Work and Potential Energy | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
6. Work and Potential Energy 
6.1 A heavy ring of mass ?? , slides on a smooth vertical rod and is attached to a 
light string which passes over a small pulley distant a from the rod and has a 
mass ?? (>?? ) fastened to its other end. Show that if the ring be dropped from a 
point in the rod in the same horizontal plane as the pulley, it will descend a 
distance 
?? ??????
?? ?? -?? ?? before coming to rest. 
(2012 : 20 Marks) 
Solution: 
 
By the law of conservation of energy, the ring will be in the rest position when loss in 
potential energy of ring is equal to the gain in potential energy of the block. 
Let the distance travelled by the ring before it comes to rest be ?? . 
Loss in potential energy of the ring =?????? (??? .?? =???? h) 
When the ring falls down by the distance ' ?? ', then the mass ?? will move up by length 
v?? 2
+?? 2
-?? . 
? Gain in P.E. of the mass ?? =???? (v?? 2
+?? 2
-?? ) 
?                             Loss in P.E. of the ring = Gain in P.E. of the block 
?                                                                    ?????? =???? (v?? 2
+?? 2
-?? ) 
?                                                                    
?? ?? ?? =v?? 2
+?? 2
-?? 
?                                                                          ?? =
2?????? ?? 2
-?? 2
 
Page 2


Edurev123 
6. Work and Potential Energy 
6.1 A heavy ring of mass ?? , slides on a smooth vertical rod and is attached to a 
light string which passes over a small pulley distant a from the rod and has a 
mass ?? (>?? ) fastened to its other end. Show that if the ring be dropped from a 
point in the rod in the same horizontal plane as the pulley, it will descend a 
distance 
?? ??????
?? ?? -?? ?? before coming to rest. 
(2012 : 20 Marks) 
Solution: 
 
By the law of conservation of energy, the ring will be in the rest position when loss in 
potential energy of ring is equal to the gain in potential energy of the block. 
Let the distance travelled by the ring before it comes to rest be ?? . 
Loss in potential energy of the ring =?????? (??? .?? =???? h) 
When the ring falls down by the distance ' ?? ', then the mass ?? will move up by length 
v?? 2
+?? 2
-?? . 
? Gain in P.E. of the mass ?? =???? (v?? 2
+?? 2
-?? ) 
?                             Loss in P.E. of the ring = Gain in P.E. of the block 
?                                                                    ?????? =???? (v?? 2
+?? 2
-?? ) 
?                                                                    
?? ?? ?? =v?? 2
+?? 2
-?? 
?                                                                          ?? =
2?????? ?? 2
-?? 2
 
6.2 A spherical shot of ?? ???? weight and radius ' ?? ' ???? , lies at the bottom of 
cylindrical bucket of radius ?? ???? . The bucket is filled with water upto a depth of ' ?? 
' ???? (?? >?? ?? ) . Show that the minimum amount of work done in lifing the shot just 
clear of the water must be [?? (?? -
?? ?? ?? ?? ?? ?? )+?? (?? -?? +
?? ?? ?? ?? ?? ?? )] ?????? .?? '
???? is the weight 
of water displaced by the shct. 
(2017 : 16 Marks) 
Solution: 
h is the final height of water when spherical solid is out of water. 
Volume of water + Volume of solid = Volume of cylinder with height h. 
?? ?? 2
h
'
+
4
3
?? ?? 3
 =?? ?? 2
h 
                                                                                          h
'
=h-
4?? 3
3?? 2
                                       (??)  
Change in potential energy of water 
 
??? ?? ?? =-?? '
[(
h+h
'
2
)-?? ]=-?? '
[h-
2?? 3
3?? 2
-?? ] 
(final centre of mass and initial centre of mass) 
Change in PE of solid, 
??? ?? ?? =?? (h
'
+?? )-?? ]=?? h
'
=?? [h-
4?? 3
3?? 3
] 
                            Work done =??? ?? ?? +??? ?? ?? 
 =?? [h-
4?? 3
3?? 2
]-?? '
[h-
2?? 3
3?? 2
-?? ]=?? [h-
4?? 3
3?? 2
]+?? '
[?? -h+
2?? 3
3?? 2
] 
6.3 The force of attraction of a particle by the earth is inversely proportional to the 
square of its distance from the earth's centre. A particle, whose weight on the 
Page 3


Edurev123 
6. Work and Potential Energy 
6.1 A heavy ring of mass ?? , slides on a smooth vertical rod and is attached to a 
light string which passes over a small pulley distant a from the rod and has a 
mass ?? (>?? ) fastened to its other end. Show that if the ring be dropped from a 
point in the rod in the same horizontal plane as the pulley, it will descend a 
distance 
?? ??????
?? ?? -?? ?? before coming to rest. 
(2012 : 20 Marks) 
Solution: 
 
By the law of conservation of energy, the ring will be in the rest position when loss in 
potential energy of ring is equal to the gain in potential energy of the block. 
Let the distance travelled by the ring before it comes to rest be ?? . 
Loss in potential energy of the ring =?????? (??? .?? =???? h) 
When the ring falls down by the distance ' ?? ', then the mass ?? will move up by length 
v?? 2
+?? 2
-?? . 
? Gain in P.E. of the mass ?? =???? (v?? 2
+?? 2
-?? ) 
?                             Loss in P.E. of the ring = Gain in P.E. of the block 
?                                                                    ?????? =???? (v?? 2
+?? 2
-?? ) 
?                                                                    
?? ?? ?? =v?? 2
+?? 2
-?? 
?                                                                          ?? =
2?????? ?? 2
-?? 2
 
6.2 A spherical shot of ?? ???? weight and radius ' ?? ' ???? , lies at the bottom of 
cylindrical bucket of radius ?? ???? . The bucket is filled with water upto a depth of ' ?? 
' ???? (?? >?? ?? ) . Show that the minimum amount of work done in lifing the shot just 
clear of the water must be [?? (?? -
?? ?? ?? ?? ?? ?? )+?? (?? -?? +
?? ?? ?? ?? ?? ?? )] ?????? .?? '
???? is the weight 
of water displaced by the shct. 
(2017 : 16 Marks) 
Solution: 
h is the final height of water when spherical solid is out of water. 
Volume of water + Volume of solid = Volume of cylinder with height h. 
?? ?? 2
h
'
+
4
3
?? ?? 3
 =?? ?? 2
h 
                                                                                          h
'
=h-
4?? 3
3?? 2
                                       (??)  
Change in potential energy of water 
 
??? ?? ?? =-?? '
[(
h+h
'
2
)-?? ]=-?? '
[h-
2?? 3
3?? 2
-?? ] 
(final centre of mass and initial centre of mass) 
Change in PE of solid, 
??? ?? ?? =?? (h
'
+?? )-?? ]=?? h
'
=?? [h-
4?? 3
3?? 3
] 
                            Work done =??? ?? ?? +??? ?? ?? 
 =?? [h-
4?? 3
3?? 2
]-?? '
[h-
2?? 3
3?? 2
-?? ]=?? [h-
4?? 3
3?? 2
]+?? '
[?? -h+
2?? 3
3?? 2
] 
6.3 The force of attraction of a particle by the earth is inversely proportional to the 
square of its distance from the earth's centre. A particle, whose weight on the 
surface of the earth is ?? , falls to the surface of the earth from a height ?? ?? above it. 
Show that the magnitude of work done by the earth's attraction force is 
?? ?? ???? , 
where ?? is the radius of the earth. 
(2019: 10 Marks) 
Solution: 
h =?? = radius of earth 
???? =h=?? ???? =3h
 
Hence, point ?? is at a distance of 4h from center of earth. The par cle at ?? has the 
weight on the surface of the earth is ?? , falls to the surface of the earth from a height 3h 
above it. 
The force of attraction of a particle by earth is inversely proportional to the square of its 
distance from earth's centers. 
 
?? 2
?? ?? ?? 2
 =
-?? ?? 2
. (?????????? )
?? 2
?? ?? ?? 2
 =-?? ;?? =h
?? =?? h
2
 
Only force acting on particle is gravity (i.e., conservative force. No fraction or drag in air 
and initial and final velocity is zero.) 
Page 4


Edurev123 
6. Work and Potential Energy 
6.1 A heavy ring of mass ?? , slides on a smooth vertical rod and is attached to a 
light string which passes over a small pulley distant a from the rod and has a 
mass ?? (>?? ) fastened to its other end. Show that if the ring be dropped from a 
point in the rod in the same horizontal plane as the pulley, it will descend a 
distance 
?? ??????
?? ?? -?? ?? before coming to rest. 
(2012 : 20 Marks) 
Solution: 
 
By the law of conservation of energy, the ring will be in the rest position when loss in 
potential energy of ring is equal to the gain in potential energy of the block. 
Let the distance travelled by the ring before it comes to rest be ?? . 
Loss in potential energy of the ring =?????? (??? .?? =???? h) 
When the ring falls down by the distance ' ?? ', then the mass ?? will move up by length 
v?? 2
+?? 2
-?? . 
? Gain in P.E. of the mass ?? =???? (v?? 2
+?? 2
-?? ) 
?                             Loss in P.E. of the ring = Gain in P.E. of the block 
?                                                                    ?????? =???? (v?? 2
+?? 2
-?? ) 
?                                                                    
?? ?? ?? =v?? 2
+?? 2
-?? 
?                                                                          ?? =
2?????? ?? 2
-?? 2
 
6.2 A spherical shot of ?? ???? weight and radius ' ?? ' ???? , lies at the bottom of 
cylindrical bucket of radius ?? ???? . The bucket is filled with water upto a depth of ' ?? 
' ???? (?? >?? ?? ) . Show that the minimum amount of work done in lifing the shot just 
clear of the water must be [?? (?? -
?? ?? ?? ?? ?? ?? )+?? (?? -?? +
?? ?? ?? ?? ?? ?? )] ?????? .?? '
???? is the weight 
of water displaced by the shct. 
(2017 : 16 Marks) 
Solution: 
h is the final height of water when spherical solid is out of water. 
Volume of water + Volume of solid = Volume of cylinder with height h. 
?? ?? 2
h
'
+
4
3
?? ?? 3
 =?? ?? 2
h 
                                                                                          h
'
=h-
4?? 3
3?? 2
                                       (??)  
Change in potential energy of water 
 
??? ?? ?? =-?? '
[(
h+h
'
2
)-?? ]=-?? '
[h-
2?? 3
3?? 2
-?? ] 
(final centre of mass and initial centre of mass) 
Change in PE of solid, 
??? ?? ?? =?? (h
'
+?? )-?? ]=?? h
'
=?? [h-
4?? 3
3?? 3
] 
                            Work done =??? ?? ?? +??? ?? ?? 
 =?? [h-
4?? 3
3?? 2
]-?? '
[h-
2?? 3
3?? 2
-?? ]=?? [h-
4?? 3
3?? 2
]+?? '
[?? -h+
2?? 3
3?? 2
] 
6.3 The force of attraction of a particle by the earth is inversely proportional to the 
square of its distance from the earth's centre. A particle, whose weight on the 
surface of the earth is ?? , falls to the surface of the earth from a height ?? ?? above it. 
Show that the magnitude of work done by the earth's attraction force is 
?? ?? ???? , 
where ?? is the radius of the earth. 
(2019: 10 Marks) 
Solution: 
h =?? = radius of earth 
???? =h=?? ???? =3h
 
Hence, point ?? is at a distance of 4h from center of earth. The par cle at ?? has the 
weight on the surface of the earth is ?? , falls to the surface of the earth from a height 3h 
above it. 
The force of attraction of a particle by earth is inversely proportional to the square of its 
distance from earth's centers. 
 
?? 2
?? ?? ?? 2
 =
-?? ?? 2
. (?????????? )
?? 2
?? ?? ?? 2
 =-?? ;?? =h
?? =?? h
2
 
Only force acting on particle is gravity (i.e., conservative force. No fraction or drag in air 
and initial and final velocity is zero.) 
 Change in K.E.  =?? (?? =
?? ?? )
?? conservative 
 =-? ?
h
4h
?
?? ?? h
·
?? ?? 2
????
 =?? h
2
(
1
?? )
4h
4
=?? h
2
[
1
h
-
1
4h
]
 =?? h
2
·
3
4h
=?? 3
4
h=
3
4
?? h
?? conservative 
 =
3
4
?? h. Hence the result.