A satellite of mass m revolves around the earth of radius R at a height x from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

  • fraction numerator gR over denominator straight R minus straight x end fraction
  • gx

  • fraction numerator gR squared over denominator straight R space plus straight x end fraction
  • fraction numerator gR squared over denominator straight R space plus straight x end fraction

D.

fraction numerator gR squared over denominator straight R space plus straight x end fraction

The gravitational force exerted on satellite at a height x is

straight F subscript straight G space equals space fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis squared end fraction
where Me = mass of earth Since, gravitational force provides the necessary centripetal force, so,

fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis squared end fraction space equals space fraction numerator mv subscript 0 superscript 2 over denominator left parenthesis straight R plus straight x right parenthesis end fraction
rightwards double arrow space fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2
rightwards double arrow space fraction numerator begin display style gR squared straight m end style over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2
rightwards double arrow space fraction numerator begin display style gR squared straight m end style over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2 space
open parentheses because space straight g space equals space fraction numerator begin display style GM subscript straight e end style over denominator straight R end fraction close parentheses
straight v subscript straight o space equals space square root of open square brackets fraction numerator gR squared over denominator left parenthesis straight R plus straight x right parenthesis end fraction close square brackets end root
space equals space open square brackets fraction numerator gR squared over denominator left parenthesis straight R plus straight x right parenthesis end fraction close square brackets to the power of 1 divided by 2 end exponent

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A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between
(you may take G = 6 . 67× 10-11 Nm2/ kg2)

  • 13.34 x 10-10 J

  • 3.33 x 10-10 J

  • 6.67 x10-9 J

  • 6.67 x10-9 J


D.

6.67 x10-9 J

increment straight W space equals space straight v subscript straight f minus straight v subscript straight i
space equals space 0 minus open square brackets fraction numerator negative GMm over denominator straight R end fraction close square brackets
straight w space equals space fraction numerator space 6.67 space straight x space 10 to the power of negative 11 end exponent space space straight x 1000 over denominator.1 end fraction space straight x 10 over 1000 space equals space 6.67 space straight x space 10 to the power of negative 10 space end exponent straight J

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A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?

  • moment of inertia

  • angular momentum

  • angular velocity

  • angular velocity


B.

angular momentum

In free space, neither acceleration due to gravity for external torque act on the rotating solid sphere. Therefore, taking the same mass of sphere if the radius is increased then a moment of inertia, rotational kinetic energy and angular velocity will change but according to the law of conservation of momentum, angular momentum will not change.

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The time period of an earth satellite in circular orbit is independent of

  • the mass of the satellite

  • radius of its orbit

  • both the mass and radius of the orbit

  • both the mass and radius of the orbit


A.

the mass of the satellite

Time period of satellite

straight T space equals space 2 straight pi space square root of fraction numerator left parenthesis straight R plus straight h right parenthesis cubed over denominator GM subscript straight e end fraction end root

where R + h = orbital radius of satellite,
Me = mass of earth Thus, time period does not depend on the mass of the satellite.

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If g is the acceleration due to gravity on the earth’s surface, the gain in the potential energy of object of mass m raised from the surface of the earth to a height equal to the radius R of the earth is

  • 2 mgR

  • mgR/2

  • mgR/4

  • mgR/4


B.

mgR/2

Gravitational potential energy of body on earth's surface

straight U space equals space minus space fraction numerator GM subscript straight e straight m over denominator straight R end fraction
At a height h from earth's surface, its value is

straight U subscript straight h space equals space minus space fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight h right parenthesis end fraction
space equals space fraction numerator GM subscript straight e straight m over denominator 2 straight R end fraction

where Me = mass of earth
m = mass of body
R = radius of earth

therefore comma space Gain space in space potential space energy space equals space straight U subscript straight h minus straight U
equals space fraction numerator negative space GM subscript straight e straight m over denominator 2 straight R end fraction minus open parentheses fraction numerator GM subscript straight e straight m over denominator straight R end fraction close parentheses
space equals space fraction numerator GM subscript straight e straight m over denominator 2 straight R end fraction space plus space fraction numerator GM subscript straight e straight m over denominator straight R end fraction
space equals space fraction numerator GM subscript straight e straight m over denominator 2 straight R end fraction
space equals space fraction numerator gR squared straight m over denominator 2 straight R end fraction space space open parentheses straight g space equals space GM subscript straight e over straight R close parentheses
space equals space 1 half space mgR

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