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Page 1
Edurev123
6. Source and Sink
6.1 Two sources each of strength ?? are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? is at the origin. Show that the streamlines are the curves :
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? )
where ?? is a variable parameter.
Show also that the fluid speed at any point is
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? , where ?? ?? ,?? ?? and ?? ?? are the
distances of the points from the sources and the sink.
(2009 : 12 Marks)
Solution:
The complex potential ?? at the any point ?? (?? ) is given by :
?? =-?? log?(?? +?? )-?? log?(?? -?? )+2?? log??? (1)
?? =-?? log??? 2
-?? 2
+?? log??? 2
?? =-?? [log?(?? +???? )
2
-?? 2
}-log?(?? +???? )
2
]
?? =?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )] (2)
Now, we know that
log?(?? +???? )=
1
2
log?(?? 2
+?? 2
)+??tan
-1
?
?? ??
Also,
?? =?? +????
where streamlines will be given by ?? .
Using the above formula and (2)
Page 2
Edurev123
6. Source and Sink
6.1 Two sources each of strength ?? are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? is at the origin. Show that the streamlines are the curves :
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? )
where ?? is a variable parameter.
Show also that the fluid speed at any point is
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? , where ?? ?? ,?? ?? and ?? ?? are the
distances of the points from the sources and the sink.
(2009 : 12 Marks)
Solution:
The complex potential ?? at the any point ?? (?? ) is given by :
?? =-?? log?(?? +?? )-?? log?(?? -?? )+2?? log??? (1)
?? =-?? log??? 2
-?? 2
+?? log??? 2
?? =-?? [log?(?? +???? )
2
-?? 2
}-log?(?? +???? )
2
]
?? =?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )] (2)
Now, we know that
log?(?? +???? )=
1
2
log?(?? 2
+?? 2
)+??tan
-1
?
?? ??
Also,
?? =?? +????
where streamlines will be given by ?? .
Using the above formula and (2)
?? =?? [tan
-1
?
2????
?? 2
-?? 2
-tan
-1
?
2????
?? 2
-?? 2
-?? 2
]
?? =?? [
tan
-1
?
2????
?? 2
-?? 2
-
2????
?? 2
-?? 2
-?? 2
1+
4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =?? [
tan
-1
?
2???? (?? 2
-?? 2
-?? 2
-?? 2
+?? 2
)
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
(?? 2
-?? 2
)
2
-(?? 2
-?? 2
)?? 2
+4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
The required streamlines will be given by taking ?? = constant.
Constant ?=
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
?=?? (-2???? ?? 2
)
( ??? is also constant and tangent of any constant is also constant)
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
-2?????? )
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
+?????? )
where ?? =-2?? and it is a variable parameter.
Now, for calculation of fluid speed at any point is given by
?? =|
????
????
|
?? =|
-?? ?? +?? -
?? ?? -?? +
2?? ?? |
?? =?? |
-(?? -?? )-(?? +?? )
(?? -?? )(?? +?? )
+
2
?? |
?? =?? |
-2?? ?? 2
-?? 2
+
2
?? |
?=2?? |
-?? 2
+?? 2
-?? 2
(?? )(?? -?? )(?? +?? )
|=
2?? ?? 2
|?? ||?? -?? ||?? +?? |
?=
2?? ?? 2
?? 1
?? 2
?? 3
?[?? 1
=|?? -?? |;?? 2
=|?? +?? |;?? 3
=|?? |]
6.2 If fluid fills the region of space on the positive side of ?? -axis which is a rigid
boundary and if there be a source ?? at the point (?? ,?? ) and an equal sink at (?? ,?? )
and if the pressure on the negative side be same as the pressure of infinity, show
that the resultant pressure on the boundary is
???? ?? ?? (?? -?? )
?? {?? ???? (?? +?? )}
where ?? is the density of
the fluid.
Page 3
Edurev123
6. Source and Sink
6.1 Two sources each of strength ?? are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? is at the origin. Show that the streamlines are the curves :
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? )
where ?? is a variable parameter.
Show also that the fluid speed at any point is
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? , where ?? ?? ,?? ?? and ?? ?? are the
distances of the points from the sources and the sink.
(2009 : 12 Marks)
Solution:
The complex potential ?? at the any point ?? (?? ) is given by :
?? =-?? log?(?? +?? )-?? log?(?? -?? )+2?? log??? (1)
?? =-?? log??? 2
-?? 2
+?? log??? 2
?? =-?? [log?(?? +???? )
2
-?? 2
}-log?(?? +???? )
2
]
?? =?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )] (2)
Now, we know that
log?(?? +???? )=
1
2
log?(?? 2
+?? 2
)+??tan
-1
?
?? ??
Also,
?? =?? +????
where streamlines will be given by ?? .
Using the above formula and (2)
?? =?? [tan
-1
?
2????
?? 2
-?? 2
-tan
-1
?
2????
?? 2
-?? 2
-?? 2
]
?? =?? [
tan
-1
?
2????
?? 2
-?? 2
-
2????
?? 2
-?? 2
-?? 2
1+
4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =?? [
tan
-1
?
2???? (?? 2
-?? 2
-?? 2
-?? 2
+?? 2
)
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
(?? 2
-?? 2
)
2
-(?? 2
-?? 2
)?? 2
+4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
The required streamlines will be given by taking ?? = constant.
Constant ?=
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
?=?? (-2???? ?? 2
)
( ??? is also constant and tangent of any constant is also constant)
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
-2?????? )
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
+?????? )
where ?? =-2?? and it is a variable parameter.
Now, for calculation of fluid speed at any point is given by
?? =|
????
????
|
?? =|
-?? ?? +?? -
?? ?? -?? +
2?? ?? |
?? =?? |
-(?? -?? )-(?? +?? )
(?? -?? )(?? +?? )
+
2
?? |
?? =?? |
-2?? ?? 2
-?? 2
+
2
?? |
?=2?? |
-?? 2
+?? 2
-?? 2
(?? )(?? -?? )(?? +?? )
|=
2?? ?? 2
|?? ||?? -?? ||?? +?? |
?=
2?? ?? 2
?? 1
?? 2
?? 3
?[?? 1
=|?? -?? |;?? 2
=|?? +?? |;?? 3
=|?? |]
6.2 If fluid fills the region of space on the positive side of ?? -axis which is a rigid
boundary and if there be a source ?? at the point (?? ,?? ) and an equal sink at (?? ,?? )
and if the pressure on the negative side be same as the pressure of infinity, show
that the resultant pressure on the boundary is
???? ?? ?? (?? -?? )
?? {?? ???? (?? +?? )}
where ?? is the density of
the fluid.
(2013 : 15 Marks)
Solution:
The image system with respect to ?? -axis in the ?? -piane is
(i) Source of strength mat (0,?? )
(ii) Source of strength ?? at (0,-?? )
(iii) Sink of strength -?? at (0,?? )
(iv) Sink of strength -?? at (0,-?? )
This system does away witht he boundary. The complex potential of this entire system is
given by
??
?? =-?? log?(?? -???? )+?? log?(?? +???? )-?? log?(?? +???? )+?? log?(?? -???? )
?? =-?? log?(?? 2
+?? 2
)+?? log?(?? 2
+?? 2
)
?
Velocity =|
????
????
|=-
2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
Velocity on the boundary, ?? =0.
?=
-2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
=
2???? (?? 2
-?? 2
)
(?? 2
+?? 2
)(?? 2
+?? 2
)
Let the pressure at infinity be ?? . By Bernoulli's theorem pressure at any point is given by
1
2
?? 2
+
?? ?? ?=
1
2
×0
2
+
?? 0
?? ?? 0
-?? ?? ?=
1
2
?? 2
Page 4
Edurev123
6. Source and Sink
6.1 Two sources each of strength ?? are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? is at the origin. Show that the streamlines are the curves :
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? )
where ?? is a variable parameter.
Show also that the fluid speed at any point is
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? , where ?? ?? ,?? ?? and ?? ?? are the
distances of the points from the sources and the sink.
(2009 : 12 Marks)
Solution:
The complex potential ?? at the any point ?? (?? ) is given by :
?? =-?? log?(?? +?? )-?? log?(?? -?? )+2?? log??? (1)
?? =-?? log??? 2
-?? 2
+?? log??? 2
?? =-?? [log?(?? +???? )
2
-?? 2
}-log?(?? +???? )
2
]
?? =?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )] (2)
Now, we know that
log?(?? +???? )=
1
2
log?(?? 2
+?? 2
)+??tan
-1
?
?? ??
Also,
?? =?? +????
where streamlines will be given by ?? .
Using the above formula and (2)
?? =?? [tan
-1
?
2????
?? 2
-?? 2
-tan
-1
?
2????
?? 2
-?? 2
-?? 2
]
?? =?? [
tan
-1
?
2????
?? 2
-?? 2
-
2????
?? 2
-?? 2
-?? 2
1+
4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =?? [
tan
-1
?
2???? (?? 2
-?? 2
-?? 2
-?? 2
+?? 2
)
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
(?? 2
-?? 2
)
2
-(?? 2
-?? 2
)?? 2
+4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
The required streamlines will be given by taking ?? = constant.
Constant ?=
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
?=?? (-2???? ?? 2
)
( ??? is also constant and tangent of any constant is also constant)
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
-2?????? )
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
+?????? )
where ?? =-2?? and it is a variable parameter.
Now, for calculation of fluid speed at any point is given by
?? =|
????
????
|
?? =|
-?? ?? +?? -
?? ?? -?? +
2?? ?? |
?? =?? |
-(?? -?? )-(?? +?? )
(?? -?? )(?? +?? )
+
2
?? |
?? =?? |
-2?? ?? 2
-?? 2
+
2
?? |
?=2?? |
-?? 2
+?? 2
-?? 2
(?? )(?? -?? )(?? +?? )
|=
2?? ?? 2
|?? ||?? -?? ||?? +?? |
?=
2?? ?? 2
?? 1
?? 2
?? 3
?[?? 1
=|?? -?? |;?? 2
=|?? +?? |;?? 3
=|?? |]
6.2 If fluid fills the region of space on the positive side of ?? -axis which is a rigid
boundary and if there be a source ?? at the point (?? ,?? ) and an equal sink at (?? ,?? )
and if the pressure on the negative side be same as the pressure of infinity, show
that the resultant pressure on the boundary is
???? ?? ?? (?? -?? )
?? {?? ???? (?? +?? )}
where ?? is the density of
the fluid.
(2013 : 15 Marks)
Solution:
The image system with respect to ?? -axis in the ?? -piane is
(i) Source of strength mat (0,?? )
(ii) Source of strength ?? at (0,-?? )
(iii) Sink of strength -?? at (0,?? )
(iv) Sink of strength -?? at (0,-?? )
This system does away witht he boundary. The complex potential of this entire system is
given by
??
?? =-?? log?(?? -???? )+?? log?(?? +???? )-?? log?(?? +???? )+?? log?(?? -???? )
?? =-?? log?(?? 2
+?? 2
)+?? log?(?? 2
+?? 2
)
?
Velocity =|
????
????
|=-
2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
Velocity on the boundary, ?? =0.
?=
-2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
=
2???? (?? 2
-?? 2
)
(?? 2
+?? 2
)(?? 2
+?? 2
)
Let the pressure at infinity be ?? . By Bernoulli's theorem pressure at any point is given by
1
2
?? 2
+
?? ?? ?=
1
2
×0
2
+
?? 0
?? ?? 0
-?? ?? ?=
1
2
?? 2
The resultant pressure on the boundary
?=? ?
8
0
?(?? 0
-?? )???? =
1
2
?? ? ?
8
0
??? 2
????
?=2?? ?? 2
? ?
8
0
?
?? 2
(?? 2
-?? 2
)
2
(?? 2
+?? 2
)
2
(?? 2
+?? 2
)
2
????
?=2?? ?? 2
? ?
8
0
?[-
?? 2
+?? 2
?? 2
-?? 2
{
1
?? 2
+?? 2
-
1
?? 2
+?? 2
}-
?? 2
(?? 2
+?? 2
)
2
-
?? 2
(?? 2
+?? 2
)
2
]????
On solving into partial fractions
?=2?? ?? 2
{
?? 2
+?? 2
?? 2
-?? 2
(
?? 2?? -
?? 2?? )-
?? 4?? -
?? 4?? }
?=
???? ?? 2
2????
·[
2(?? 2
+?? 2
)-(?? +?? )
2
(?? +?? )
]=
???? ?? 2
(?? -?? )
2
2???? (?? +?? )
6.3 Two sources, each of strength ?? , are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? at origin. Show that the stream lines are the curves
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? ) , where ?? is a variable parameter.
Show also that the fluid speed at any point is (?? ?? ?? ?? )/(?? ?? ?? ?? ?? ?? ) , where ?? ?? ,?? ?? and ?? ??
are the distances of the points from the sources and the sink, respectively.
(2019 : 20 Marks)
Solution:
First Part: The complex potential ?? at any point ?? (?? ) given by
?? ?=-?? log?(2-?? )-?? log?(2+?? )+2?? log?2 (1)
?? ?=?? [log?2
2
-log?(2
2
-?? 2
)]
???? +???? ?=?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )], as =?? +????
Equating the imaginary parts, we have
Page 5
Edurev123
6. Source and Sink
6.1 Two sources each of strength ?? are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? is at the origin. Show that the streamlines are the curves :
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? )
where ?? is a variable parameter.
Show also that the fluid speed at any point is
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? , where ?? ?? ,?? ?? and ?? ?? are the
distances of the points from the sources and the sink.
(2009 : 12 Marks)
Solution:
The complex potential ?? at the any point ?? (?? ) is given by :
?? =-?? log?(?? +?? )-?? log?(?? -?? )+2?? log??? (1)
?? =-?? log??? 2
-?? 2
+?? log??? 2
?? =-?? [log?(?? +???? )
2
-?? 2
}-log?(?? +???? )
2
]
?? =?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )] (2)
Now, we know that
log?(?? +???? )=
1
2
log?(?? 2
+?? 2
)+??tan
-1
?
?? ??
Also,
?? =?? +????
where streamlines will be given by ?? .
Using the above formula and (2)
?? =?? [tan
-1
?
2????
?? 2
-?? 2
-tan
-1
?
2????
?? 2
-?? 2
-?? 2
]
?? =?? [
tan
-1
?
2????
?? 2
-?? 2
-
2????
?? 2
-?? 2
-?? 2
1+
4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =?? [
tan
-1
?
2???? (?? 2
-?? 2
-?? 2
-?? 2
+?? 2
)
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
(?? 2
-?? 2
)
2
-(?? 2
-?? 2
)?? 2
+4?? 2
?? 2
(?? 2
-?? 2
)(?? 2
-?? 2
-?? 2
)
]
?? =
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
The required streamlines will be given by taking ?? = constant.
Constant ?=
?? tan
-1
-2???? ?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
(?? 2
+?? 2
)
2
-(?? 2
-?? 2
)?? 2
?=?? (-2???? ?? 2
)
( ??? is also constant and tangent of any constant is also constant)
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
-2?????? )
(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
+?????? )
where ?? =-2?? and it is a variable parameter.
Now, for calculation of fluid speed at any point is given by
?? =|
????
????
|
?? =|
-?? ?? +?? -
?? ?? -?? +
2?? ?? |
?? =?? |
-(?? -?? )-(?? +?? )
(?? -?? )(?? +?? )
+
2
?? |
?? =?? |
-2?? ?? 2
-?? 2
+
2
?? |
?=2?? |
-?? 2
+?? 2
-?? 2
(?? )(?? -?? )(?? +?? )
|=
2?? ?? 2
|?? ||?? -?? ||?? +?? |
?=
2?? ?? 2
?? 1
?? 2
?? 3
?[?? 1
=|?? -?? |;?? 2
=|?? +?? |;?? 3
=|?? |]
6.2 If fluid fills the region of space on the positive side of ?? -axis which is a rigid
boundary and if there be a source ?? at the point (?? ,?? ) and an equal sink at (?? ,?? )
and if the pressure on the negative side be same as the pressure of infinity, show
that the resultant pressure on the boundary is
???? ?? ?? (?? -?? )
?? {?? ???? (?? +?? )}
where ?? is the density of
the fluid.
(2013 : 15 Marks)
Solution:
The image system with respect to ?? -axis in the ?? -piane is
(i) Source of strength mat (0,?? )
(ii) Source of strength ?? at (0,-?? )
(iii) Sink of strength -?? at (0,?? )
(iv) Sink of strength -?? at (0,-?? )
This system does away witht he boundary. The complex potential of this entire system is
given by
??
?? =-?? log?(?? -???? )+?? log?(?? +???? )-?? log?(?? +???? )+?? log?(?? -???? )
?? =-?? log?(?? 2
+?? 2
)+?? log?(?? 2
+?? 2
)
?
Velocity =|
????
????
|=-
2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
Velocity on the boundary, ?? =0.
?=
-2????
?? 2
+?? 2
+
2????
?? 2
+?? 2
=
2???? (?? 2
-?? 2
)
(?? 2
+?? 2
)(?? 2
+?? 2
)
Let the pressure at infinity be ?? . By Bernoulli's theorem pressure at any point is given by
1
2
?? 2
+
?? ?? ?=
1
2
×0
2
+
?? 0
?? ?? 0
-?? ?? ?=
1
2
?? 2
The resultant pressure on the boundary
?=? ?
8
0
?(?? 0
-?? )???? =
1
2
?? ? ?
8
0
??? 2
????
?=2?? ?? 2
? ?
8
0
?
?? 2
(?? 2
-?? 2
)
2
(?? 2
+?? 2
)
2
(?? 2
+?? 2
)
2
????
?=2?? ?? 2
? ?
8
0
?[-
?? 2
+?? 2
?? 2
-?? 2
{
1
?? 2
+?? 2
-
1
?? 2
+?? 2
}-
?? 2
(?? 2
+?? 2
)
2
-
?? 2
(?? 2
+?? 2
)
2
]????
On solving into partial fractions
?=2?? ?? 2
{
?? 2
+?? 2
?? 2
-?? 2
(
?? 2?? -
?? 2?? )-
?? 4?? -
?? 4?? }
?=
???? ?? 2
2????
·[
2(?? 2
+?? 2
)-(?? +?? )
2
(?? +?? )
]=
???? ?? 2
(?? -?? )
2
2???? (?? +?? )
6.3 Two sources, each of strength ?? , are placed at the points (-?? ,?? ),(?? ,?? ) and a
sink of strength ?? ?? at origin. Show that the stream lines are the curves
(?? ?? +?? ?? )
?? =?? ?? (?? ?? -?? ?? +?????? ) , where ?? is a variable parameter.
Show also that the fluid speed at any point is (?? ?? ?? ?? )/(?? ?? ?? ?? ?? ?? ) , where ?? ?? ,?? ?? and ?? ??
are the distances of the points from the sources and the sink, respectively.
(2019 : 20 Marks)
Solution:
First Part: The complex potential ?? at any point ?? (?? ) given by
?? ?=-?? log?(2-?? )-?? log?(2+?? )+2?? log?2 (1)
?? ?=?? [log?2
2
-log?(2
2
-?? 2
)]
???? +???? ?=?? [log?(?? 2
-?? 2
+2?????? )-log?(?? 2
-?? 2
-?? 2
+2?????? )], as =?? +????
Equating the imaginary parts, we have
?? =[tan
-1
?{
2????
(?? 2
-?? 2
)
}-tan
-1
?{2???? |(?? 2
-?? 2
-?? 2
)}]
?? =?? tan
-1
?[
-2?? 2
?? ?? (?? 2
+?? 2
)-?? 2
(?? 2
-?? 2
)
] on simplification.
The detailed streamlines are given by u= Constant =?? tan
-1
?(
-2
?? )
Then we obtain
(
-2
?? )=
(-2?? 2
???? )
[(?? 2
+?? 2
)
2
-?? 2
(?? 2
-?? 2
)]
??????????????????????????????????????????(?? 2
+?? 2
)
2
=?? 2
(?? 2
-?? 2
+?? 2?? )
Second Part : From 0, we have
????
????
?=-
?? 2-?? -
?? 2+?? +
2?? 2
?=
2?? 2
?? 2(2-?? )(2+?? )
?? ?=|
????
?? 2
|=
2?? 2
?? |2|2-?? ||2+?? |
=
2?? 2
?? ?? 1
?? 2
?? 3
where ??? 1
=|?? -?? |,?? 2
=|?? +?? | and ?? 3
=|?? |
6.4 Two sources of strength ?? /?? are placed at the points (±?? ,?? ) . Show that at any
point on the circle ?? ?? +?? ?? =?? ?? , the velocity is parallel to the ?? -axis and is
inversely proportional to ?? .
(2020 : 15 Marks)
Solution:
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