21. A body of mass ‘m’ is taken from the earth’s surface to the height equal to twice the radius (R) of the earth. The change in potential energy of body will be

1. 1. 	2	mgR
      	3

   
   
   

   2. 3 mgR

   
   
   

   3. 	1	mgR
      	3

   
   
   

   4. mg²R

   
   
   

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1. View Hint View Answer Discuss in Forum

   Initial P. E., Ui =	-GMm
   	R

   
   

   Final P.E., Uf =	-GMm	[∵ R' = R + 2R = 3R]
   	3R

   
   

   ∴ Change in potential energy, ∆U =	-GMm	+	GMm
   	3R		R

   
   

   =	GMm			1 -	1		=	2	GMm	=	2	mgR
   	R				3			3	R		3

   
   

   	∵	GMm	= mgR
   		R

   
   

   ALTERNATE : ∆U =	mgh
   	1 +	h
   		R

   
   

   By placing the value of h = 2R we get ​∆U =	2	mgR
   	3

   
   

   Correct Option: A

   Initial P. E., Ui =	-GMm
   	R

   
   

   Final P.E., Uf =	-GMm	[∵ R' = R + 2R = 3R]
   	3R

   
   

   ∴ Change in potential energy, ∆U =	-GMm	+	GMm
   	3R		R

   
   

   =	GMm			1 -	1		=	2	GMm	=	2	mgR
   	R				3			3	R		3

   
   

   	∵	GMm	= mgR
   		R

   
   

   ALTERNATE : ∆U =	mgh
   	1 +	h
   		R

   
   

   By placing the value of h = 2R we get ​∆U =	2	mgR
   	3

   
   

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22. Infinite number of bodies, each of mass 2 kg are situated on x-axis at distances 1m, 2m, 4m, 8m, ..... respectively, from the origin. The resulting gravitational potential due to this system at the origin will be​​

1. 1. -	8	G
      	3

   
   
   

   2. -	4	G
      	3

   
   
   

   3. - 4 G
      

   
   
   

   4. - G

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   Gravitational potential V =	-Gm
   	r

   
   

   V0 = -	G × 2	-	G × 2	-	G × 2	-	G × 2
   	1		2		4		8

   
   

   = -2G		1 +	1	+	1	+	1	+	1	+ .......∞
   			1		2		4		8

   
   

   = -2G ×	1				= -2G ×	1				= -4 G
   	1 -	1					1
   		2					2

   Correct Option: C

   
   

   Gravitational potential V =	-Gm
   	r

   
   

   V0 = -	G × 2	-	G × 2	-	G × 2	-	G × 2
   	1		2		4		8

   
   

   = -2G		1 +	1	+	1	+	1	+	1	+ .......∞
   			1		2		4		8

   
   

   = -2G ×	1				= -2G ×	1				= -4 G
   	1 -	1					1
   		2					2

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23. Which one of the following plots represents the variation of gravitational field on a particle with distance r due to a thin spherical shell of radius R ? (r is measured from the centre of the spherical shell)
    

1. 1. 
      

   
   
   

   2. 
      

   
   
   

   3. 
      

   
   
   

   4. 

   
   
   

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1. View Hint View Answer Discuss in Forum

   The Gravitational field due to a thin spherical shell of radius R at distance r.
   

   E =	GM	(If r > R)
   	r2

   
   For   r = R i.e. on the surface of the shell
   

   E =	GM
   	R2

   
   For r < R  i.e. inside the shell ​E = 0
   

   Correct Option: B

   The Gravitational field due to a thin spherical shell of radius R at distance r.
   

   E =	GM	(If r > R)
   	r2

   
   For   r = R i.e. on the surface of the shell
   

   E =	GM
   	R2

   
   For r < R  i.e. inside the shell ​E = 0
   

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24. A particle of mass M is situated at the centre of spherical shell of mass M and radius a. The magnitude of the gravitational potential at a point situated at a/2 distance from the centre, will be​​​

1. 1. 	2GM
      	a

   
   
   

   2. 	3GM
      	a

   
   
   

   3. 	4GM
      	a

   
   
   

   4. 	GM
      	a

   
   
   

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1. View Hint View Answer Discuss in Forum

   V_P = V_{spherical shell} + V_particle
   

   =	GM	=	GM	=	3GM
   	a		a / 2		a

   
   

   Correct Option: B

   V_P = V_{spherical shell} + V_particle
   

   =	GM	=	GM	=	3GM
   	a		a / 2		a

   
   

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25. A particle of mass M is situated at the centre of a spherical shell of same mass and radius a.
    

    The gravitational potential at a point situated at	a	distance from the centre, will be :
    	2

1. 1. -	3GM
      	a

   
   
   

   2. -	2GM
      	a

   
   
   

   3. -	GM
      	a

   
   
   

   4. -	4GM
      	a

   
   
   

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1. View Hint View Answer Discuss in Forum

   Potential at the given point = Potential at the point due to the shell + Potential due to the particle
   

   = -	GM	-	2GM	= -	3GM
   	a		a		a

   Correct Option: A

   Potential at the given point = Potential at the point due to the shell + Potential due to the particle
   

   = -	GM	-	2GM	= -	3GM
   	a		a		a

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