? ?? )( ?? ?? ?? ?? ) = 1 6 × 5 3 ? ?? ?? = 5 18 ?? ?? Page 2 Solved Examples on Gravitation JEE Mains Single Correct Type Q1. Three identical point masses, each of mass ?? ???? lie in the ?? - ?? plane at points (?? ,?? ),(?? , ?? .?? ?? ) and (?? .?? ?? ,?? ) . The net gravitational force on the mass at the origin is (a) ?? .???? × ???? -?? (??ˆ + ??ˆ)?? (b) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? (c) ?? .???? × ???? -?? (??ˆ - ??ˆ)?? (d) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? Ans: (a) Let particle ?? lies at origin, particle ?? and ?? on ?? and ?? -axis respectively ?? ???? ?????? = ?? ?? ?? ?? ?? ?? ???? 2 ??ˆ = 6.67× 10 -11 × 1× 1 (0.2) 2 ??ˆ = 1.67× 10 -9 ??ˆ?? Similarly ?? ???? ?????? = 1.67× 10 -9 ??ˆ N ? Net force on particle ?? ?? = ?? ???? + ?? ???? = 1.67× 10 -9 (??ˆ + ??ˆ)?? Q2. Acceleration due to gravity on moon is ?? /?? of the acceleration due to gravity on earth. If the ratio of densities of earth (?? ?? ) and moon (?? ?? ) is ( ?? ?? ?? ?? ) = ?? ?? then radius of moon ?? ?? in terms of ?? ?? will be (a) ?? ???? ?? ?? (b) ?? ?? ?? ?? (c) ?? ???? ?? ?? (d) ?? ?? v?? ?? ?? Ans: (a) Acceleration due to gravity ?? = 4 3 ???????? ? ?? ? ???? or ?? ?? ?? ?? = ?? ?? ?? ?? · ?? ?? ?? ?? [ As ?? ?? ?? ?? = 1 6 and ?? ?? ?? ?? = 5 3 (given)] ? ?? ?? ?? ?? = ( ?? ?? ?? ?? )( ?? ?? ?? ?? ) = 1 6 × 5 3 ? ?? ?? = 5 18 ?? ?? Q3. The moon's radius is ?? /?? that of the earth and its mass is ?? /???? times that of the earth. If ?? represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? ?? Ans: (b) Acceleration due to gravity ?? = ???? ?? 2 ? ?? moon ?? earth = ?? moon ?? earth · ?? earth 2 ?? moon 2 = ( 1 80 )( 4 1 ) 2 ?? moon = ?? earth × 16 80 = ?? 5 . Q4. The radii of two planets are respectively ?? ?? and ?? ?? and their densities are respectively ?? ?? and ?? ?? . The ratio of the accelerations due to gravity at their surfaces is (a) ?? ?? :?? ?? = ?? ?? ?? ?? ?? : ?? ?? ?? ?? ?? (b) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (c) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (d) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? Ans: (d) Acceleration due to gravity ?? = 4 3 ???????? ? ?? 1 :?? 2 = ?? 1 ?? 1 :?? 2 ?? 2 . Q5. The height of the point vertically above the earth's surface, at which acceleration due to gravity becomes ?? % of its value at the surface is (Radius of the earth = ?? ) (a) ?? ?? (b) ?? ?? (c) ???? ?? (d) ???? ?? Ans : (b) Acceleration due to gravity at height h is given by ?? ' = ?? ( ?? ?? +h ) 2 ? ?? 100 = ?? ( ?? ?? + h ) 2 ? ?? ?? + h = 1 10 ? h = 9?? . Page 3 Solved Examples on Gravitation JEE Mains Single Correct Type Q1. Three identical point masses, each of mass ?? ???? lie in the ?? - ?? plane at points (?? ,?? ),(?? , ?? .?? ?? ) and (?? .?? ?? ,?? ) . The net gravitational force on the mass at the origin is (a) ?? .???? × ???? -?? (??ˆ + ??ˆ)?? (b) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? (c) ?? .???? × ???? -?? (??ˆ - ??ˆ)?? (d) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? Ans: (a) Let particle ?? lies at origin, particle ?? and ?? on ?? and ?? -axis respectively ?? ???? ?????? = ?? ?? ?? ?? ?? ?? ???? 2 ??ˆ = 6.67× 10 -11 × 1× 1 (0.2) 2 ??ˆ = 1.67× 10 -9 ??ˆ?? Similarly ?? ???? ?????? = 1.67× 10 -9 ??ˆ N ? Net force on particle ?? ?? = ?? ???? + ?? ???? = 1.67× 10 -9 (??ˆ + ??ˆ)?? Q2. Acceleration due to gravity on moon is ?? /?? of the acceleration due to gravity on earth. If the ratio of densities of earth (?? ?? ) and moon (?? ?? ) is ( ?? ?? ?? ?? ) = ?? ?? then radius of moon ?? ?? in terms of ?? ?? will be (a) ?? ???? ?? ?? (b) ?? ?? ?? ?? (c) ?? ???? ?? ?? (d) ?? ?? v?? ?? ?? Ans: (a) Acceleration due to gravity ?? = 4 3 ???????? ? ?? ? ???? or ?? ?? ?? ?? = ?? ?? ?? ?? · ?? ?? ?? ?? [ As ?? ?? ?? ?? = 1 6 and ?? ?? ?? ?? = 5 3 (given)] ? ?? ?? ?? ?? = ( ?? ?? ?? ?? )( ?? ?? ?? ?? ) = 1 6 × 5 3 ? ?? ?? = 5 18 ?? ?? Q3. The moon's radius is ?? /?? that of the earth and its mass is ?? /???? times that of the earth. If ?? represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? ?? Ans: (b) Acceleration due to gravity ?? = ???? ?? 2 ? ?? moon ?? earth = ?? moon ?? earth · ?? earth 2 ?? moon 2 = ( 1 80 )( 4 1 ) 2 ?? moon = ?? earth × 16 80 = ?? 5 . Q4. The radii of two planets are respectively ?? ?? and ?? ?? and their densities are respectively ?? ?? and ?? ?? . The ratio of the accelerations due to gravity at their surfaces is (a) ?? ?? :?? ?? = ?? ?? ?? ?? ?? : ?? ?? ?? ?? ?? (b) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (c) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (d) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? Ans: (d) Acceleration due to gravity ?? = 4 3 ???????? ? ?? 1 :?? 2 = ?? 1 ?? 1 :?? 2 ?? 2 . Q5. The height of the point vertically above the earth's surface, at which acceleration due to gravity becomes ?? % of its value at the surface is (Radius of the earth = ?? ) (a) ?? ?? (b) ?? ?? (c) ???? ?? (d) ???? ?? Ans : (b) Acceleration due to gravity at height h is given by ?? ' = ?? ( ?? ?? +h ) 2 ? ?? 100 = ?? ( ?? ?? + h ) 2 ? ?? ?? + h = 1 10 ? h = 9?? . Q6. At surface of earth weight of a person is ???? ?? then his weight at height ?? /?? from surface of earth is ( ?? = radius of earth) (a) ???? ?? (b) ???? ?? (c) ???? ?? (d) ???? ?? Ans : (c) Weight of the body at height ?? , ?? ' = ?? ( ?? ?? +h ) 2 = ?? ( ?? ?? + ?? 2 ) 2 = ?? ( 2 3 ) 2 = 4 9 ?? = 4 9 × 72 = 32?? . Q7. Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine ?????? ???? below the earth's surface (Given ?? = ???????? ???? ) (a) ?? .???? ?? /?? ?? (b) ?? .???? ?? /?? ?? (c) ?? .???? ?? /?? ?? (d) ?? .???? ?? /?? ?? Ans: (a) Acceleration due to gravity at depth ?? , ?? ' = ?? [1- ?? ?? ] = ?? [1- 100 6400 ] = 9.8[1- 1 64 ] = 9.8 × 63 64 = 9.66 m/s 2 . Q8. The depth ?? at which the value of acceleration due to gravity becomes ?? ?? times the value at the surface, is [?? = radius of the earth ] (a) ?? ?? (b) ?? ( ?? -?? ?? ) (c) ?? ?? ?? (d) ?? ( ?? ?? +?? ) Ans : (b) ?? ' = ?? (1- ?? ?? ) ? ?? ?? = ?? (1- ?? ?? ) ? ?? ?? = 1 - 1 ?? ? ?? = ( ?? -1 ?? )?? Q9. The angular velocity of the earth with which it has to rotate so that acceleration due to gravity on ???? ° latitude becomes zero is (Radius of earth = ???????? ???? . At the poles ?? = ???? ???? - ?? ) (a) ?? .?? × ???? -?? ?????? /?????? (b) ?? .?? × ???? -?? ?????? /?????? (c) ???? × ???? ?? ?????? /?????? (d) ?? .?? × ???? -?? ?????? /?????? Ans : (a) Effective acceleration due to gravity due to rotation of earth ?? ' = ?? - ?? 2 ?? cos 2 ?? Page 4 Solved Examples on Gravitation JEE Mains Single Correct Type Q1. Three identical point masses, each of mass ?? ???? lie in the ?? - ?? plane at points (?? ,?? ),(?? , ?? .?? ?? ) and (?? .?? ?? ,?? ) . The net gravitational force on the mass at the origin is (a) ?? .???? × ???? -?? (??ˆ + ??ˆ)?? (b) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? (c) ?? .???? × ???? -?? (??ˆ - ??ˆ)?? (d) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? Ans: (a) Let particle ?? lies at origin, particle ?? and ?? on ?? and ?? -axis respectively ?? ???? ?????? = ?? ?? ?? ?? ?? ?? ???? 2 ??ˆ = 6.67× 10 -11 × 1× 1 (0.2) 2 ??ˆ = 1.67× 10 -9 ??ˆ?? Similarly ?? ???? ?????? = 1.67× 10 -9 ??ˆ N ? Net force on particle ?? ?? = ?? ???? + ?? ???? = 1.67× 10 -9 (??ˆ + ??ˆ)?? Q2. Acceleration due to gravity on moon is ?? /?? of the acceleration due to gravity on earth. If the ratio of densities of earth (?? ?? ) and moon (?? ?? ) is ( ?? ?? ?? ?? ) = ?? ?? then radius of moon ?? ?? in terms of ?? ?? will be (a) ?? ???? ?? ?? (b) ?? ?? ?? ?? (c) ?? ???? ?? ?? (d) ?? ?? v?? ?? ?? Ans: (a) Acceleration due to gravity ?? = 4 3 ???????? ? ?? ? ???? or ?? ?? ?? ?? = ?? ?? ?? ?? · ?? ?? ?? ?? [ As ?? ?? ?? ?? = 1 6 and ?? ?? ?? ?? = 5 3 (given)] ? ?? ?? ?? ?? = ( ?? ?? ?? ?? )( ?? ?? ?? ?? ) = 1 6 × 5 3 ? ?? ?? = 5 18 ?? ?? Q3. The moon's radius is ?? /?? that of the earth and its mass is ?? /???? times that of the earth. If ?? represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? ?? Ans: (b) Acceleration due to gravity ?? = ???? ?? 2 ? ?? moon ?? earth = ?? moon ?? earth · ?? earth 2 ?? moon 2 = ( 1 80 )( 4 1 ) 2 ?? moon = ?? earth × 16 80 = ?? 5 . Q4. The radii of two planets are respectively ?? ?? and ?? ?? and their densities are respectively ?? ?? and ?? ?? . The ratio of the accelerations due to gravity at their surfaces is (a) ?? ?? :?? ?? = ?? ?? ?? ?? ?? : ?? ?? ?? ?? ?? (b) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (c) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (d) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? Ans: (d) Acceleration due to gravity ?? = 4 3 ???????? ? ?? 1 :?? 2 = ?? 1 ?? 1 :?? 2 ?? 2 . Q5. The height of the point vertically above the earth's surface, at which acceleration due to gravity becomes ?? % of its value at the surface is (Radius of the earth = ?? ) (a) ?? ?? (b) ?? ?? (c) ???? ?? (d) ???? ?? Ans : (b) Acceleration due to gravity at height h is given by ?? ' = ?? ( ?? ?? +h ) 2 ? ?? 100 = ?? ( ?? ?? + h ) 2 ? ?? ?? + h = 1 10 ? h = 9?? . Q6. At surface of earth weight of a person is ???? ?? then his weight at height ?? /?? from surface of earth is ( ?? = radius of earth) (a) ???? ?? (b) ???? ?? (c) ???? ?? (d) ???? ?? Ans : (c) Weight of the body at height ?? , ?? ' = ?? ( ?? ?? +h ) 2 = ?? ( ?? ?? + ?? 2 ) 2 = ?? ( 2 3 ) 2 = 4 9 ?? = 4 9 × 72 = 32?? . Q7. Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine ?????? ???? below the earth's surface (Given ?? = ???????? ???? ) (a) ?? .???? ?? /?? ?? (b) ?? .???? ?? /?? ?? (c) ?? .???? ?? /?? ?? (d) ?? .???? ?? /?? ?? Ans: (a) Acceleration due to gravity at depth ?? , ?? ' = ?? [1- ?? ?? ] = ?? [1- 100 6400 ] = 9.8[1- 1 64 ] = 9.8 × 63 64 = 9.66 m/s 2 . Q8. The depth ?? at which the value of acceleration due to gravity becomes ?? ?? times the value at the surface, is [?? = radius of the earth ] (a) ?? ?? (b) ?? ( ?? -?? ?? ) (c) ?? ?? ?? (d) ?? ( ?? ?? +?? ) Ans : (b) ?? ' = ?? (1- ?? ?? ) ? ?? ?? = ?? (1- ?? ?? ) ? ?? ?? = 1 - 1 ?? ? ?? = ( ?? -1 ?? )?? Q9. The angular velocity of the earth with which it has to rotate so that acceleration due to gravity on ???? ° latitude becomes zero is (Radius of earth = ???????? ???? . At the poles ?? = ???? ???? - ?? ) (a) ?? .?? × ???? -?? ?????? /?????? (b) ?? .?? × ???? -?? ?????? /?????? (c) ???? × ???? ?? ?????? /?????? (d) ?? .?? × ???? -?? ?????? /?????? Ans : (a) Effective acceleration due to gravity due to rotation of earth ?? ' = ?? - ?? 2 ?? cos 2 ?? ? 0 = ?? - ?? 2 ?? cos 2 60 ° ? ?? 2 ?? 4 = ?? ? ?? = v 4?? ?? = 2v ?? ?? = 2 800 rad sec [ As ?? ' = 0 and ?? = 60 ° ] ? ?? = 1 400 = 2.5× 10 -3 rad sec . [As ?? ' = 0 and ?? = 60 ° ] Q10. The gravitational potential in a region is given by ?? = (?? ?? + ?? ?? + ???? ?? )?? /???? . The modulus of the gravitational field at (?? = ?? ,?? = ?? ,?? = ?? ) is (a) ???? ?????? -?? (b) ???? ?????? -?? (c) ???? ?????? -?? (d) ?? ?????? -?? Ans : (b) ?? = -( ??? ??? ??ˆ + ??? ??? ??ˆ + ??? ??? ?? ˆ ) = -(3??ˆ + 4??ˆ + 12?? ˆ ) [As ?? = (3?? + 4?? + 12?? ) (given)] It is uniform field Hence its value is same every where |?? | = v3 2 + 4 2 + 12 2 = 13Nkg -1 . Q11. Two concentric shells of mass ?? ?? and ?? ?? are having radii ?? ?? and ?? ?? . Which of the following is the correct expression for the gravitational field on a mass ?? . (a) ?? = ?? (?? ?? +?? ?? ) ?? ?? for ?? < ?? ?? (b) ?? = ?? (?? ?? +?? ?? ) ?? ?? for ?? < ?? ?? (c) ?? = ?? ?? ?? ?? ?? for ?? ?? < ?? < ?? ?? (d) ?? = ?? ?? ?? ?? ?? for ?? ?? < ?? < ?? ?? Ans: (d) Gravitational field on a mass ?? due to outer shell (radius ?? 2 ) will be zero because the mass is placed inside this shell. But the inner shell (radius ?? 1 ) behaves like point mass placed at the centre so ?? = ?? ?? 1 ?? 2 for ?? 1 < ?? < ?? 2 Q12 A uniform ring of mass ?? is lying at a distance 1.73 a from the centre of a sphere of mass ?? just over the sphere where ?? is the small radius of the ring as well as that of the sphere. Then gravitational force exerted is (a) ?????? ?? ?? ?? (b) ?????? (?? .???? ?? ) ?? Page 5 Solved Examples on Gravitation JEE Mains Single Correct Type Q1. Three identical point masses, each of mass ?? ???? lie in the ?? - ?? plane at points (?? ,?? ),(?? , ?? .?? ?? ) and (?? .?? ?? ,?? ) . The net gravitational force on the mass at the origin is (a) ?? .???? × ???? -?? (??ˆ + ??ˆ)?? (b) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? (c) ?? .???? × ???? -?? (??ˆ - ??ˆ)?? (d) ?? .???? × ???? -???? (??ˆ + ??ˆ)?? Ans: (a) Let particle ?? lies at origin, particle ?? and ?? on ?? and ?? -axis respectively ?? ???? ?????? = ?? ?? ?? ?? ?? ?? ???? 2 ??ˆ = 6.67× 10 -11 × 1× 1 (0.2) 2 ??ˆ = 1.67× 10 -9 ??ˆ?? Similarly ?? ???? ?????? = 1.67× 10 -9 ??ˆ N ? Net force on particle ?? ?? = ?? ???? + ?? ???? = 1.67× 10 -9 (??ˆ + ??ˆ)?? Q2. Acceleration due to gravity on moon is ?? /?? of the acceleration due to gravity on earth. If the ratio of densities of earth (?? ?? ) and moon (?? ?? ) is ( ?? ?? ?? ?? ) = ?? ?? then radius of moon ?? ?? in terms of ?? ?? will be (a) ?? ???? ?? ?? (b) ?? ?? ?? ?? (c) ?? ???? ?? ?? (d) ?? ?? v?? ?? ?? Ans: (a) Acceleration due to gravity ?? = 4 3 ???????? ? ?? ? ???? or ?? ?? ?? ?? = ?? ?? ?? ?? · ?? ?? ?? ?? [ As ?? ?? ?? ?? = 1 6 and ?? ?? ?? ?? = 5 3 (given)] ? ?? ?? ?? ?? = ( ?? ?? ?? ?? )( ?? ?? ?? ?? ) = 1 6 × 5 3 ? ?? ?? = 5 18 ?? ?? Q3. The moon's radius is ?? /?? that of the earth and its mass is ?? /???? times that of the earth. If ?? represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? ?? Ans: (b) Acceleration due to gravity ?? = ???? ?? 2 ? ?? moon ?? earth = ?? moon ?? earth · ?? earth 2 ?? moon 2 = ( 1 80 )( 4 1 ) 2 ?? moon = ?? earth × 16 80 = ?? 5 . Q4. The radii of two planets are respectively ?? ?? and ?? ?? and their densities are respectively ?? ?? and ?? ?? . The ratio of the accelerations due to gravity at their surfaces is (a) ?? ?? :?? ?? = ?? ?? ?? ?? ?? : ?? ?? ?? ?? ?? (b) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (c) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? (d) ?? ?? :?? ?? = ?? ?? ?? ?? :?? ?? ?? ?? Ans: (d) Acceleration due to gravity ?? = 4 3 ???????? ? ?? 1 :?? 2 = ?? 1 ?? 1 :?? 2 ?? 2 . Q5. The height of the point vertically above the earth's surface, at which acceleration due to gravity becomes ?? % of its value at the surface is (Radius of the earth = ?? ) (a) ?? ?? (b) ?? ?? (c) ???? ?? (d) ???? ?? Ans : (b) Acceleration due to gravity at height h is given by ?? ' = ?? ( ?? ?? +h ) 2 ? ?? 100 = ?? ( ?? ?? + h ) 2 ? ?? ?? + h = 1 10 ? h = 9?? . Q6. At surface of earth weight of a person is ???? ?? then his weight at height ?? /?? from surface of earth is ( ?? = radius of earth) (a) ???? ?? (b) ???? ?? (c) ???? ?? (d) ???? ?? Ans : (c) Weight of the body at height ?? , ?? ' = ?? ( ?? ?? +h ) 2 = ?? ( ?? ?? + ?? 2 ) 2 = ?? ( 2 3 ) 2 = 4 9 ?? = 4 9 × 72 = 32?? . Q7. Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine ?????? ???? below the earth's surface (Given ?? = ???????? ???? ) (a) ?? .???? ?? /?? ?? (b) ?? .???? ?? /?? ?? (c) ?? .???? ?? /?? ?? (d) ?? .???? ?? /?? ?? Ans: (a) Acceleration due to gravity at depth ?? , ?? ' = ?? [1- ?? ?? ] = ?? [1- 100 6400 ] = 9.8[1- 1 64 ] = 9.8 × 63 64 = 9.66 m/s 2 . Q8. The depth ?? at which the value of acceleration due to gravity becomes ?? ?? times the value at the surface, is [?? = radius of the earth ] (a) ?? ?? (b) ?? ( ?? -?? ?? ) (c) ?? ?? ?? (d) ?? ( ?? ?? +?? ) Ans : (b) ?? ' = ?? (1- ?? ?? ) ? ?? ?? = ?? (1- ?? ?? ) ? ?? ?? = 1 - 1 ?? ? ?? = ( ?? -1 ?? )?? Q9. The angular velocity of the earth with which it has to rotate so that acceleration due to gravity on ???? ° latitude becomes zero is (Radius of earth = ???????? ???? . At the poles ?? = ???? ???? - ?? ) (a) ?? .?? × ???? -?? ?????? /?????? (b) ?? .?? × ???? -?? ?????? /?????? (c) ???? × ???? ?? ?????? /?????? (d) ?? .?? × ???? -?? ?????? /?????? Ans : (a) Effective acceleration due to gravity due to rotation of earth ?? ' = ?? - ?? 2 ?? cos 2 ?? ? 0 = ?? - ?? 2 ?? cos 2 60 ° ? ?? 2 ?? 4 = ?? ? ?? = v 4?? ?? = 2v ?? ?? = 2 800 rad sec [ As ?? ' = 0 and ?? = 60 ° ] ? ?? = 1 400 = 2.5× 10 -3 rad sec . [As ?? ' = 0 and ?? = 60 ° ] Q10. The gravitational potential in a region is given by ?? = (?? ?? + ?? ?? + ???? ?? )?? /???? . The modulus of the gravitational field at (?? = ?? ,?? = ?? ,?? = ?? ) is (a) ???? ?????? -?? (b) ???? ?????? -?? (c) ???? ?????? -?? (d) ?? ?????? -?? Ans : (b) ?? = -( ??? ??? ??ˆ + ??? ??? ??ˆ + ??? ??? ?? ˆ ) = -(3??ˆ + 4??ˆ + 12?? ˆ ) [As ?? = (3?? + 4?? + 12?? ) (given)] It is uniform field Hence its value is same every where |?? | = v3 2 + 4 2 + 12 2 = 13Nkg -1 . Q11. Two concentric shells of mass ?? ?? and ?? ?? are having radii ?? ?? and ?? ?? . Which of the following is the correct expression for the gravitational field on a mass ?? . (a) ?? = ?? (?? ?? +?? ?? ) ?? ?? for ?? < ?? ?? (b) ?? = ?? (?? ?? +?? ?? ) ?? ?? for ?? < ?? ?? (c) ?? = ?? ?? ?? ?? ?? for ?? ?? < ?? < ?? ?? (d) ?? = ?? ?? ?? ?? ?? for ?? ?? < ?? < ?? ?? Ans: (d) Gravitational field on a mass ?? due to outer shell (radius ?? 2 ) will be zero because the mass is placed inside this shell. But the inner shell (radius ?? 1 ) behaves like point mass placed at the centre so ?? = ?? ?? 1 ?? 2 for ?? 1 < ?? < ?? 2 Q12 A uniform ring of mass ?? is lying at a distance 1.73 a from the centre of a sphere of mass ?? just over the sphere where ?? is the small radius of the ring as well as that of the sphere. Then gravitational force exerted is (a) ?????? ?? ?? ?? (b) ?????? (?? .???? ?? ) ?? (c) v?? ?????? ?? ?? (d) ?? .???? ?????? ?? ?? ?? Ans: (d) Intensity due to uniform circular ring at a point on its axis ?? = ?????? (?? 2 +?? 2 ) 3/2 ? Force on sphere ?? = ???????? (?? 2 +?? 2 ) 3/2 = ?????? v3?? (?? 2 +(v3?? ) 2 ) 3/2 = ?????? v3?? (4?? 2 ) 3/2 = v3?????? 8?? 2 [As ?? = v3?? ] Q13 The gravitational field due to a mass distribution is ?? = ?? /?? ?? in the ?? -direction ( ?? is a constant). Taking the gravitational potential to be zero at infinity, its value at a distance ?? is (a) ?? /?? (b) ?? /?? ?? (c) ?? /?? ?? (d) ?? /?? ?? ?? Ans: (d) ?? = -? ?????? = -? ?? ?? 3 ???? = ?? 2?? 2 . Q14. Two bodies of masses ?? and ?? are placed a distance ?? apart. The gravitational potential at the position where the gravitational field due to them is zero is ?? , then (a) ?? = - ?? ?? (?? + ?? ) (b) ?? = - ???? ?? (c) ?? = - ???? ?? (d) ?? = - ?? ?? (v?? + v?? ) ?? Ans: (d) If ?? is the point of zero intensity, then ?? = v?? v?? +v?? · ?? and ?? - ?? = v?? v?? +v?? ?? Now potential at point ?? ,?? = ?? 1 + ?? 2 = - ???? ?? - ???? ?? -?? Substituting the value of ?? and ?? - ?? we get ?? = - ?? ?? (v?? + v?? ) 2 . Q15. A body of mass ?? is taken from earth surface to the height ?? equal to radius of earth, the increase in potential energy will beRead More
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FAQs on Gravitation Solved Examples - Physics for JEE Main & Advanced
| 1. What is the formula for gravitational force? |  |
Ans. The formula for gravitational force is given by Newton's law of universal gravitation: F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
| 2. How does the mass of an object affect the gravitational force? | |
Ans. The gravitational force between two objects is directly proportional to the product of their masses. This means that as the mass of an object increases, the gravitational force it exerts also increases.
| 3. How does the distance between two objects affect the gravitational force? | |
Ans. The gravitational force between two objects is inversely proportional to the square of the distance between their centers. This means that as the distance between two objects increases, the gravitational force between them decreases.
| 4. What is the difference between weight and mass in the context of gravitation? | |
Ans. Mass is a measure of the amount of matter in an object, while weight is the force of gravity acting on an object's mass. Mass is constant, while weight can vary depending on the strength of the gravitational field.
| 5. How does gravitational force vary with altitude on Earth? | |
Ans. Gravitational force decreases with altitude on Earth because the distance between an object and the center of the Earth increases as altitude increases. This decrease in distance causes a decrease in gravitational force.
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Gravitation Solved Examples Notes
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The content of Gravitation Solved Examples is prepared as per the latest JEE syllabus.
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