31. A satellite of mass m is orbiting around the earth in a circular orbit with a velocity v. What will be its total energy?​​
    

1. 1. ​ (3/4) mv²

   
   
   

   2. (1/2) mv²

   
   
   

   3. mv²

   
   
   

   4. – (1/2) m v²

   
   
   

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   Total energy = – K E =	PE
   	2

   
   

   =	1	mv2
   	2

   
   

   Correct Option: D

   Total energy = – K E =	PE
   	2

   
   

   =	1	mv2
   	2

   
   

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32. The mean radius of earth is R, its angular speed on its own axis is ω and the acceleration due to gravity at earth's surface is g. What will be the radius of the orbit of a geostationary satellite ?

1. 1. (​R²g / ω​²)^{1 / 3}
      

   
   
   

   2. (​Rg / ω​²)^{1 / 3}
      

   
   
   

   3. (​R² ω​² / g)^{1 / 3}
      

   
   
   

   4. (​R²g / ω​)^{1 / 3}

   
   
   

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1. View Hint View Answer Discuss in Forum

   T =	2πr	=	2πr	=	2πr3 / 2	=	2π
   	v0		(gR2 / r)1 / 2		(gR2)1 / 2		ω

   
   

   Hence , r3 / 2 =	√gR²	or r3 =	gR2
   	ω		ω2

   
   or r = (gR² / ω²)^{1 / 3}

   Correct Option: A

   T =	2πr	=	2πr	=	2πr3 / 2	=	2π
   	v0		(gR2 / r)1 / 2		(gR2)1 / 2		ω

   
   

   Hence , r3 / 2 =	√gR²	or r3 =	gR2
   	ω		ω2

   
   or r = (gR² / ω²)^{1 / 3}

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33. The escape velocity from earth is 11.2 km/s. If a body is to be projected in a direction making an angle 45° to the vertical, then the escape velocity is
    

1. 1. ​11.2 × 2  km/s

   
   
   

   2. 11.2 km/s

   
   
   

   3. 11.2 / √2 km/s ​

   
   
   

   4. 11.2 √2 km/s

   
   
   

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1. View Hint View Answer Discuss in Forum

   Escape velocity does not depend on the angle of projection.

   Correct Option: B

   Escape velocity does not depend on the angle of projection.

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34. A satellite in force free space sweeps stationary interplanetary dust at a rate dM/dt = αv where M is the mass and v is the velocity of the satellite and α is a constant. What is the deceleration of the satellite?​​​
    

1. 1. - αv²

   
   
   

   2. ​​- αv² / 2 M

   
   
   

   3. ​​- αv² / M

   
   
   

   4. ​- 2αv² / M

   
   
   

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1. View Hint View Answer Discuss in Forum

   F =		dM		v = αv2		∵	dM	= αv
   		dt					dt

   
   

   ∴ Retardation =	-F	= -	αv2
   	M		M

   
   

   Correct Option: C

   F =		dM		v = αv2		∵	dM	= αv
   		dt					dt

   
   

   ∴ Retardation =	-F	= -	αv2
   	M		M

   
   

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35. The escape velocity from the surface of the earth is ve.The escape velocity from the surface of a planet whose mass and radius are three times those of the earth, will be
    

1. 1. ​v_e

   
   
   

   2. 3v_e

   
   
   

   3. 9v_e

   
   
   

   4. 1/ 3v_e

   
   
   

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1. View Hint View Answer Discuss in Forum

   Escape velocity on surface of earth
   

   (ve) = √	2GMe	∝ √	Me
   	Re		Re

   
   
   or, v_P= v_e.

   Correct Option: A

   Escape velocity on surface of earth
   

   (ve) = √	2GMe	∝ √	Me
   	Re		Re

   
   
   or, v_P= v_e.

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