Hydrostatics | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
HYDROSTATICS 
Equilibrium of Fluids under Pressure 
Ideal fluid exerts a force normal to the area with which it is in contact. This force is called 
the (fluid) thrust on the area. Thrust per unit area is pressure. 
The pressure at a point inside a liquid at res is same in all directions. 
A fluid is at rest under the action of given forces. Let ?? ,?? and ?? be the components 
along ???? ,???? and ???? of the resultant force on unit mass of the fluid at any point. Then if 
?? denotes pressure 
???? =?? (?????? +?????? +?????? ) 
?? being the density of fluid. 
in the case of cylindrical polar coordinates 
???? =?? [?????? +???????? +?????? ] 
where ?? ,?? and ?? denote components of the force on unit mass in the directions of 
increasing ?? increasing ?? and increasing ?? respectively. 
In the case of spherical polar coordinates 
???? =?? [?????? +???????? +?? ·
sin ???????? ] 
where R,N and ?? denote components of the force per unit mass in the direction of 
increasing ?? , increasing ???
 and increasing ?? respectively. 
The necessary and sufficient condition of equilibrium for a fluid acted on by forces is 
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )=0 
 
In the case of homogeneous liquids. ?? is constant, the above condition becomes 
??? ??? =
??? ??? ·
??? ??? =
??? ??? and 
??? ??? =
??? ??? 
For heterogeneous liquid the above condition becomes 
?(???? )
??? =
?(???? )
??? ,
?(???? )
??? =
?(???? )
??? and 
?(???? )
??? =
?(???? )
??? 
The surfaces of equal pressure are given by 
Page 2


Edurev123 
HYDROSTATICS 
Equilibrium of Fluids under Pressure 
Ideal fluid exerts a force normal to the area with which it is in contact. This force is called 
the (fluid) thrust on the area. Thrust per unit area is pressure. 
The pressure at a point inside a liquid at res is same in all directions. 
A fluid is at rest under the action of given forces. Let ?? ,?? and ?? be the components 
along ???? ,???? and ???? of the resultant force on unit mass of the fluid at any point. Then if 
?? denotes pressure 
???? =?? (?????? +?????? +?????? ) 
?? being the density of fluid. 
in the case of cylindrical polar coordinates 
???? =?? [?????? +???????? +?????? ] 
where ?? ,?? and ?? denote components of the force on unit mass in the directions of 
increasing ?? increasing ?? and increasing ?? respectively. 
In the case of spherical polar coordinates 
???? =?? [?????? +???????? +?? ·
sin ???????? ] 
where R,N and ?? denote components of the force per unit mass in the direction of 
increasing ?? , increasing ???
 and increasing ?? respectively. 
The necessary and sufficient condition of equilibrium for a fluid acted on by forces is 
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )=0 
 
In the case of homogeneous liquids. ?? is constant, the above condition becomes 
??? ??? =
??? ??? ·
??? ??? =
??? ??? and 
??? ??? =
??? ??? 
For heterogeneous liquid the above condition becomes 
?(???? )
??? =
?(???? )
??? ,
?(???? )
??? =
?(???? )
??? and 
?(???? )
??? =
?(???? )
??? 
The surfaces of equal pressure are given by 
?????? +?????? +?????? =0 
The surfaces of equal density are 
??? ??? ???? +
??? ??? ???? +
??? ??? ???? =0 
The curves of equal pressure and equal density are given by 
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
Note 1: In all cases in which equilibrium of the fluid is possible, by integrating the 
pressure equation. 
?? =?? (?? ,?? ,?? ) 
If ?? = constant, the surface of equal pressure is ?? .(?? ,?? ;?? )= constant ...(1) 
If the external pressure be zero, the free surface is ?? (?? ,?? ,?? )=0 
Direction cosines of the normal at (?? ,?? ,?? ) of surface (1) are 
??? ??? ,
??? ??? ,
??? ??? i.e., ???? ,???? ,?? ?? 1
  i.e., ?? ,?? ,?? .  
Hence the resultant external force at any point cuts the surface of equal pressure 
passing through the point orthogonally. 
Note 2: If the force system is conservative ? ?????? is a perfect differentia equal to -???? , 
where ?? is the poten tial of the system. 
???? =-?????? . 
As the first member is an exact differential ?? must be a function of ?? . Hence ?? and 
therefore ?? is a function of ?? . Surfaces of equal pressure are equi-potential! surfaces and 
are also surfaces of equal density. 
Example 1 
If forces per unit mass at the point (?? ,?? ,?? ) Jarailel to the axes are ?? (?? -?? ),?? (?? -?? ),???? , 
:How that the surfaces of equal pressure are hyperbolic paraboloids and the curves of 
equal pressure and density are rectangular hyperbolas.      
  (1979) 
?? =?? (?? -?? ),?? =?? (?? -?? ) ?????? ?? =???? 
Surfaces of equal pressure are given by 
Page 3


Edurev123 
HYDROSTATICS 
Equilibrium of Fluids under Pressure 
Ideal fluid exerts a force normal to the area with which it is in contact. This force is called 
the (fluid) thrust on the area. Thrust per unit area is pressure. 
The pressure at a point inside a liquid at res is same in all directions. 
A fluid is at rest under the action of given forces. Let ?? ,?? and ?? be the components 
along ???? ,???? and ???? of the resultant force on unit mass of the fluid at any point. Then if 
?? denotes pressure 
???? =?? (?????? +?????? +?????? ) 
?? being the density of fluid. 
in the case of cylindrical polar coordinates 
???? =?? [?????? +???????? +?????? ] 
where ?? ,?? and ?? denote components of the force on unit mass in the directions of 
increasing ?? increasing ?? and increasing ?? respectively. 
In the case of spherical polar coordinates 
???? =?? [?????? +???????? +?? ·
sin ???????? ] 
where R,N and ?? denote components of the force per unit mass in the direction of 
increasing ?? , increasing ???
 and increasing ?? respectively. 
The necessary and sufficient condition of equilibrium for a fluid acted on by forces is 
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )=0 
 
In the case of homogeneous liquids. ?? is constant, the above condition becomes 
??? ??? =
??? ??? ·
??? ??? =
??? ??? and 
??? ??? =
??? ??? 
For heterogeneous liquid the above condition becomes 
?(???? )
??? =
?(???? )
??? ,
?(???? )
??? =
?(???? )
??? and 
?(???? )
??? =
?(???? )
??? 
The surfaces of equal pressure are given by 
?????? +?????? +?????? =0 
The surfaces of equal density are 
??? ??? ???? +
??? ??? ???? +
??? ??? ???? =0 
The curves of equal pressure and equal density are given by 
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
Note 1: In all cases in which equilibrium of the fluid is possible, by integrating the 
pressure equation. 
?? =?? (?? ,?? ,?? ) 
If ?? = constant, the surface of equal pressure is ?? .(?? ,?? ;?? )= constant ...(1) 
If the external pressure be zero, the free surface is ?? (?? ,?? ,?? )=0 
Direction cosines of the normal at (?? ,?? ,?? ) of surface (1) are 
??? ??? ,
??? ??? ,
??? ??? i.e., ???? ,???? ,?? ?? 1
  i.e., ?? ,?? ,?? .  
Hence the resultant external force at any point cuts the surface of equal pressure 
passing through the point orthogonally. 
Note 2: If the force system is conservative ? ?????? is a perfect differentia equal to -???? , 
where ?? is the poten tial of the system. 
???? =-?????? . 
As the first member is an exact differential ?? must be a function of ?? . Hence ?? and 
therefore ?? is a function of ?? . Surfaces of equal pressure are equi-potential! surfaces and 
are also surfaces of equal density. 
Example 1 
If forces per unit mass at the point (?? ,?? ,?? ) Jarailel to the axes are ?? (?? -?? ),?? (?? -?? ),???? , 
:How that the surfaces of equal pressure are hyperbolic paraboloids and the curves of 
equal pressure and density are rectangular hyperbolas.      
  (1979) 
?? =?? (?? -?? ),?? =?? (?? -?? ) ?????? ?? =???? 
Surfaces of equal pressure are given by 
?? (?? -?? )???? +?? (?? -?? )?? ?? +???????? =0 or 
?????? +?????? ????
+
????
?? -?? =0 
Integrating, 
????
?? -?? = constant  
which are hyperbolic paraboloids. 
Note: The condition for 
equilibrium
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? ) +?? (
??? ??? -
??? ??? )=0 is satisfied.
 
 
The curves of equal pressure and equal density are given by, 
??? ??? ??? -
????
??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
????
-2?? =
????
2?? =
????
0
 
or ???? = constant and ?? = constant which are rectangular hyperbola. 
 
Example 2 
A liquid of given volume ?? is at rest under the forces per unit mass ?? =-
?? ???
?? 2
;?? =-
????
?? 2
 
and ?? =-
????
?? 2
. Find the pressure at any point of the liquid and the surfaces of equal 
pressure. Solution 
???? =?? (?????? +?????? +?????? )=-???? (
?????? ?? 2
+
?????? ?? 2
+
?????? ?? 2
) or ?? =?? -
?? 2
?? (
?? 2
?? 2
+
?? 2
?? 2
+
?? 1
2
?? 2
) 
 
The free surface is given by ?? =0i.e. 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=
2?? ????
 
The volume of this ellipsoid is 
4?? 3
abc(
2c
?? p
)H
3/2
v  
&??? =
????
2
(
3?? 4???????? )
2/3
 and     ?? =
????
2
[(
3?? 4???????? )
3/2
-(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)] 
Surface of equal pressure ?? is given by 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(
3?? 4???? ????
)
3/2
-
2?? 11?? 
Page 4


Edurev123 
HYDROSTATICS 
Equilibrium of Fluids under Pressure 
Ideal fluid exerts a force normal to the area with which it is in contact. This force is called 
the (fluid) thrust on the area. Thrust per unit area is pressure. 
The pressure at a point inside a liquid at res is same in all directions. 
A fluid is at rest under the action of given forces. Let ?? ,?? and ?? be the components 
along ???? ,???? and ???? of the resultant force on unit mass of the fluid at any point. Then if 
?? denotes pressure 
???? =?? (?????? +?????? +?????? ) 
?? being the density of fluid. 
in the case of cylindrical polar coordinates 
???? =?? [?????? +???????? +?????? ] 
where ?? ,?? and ?? denote components of the force on unit mass in the directions of 
increasing ?? increasing ?? and increasing ?? respectively. 
In the case of spherical polar coordinates 
???? =?? [?????? +???????? +?? ·
sin ???????? ] 
where R,N and ?? denote components of the force per unit mass in the direction of 
increasing ?? , increasing ???
 and increasing ?? respectively. 
The necessary and sufficient condition of equilibrium for a fluid acted on by forces is 
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )=0 
 
In the case of homogeneous liquids. ?? is constant, the above condition becomes 
??? ??? =
??? ??? ·
??? ??? =
??? ??? and 
??? ??? =
??? ??? 
For heterogeneous liquid the above condition becomes 
?(???? )
??? =
?(???? )
??? ,
?(???? )
??? =
?(???? )
??? and 
?(???? )
??? =
?(???? )
??? 
The surfaces of equal pressure are given by 
?????? +?????? +?????? =0 
The surfaces of equal density are 
??? ??? ???? +
??? ??? ???? +
??? ??? ???? =0 
The curves of equal pressure and equal density are given by 
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
Note 1: In all cases in which equilibrium of the fluid is possible, by integrating the 
pressure equation. 
?? =?? (?? ,?? ,?? ) 
If ?? = constant, the surface of equal pressure is ?? .(?? ,?? ;?? )= constant ...(1) 
If the external pressure be zero, the free surface is ?? (?? ,?? ,?? )=0 
Direction cosines of the normal at (?? ,?? ,?? ) of surface (1) are 
??? ??? ,
??? ??? ,
??? ??? i.e., ???? ,???? ,?? ?? 1
  i.e., ?? ,?? ,?? .  
Hence the resultant external force at any point cuts the surface of equal pressure 
passing through the point orthogonally. 
Note 2: If the force system is conservative ? ?????? is a perfect differentia equal to -???? , 
where ?? is the poten tial of the system. 
???? =-?????? . 
As the first member is an exact differential ?? must be a function of ?? . Hence ?? and 
therefore ?? is a function of ?? . Surfaces of equal pressure are equi-potential! surfaces and 
are also surfaces of equal density. 
Example 1 
If forces per unit mass at the point (?? ,?? ,?? ) Jarailel to the axes are ?? (?? -?? ),?? (?? -?? ),???? , 
:How that the surfaces of equal pressure are hyperbolic paraboloids and the curves of 
equal pressure and density are rectangular hyperbolas.      
  (1979) 
?? =?? (?? -?? ),?? =?? (?? -?? ) ?????? ?? =???? 
Surfaces of equal pressure are given by 
?? (?? -?? )???? +?? (?? -?? )?? ?? +???????? =0 or 
?????? +?????? ????
+
????
?? -?? =0 
Integrating, 
????
?? -?? = constant  
which are hyperbolic paraboloids. 
Note: The condition for 
equilibrium
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? ) +?? (
??? ??? -
??? ??? )=0 is satisfied.
 
 
The curves of equal pressure and equal density are given by, 
??? ??? ??? -
????
??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
????
-2?? =
????
2?? =
????
0
 
or ???? = constant and ?? = constant which are rectangular hyperbola. 
 
Example 2 
A liquid of given volume ?? is at rest under the forces per unit mass ?? =-
?? ???
?? 2
;?? =-
????
?? 2
 
and ?? =-
????
?? 2
. Find the pressure at any point of the liquid and the surfaces of equal 
pressure. Solution 
???? =?? (?????? +?????? +?????? )=-???? (
?????? ?? 2
+
?????? ?? 2
+
?????? ?? 2
) or ?? =?? -
?? 2
?? (
?? 2
?? 2
+
?? 2
?? 2
+
?? 1
2
?? 2
) 
 
The free surface is given by ?? =0i.e. 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=
2?? ????
 
The volume of this ellipsoid is 
4?? 3
abc(
2c
?? p
)H
3/2
v  
&??? =
????
2
(
3?? 4???????? )
2/3
 and     ?? =
????
2
[(
3?? 4???????? )
3/2
-(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)] 
Surface of equal pressure ?? is given by 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(
3?? 4???? ????
)
3/2
-
2?? 11?? 
 
Example 3 
A mass ?? of gas at uniform temperature is diffused through all space, and at each point 
(?? ,?? ,?? ) the components of the force per unit mass are -?? ?? 1
-?? ?? 1
-???? . The pressure 
and density at the origin are ?? 0
 and ?? 0
. Prove that 
?????? ?? 0
?? 2
=8?? 3
?? 0
3
 
 
By Boyle's Law, ?? =????  … (2) 
As ?? =?? 0
 when ?? =?? 0
 ,?? =
?? 0
?? 0
 
The pressure equation is ???? =?? (-???????? -???????? -???????? ) 
?
?????? ?? =-???????? -???????? -???????? 
Integrating, ?? log ?? =-
1
2
? ?? ?? 2
+?? 1
 
At the origin ?? =?? 0
     ?K log?? 0
=?? 1
 
 
Hence log 
?? ?? 0
=
1
2?? ? ?? ?? 2
 
??? =?? 0
e
-
1
2?? (?? ?? 2
+?? ?? 2
+?? ?? 2
)
(3)
 
?? =?
-8
8
??
-8
8
??
8
?????????????? 
=?? 0
?
-8
8
?e
-
?? ?? 2
2?? ???? ?
-8
8
??? -
B?? 2
2?? d?? ?
-8
8
??? -
?? ?? 2
2?? ???? 
=
?? 0
(?? )
3/2
v8?? 3
v??????
  as ?
-8
8
??? -?? ?? 2
???? 
=v
?? ?? (?? >0) 
??? 2
?????? =?? ?? 3
?? 0
2
?? 3
 
=
8?? 2
?? 0
3
?? 0
( using (2))  
 
Example 4 
Page 5


Edurev123 
HYDROSTATICS 
Equilibrium of Fluids under Pressure 
Ideal fluid exerts a force normal to the area with which it is in contact. This force is called 
the (fluid) thrust on the area. Thrust per unit area is pressure. 
The pressure at a point inside a liquid at res is same in all directions. 
A fluid is at rest under the action of given forces. Let ?? ,?? and ?? be the components 
along ???? ,???? and ???? of the resultant force on unit mass of the fluid at any point. Then if 
?? denotes pressure 
???? =?? (?????? +?????? +?????? ) 
?? being the density of fluid. 
in the case of cylindrical polar coordinates 
???? =?? [?????? +???????? +?????? ] 
where ?? ,?? and ?? denote components of the force on unit mass in the directions of 
increasing ?? increasing ?? and increasing ?? respectively. 
In the case of spherical polar coordinates 
???? =?? [?????? +???????? +?? ·
sin ???????? ] 
where R,N and ?? denote components of the force per unit mass in the direction of 
increasing ?? , increasing ???
 and increasing ?? respectively. 
The necessary and sufficient condition of equilibrium for a fluid acted on by forces is 
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? )=0 
 
In the case of homogeneous liquids. ?? is constant, the above condition becomes 
??? ??? =
??? ??? ·
??? ??? =
??? ??? and 
??? ??? =
??? ??? 
For heterogeneous liquid the above condition becomes 
?(???? )
??? =
?(???? )
??? ,
?(???? )
??? =
?(???? )
??? and 
?(???? )
??? =
?(???? )
??? 
The surfaces of equal pressure are given by 
?????? +?????? +?????? =0 
The surfaces of equal density are 
??? ??? ???? +
??? ??? ???? +
??? ??? ???? =0 
The curves of equal pressure and equal density are given by 
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
Note 1: In all cases in which equilibrium of the fluid is possible, by integrating the 
pressure equation. 
?? =?? (?? ,?? ,?? ) 
If ?? = constant, the surface of equal pressure is ?? .(?? ,?? ;?? )= constant ...(1) 
If the external pressure be zero, the free surface is ?? (?? ,?? ,?? )=0 
Direction cosines of the normal at (?? ,?? ,?? ) of surface (1) are 
??? ??? ,
??? ??? ,
??? ??? i.e., ???? ,???? ,?? ?? 1
  i.e., ?? ,?? ,?? .  
Hence the resultant external force at any point cuts the surface of equal pressure 
passing through the point orthogonally. 
Note 2: If the force system is conservative ? ?????? is a perfect differentia equal to -???? , 
where ?? is the poten tial of the system. 
???? =-?????? . 
As the first member is an exact differential ?? must be a function of ?? . Hence ?? and 
therefore ?? is a function of ?? . Surfaces of equal pressure are equi-potential! surfaces and 
are also surfaces of equal density. 
Example 1 
If forces per unit mass at the point (?? ,?? ,?? ) Jarailel to the axes are ?? (?? -?? ),?? (?? -?? ),???? , 
:How that the surfaces of equal pressure are hyperbolic paraboloids and the curves of 
equal pressure and density are rectangular hyperbolas.      
  (1979) 
?? =?? (?? -?? ),?? =?? (?? -?? ) ?????? ?? =???? 
Surfaces of equal pressure are given by 
?? (?? -?? )???? +?? (?? -?? )?? ?? +???????? =0 or 
?????? +?????? ????
+
????
?? -?? =0 
Integrating, 
????
?? -?? = constant  
which are hyperbolic paraboloids. 
Note: The condition for 
equilibrium
?? (
??? ??? -
??? ??? )+?? (
??? ??? -
??? ??? ) +?? (
??? ??? -
??? ??? )=0 is satisfied.
 
 
The curves of equal pressure and equal density are given by, 
??? ??? ??? -
????
??? =
????
??? ??? -
??? ??? =
????
??? ??? -
??? ??? 
????
-2?? =
????
2?? =
????
0
 
or ???? = constant and ?? = constant which are rectangular hyperbola. 
 
Example 2 
A liquid of given volume ?? is at rest under the forces per unit mass ?? =-
?? ???
?? 2
;?? =-
????
?? 2
 
and ?? =-
????
?? 2
. Find the pressure at any point of the liquid and the surfaces of equal 
pressure. Solution 
???? =?? (?????? +?????? +?????? )=-???? (
?????? ?? 2
+
?????? ?? 2
+
?????? ?? 2
) or ?? =?? -
?? 2
?? (
?? 2
?? 2
+
?? 2
?? 2
+
?? 1
2
?? 2
) 
 
The free surface is given by ?? =0i.e. 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=
2?? ????
 
The volume of this ellipsoid is 
4?? 3
abc(
2c
?? p
)H
3/2
v  
&??? =
????
2
(
3?? 4???????? )
2/3
 and     ?? =
????
2
[(
3?? 4???????? )
3/2
-(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)] 
Surface of equal pressure ?? is given by 
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(
3?? 4???? ????
)
3/2
-
2?? 11?? 
 
Example 3 
A mass ?? of gas at uniform temperature is diffused through all space, and at each point 
(?? ,?? ,?? ) the components of the force per unit mass are -?? ?? 1
-?? ?? 1
-???? . The pressure 
and density at the origin are ?? 0
 and ?? 0
. Prove that 
?????? ?? 0
?? 2
=8?? 3
?? 0
3
 
 
By Boyle's Law, ?? =????  … (2) 
As ?? =?? 0
 when ?? =?? 0
 ,?? =
?? 0
?? 0
 
The pressure equation is ???? =?? (-???????? -???????? -???????? ) 
?
?????? ?? =-???????? -???????? -???????? 
Integrating, ?? log ?? =-
1
2
? ?? ?? 2
+?? 1
 
At the origin ?? =?? 0
     ?K log?? 0
=?? 1
 
 
Hence log 
?? ?? 0
=
1
2?? ? ?? ?? 2
 
??? =?? 0
e
-
1
2?? (?? ?? 2
+?? ?? 2
+?? ?? 2
)
(3)
 
?? =?
-8
8
??
-8
8
??
8
?????????????? 
=?? 0
?
-8
8
?e
-
?? ?? 2
2?? ???? ?
-8
8
??? -
B?? 2
2?? d?? ?
-8
8
??? -
?? ?? 2
2?? ???? 
=
?? 0
(?? )
3/2
v8?? 3
v??????
  as ?
-8
8
??? -?? ?? 2
???? 
=v
?? ?? (?? >0) 
??? 2
?????? =?? ?? 3
?? 0
2
?? 3
 
=
8?? 2
?? 0
3
?? 0
( using (2))  
 
Example 4 
A mass of elastic fluid is rotating about a vertical axis with uniform angular velocity ?? and 
is acted on by an attraction towards a poirt on that axis equal ?? times the distance 
(?? >0 
2
) . Prove that the surface of actual density ?? is 
?? (?? 2
+?? 2
+?? 2
)-?? (?? 2
+?? 2
)=?? log {
?? (?? -?? 2
)
2
8?? 3
?? 2
?? !
3
} 
 
The centre of attraction as the origin and vertical axis as oz. As the fluid is elastic, ?? by 
Boyle's law. 
???? =?? [-?? (?????? +?????? +?????? )+?? 2
(?????? +?????? )]
?? ?? ?? =-?? ? ???? ?? +?? 2
(?????? +?????? )
 
Integrating 2?? log ?? =-?? (?? 2
+?? 2
+?? 2
)+?? 2
(?? 2
+?? 2
)+?? 
?-?? (?? 2
+?? 2
+?? 2
)+1
2
(?? 2
+?? 2
)=?? log 
?? 2
?? 2
    … (1) 
Hence ?? =?? ?? -
(1-?? ?? 2
)?? 2
+1
2?? -?? 2
 where ?? 2
=(?? 2
+?? 2
) 
?? = the mass of filuid ?? vich pervades all space 
 =? ?
8
0
??? 8
2?????????????? =???? ? ?
8
0
??? -
?? ?? 2
2?? 2
???? ? ?
8
0
??? -
(?? -?? )
2
)?? 2
2?? ?????? =???? (
2?? v
?? v2?? )
2?? ?? -1
2
?? =
?? (1-?? 2
)
v
?? ?? 3/2
(2?? )
3/2
 
From (1), the surface of equal density  
?? (?? 2
+?? 2
+?? 2
)-?? 2
(?? 2
+?? 2
)=?? log {
?? (?? -?? 2
)
2
8?? 3
?? 2
?? 3
} 
 
Example 5 
A given mass of air is contained within a closed air tight cylinder with its axis vertical. 
The air is rotating in relative equilibrium about the axis of the cylinder. The pressure at 
the highest point of its axis is ?? and the pressure at the highest points of its curved 
surface is ?? . Prove that, if the fiuld were absolutely at rest, the pressure at the upper end 
of the axis would be 
?? -?? log ?? -log ?? '
 where the weight of the air is taken into account.