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Edurev123
7. Principle of Virtual Work
7.1 A solid hemisphere is supported by a string fixed to a point on its rim and to a
point on a smooth vertical wall with which the curved surface of the hemisphere is
in contact. If ?? and ?? are inclinations of the string and the plane base of the
hemisphere to the vertical, prove by using the principle of virtual work that
?????? ?? =
?? ?? +?????? ??
(2010: 20 marks)
Solution:
?? = radius of solid hemisphere
?? = weight of solid hemisphere
?? = length of string
??sin ?? +?? sin ?? =?? ? ??cos ?????? +?? cos ?????? =0
? ??cos ?????? =-?? cos ?????? The weight ?? acts at a distance
3?? 8
from centre.
? By principle of virtual work
Page 2
Edurev123
7. Principle of Virtual Work
7.1 A solid hemisphere is supported by a string fixed to a point on its rim and to a
point on a smooth vertical wall with which the curved surface of the hemisphere is
in contact. If ?? and ?? are inclinations of the string and the plane base of the
hemisphere to the vertical, prove by using the principle of virtual work that
?????? ?? =
?? ?? +?????? ??
(2010: 20 marks)
Solution:
?? = radius of solid hemisphere
?? = weight of solid hemisphere
?? = length of string
??sin ?? +?? sin ?? =?? ? ??cos ?????? +?? cos ?????? =0
? ??cos ?????? =-?? cos ?????? The weight ?? acts at a distance
3?? 8
from centre.
? By principle of virtual work
???? (??cos ?? +?? cos ?? +
3?? 8
sin ?? )=0
? -??sin ???? ??ˆ-?? sin ?? ???? +
3?? 8
cos ?????? =0
? cos ?????? +tan ?? -?? sin ?????? +
3?? 8
cos ?????? =0
? tan ?? -tan ?? +
3
8
=0
? tan ?? =
3
8
+tan ??
7.2 Six rods ???? ,???? ,???? ,???? ,???? and ???? are each of weight ?? and are freely jointed
at their extremities so as to form a hexagon; the rod ???? is fixed in a horizontal
position and the middle points of ???? and ???? are joined by a string. Find the
tension in the string.
(2013 : 15 Marks)
Solution:
The middle points ?? and ?? are connected ?? by the string.
Let ?? be the tension in the string. Also let length of the rods be 2?? .
The weight of all the rods can be assumed to be acting at the middle point ?? .
Let ??????? =??
Also for a hexagon interior angle
=
6×?? -2?? 6
=
4?? 6
=
2?? 3
? ?? =?? -
2?? 3
=
?? 3
Let it be given a small displacement in which the angle changes from ?? to ?? +???? .
Page 3
Edurev123
7. Principle of Virtual Work
7.1 A solid hemisphere is supported by a string fixed to a point on its rim and to a
point on a smooth vertical wall with which the curved surface of the hemisphere is
in contact. If ?? and ?? are inclinations of the string and the plane base of the
hemisphere to the vertical, prove by using the principle of virtual work that
?????? ?? =
?? ?? +?????? ??
(2010: 20 marks)
Solution:
?? = radius of solid hemisphere
?? = weight of solid hemisphere
?? = length of string
??sin ?? +?? sin ?? =?? ? ??cos ?????? +?? cos ?????? =0
? ??cos ?????? =-?? cos ?????? The weight ?? acts at a distance
3?? 8
from centre.
? By principle of virtual work
???? (??cos ?? +?? cos ?? +
3?? 8
sin ?? )=0
? -??sin ???? ??ˆ-?? sin ?? ???? +
3?? 8
cos ?????? =0
? cos ?????? +tan ?? -?? sin ?????? +
3?? 8
cos ?????? =0
? tan ?? -tan ?? +
3
8
=0
? tan ?? =
3
8
+tan ??
7.2 Six rods ???? ,???? ,???? ,???? ,???? and ???? are each of weight ?? and are freely jointed
at their extremities so as to form a hexagon; the rod ???? is fixed in a horizontal
position and the middle points of ???? and ???? are joined by a string. Find the
tension in the string.
(2013 : 15 Marks)
Solution:
The middle points ?? and ?? are connected ?? by the string.
Let ?? be the tension in the string. Also let length of the rods be 2?? .
The weight of all the rods can be assumed to be acting at the middle point ?? .
Let ??????? =??
Also for a hexagon interior angle
=
6×?? -2?? 6
=
4?? 6
=
2?? 3
? ?? =?? -
2?? 3
=
?? 3
Let it be given a small displacement in which the angle changes from ?? to ?? +???? .
The line ???? is fixed and so we measure all distances from it. The length ???? and ????
changes.
???? =2???? =2·2?? sin ?? =4?? sin ??
The depth of ?? above the fixed line =???? =2?? sin ?? .
By principle of virtual work
-???? (4?? sin ?? )+6???? (2?? sin ?? ) =0
-4???? cos ?????? +16???? cos ?????? =0
4?? [-?? +3?? ]cos ?????? =0
? ?? =3??
7.3 Two equal uniform rods ???? and ?? ?? ?? each of length ?? , are freely jointed at ?? and
rest on a smooth fixed verticai circle of radius ?? . If ?? ?? is the angle between the
rods, then find the relation between ?? ,?? and ?? by using the principle of virtual
work.
(2014 : 10 Marks)
Solution:
Let ' ?? ' be the centre of the given fixed circle and ' ?? ' be the weight each of the rods
???? and ???? . If ' ?? ' and ' ?? ' are the middle points of ???? and ???? , then the total weight ' 2??
' of the two rods can be taken as acting at ' ?? '. Middle point of ???? . The line ???? is vertical
we have
??? ?? ?? =??????? =??
Also, ???? =??,???? =??/2. If the rods ???? touches the circle at ?? ; then ??????? =90
°
and
???? =?? the radius of circle.
Give the rods a small symmetrical displacement in which ' ?? ' changes to ?? +???? . The
point ?? , remains fixed and the point ' ?? ' is slightly displaced.
The ??????? remains 90
°
, we have the height of ?? above the fixed point ' ?? '.
???? =???? -???? =???? cosec ?? -???? cos ?? ???? =?? cosec ?? -?? 2
cos ??
Equation of virtual work is
-2?? (???? )
ˆ
=?? ?? =(?? cosec ?? -??/2cos ?? )=0
=(-?? cosec ?? cot ?? +??/2)???? =0
?? cosec ?? cot ?? =??/2sin ??
Page 4
Edurev123
7. Principle of Virtual Work
7.1 A solid hemisphere is supported by a string fixed to a point on its rim and to a
point on a smooth vertical wall with which the curved surface of the hemisphere is
in contact. If ?? and ?? are inclinations of the string and the plane base of the
hemisphere to the vertical, prove by using the principle of virtual work that
?????? ?? =
?? ?? +?????? ??
(2010: 20 marks)
Solution:
?? = radius of solid hemisphere
?? = weight of solid hemisphere
?? = length of string
??sin ?? +?? sin ?? =?? ? ??cos ?????? +?? cos ?????? =0
? ??cos ?????? =-?? cos ?????? The weight ?? acts at a distance
3?? 8
from centre.
? By principle of virtual work
???? (??cos ?? +?? cos ?? +
3?? 8
sin ?? )=0
? -??sin ???? ??ˆ-?? sin ?? ???? +
3?? 8
cos ?????? =0
? cos ?????? +tan ?? -?? sin ?????? +
3?? 8
cos ?????? =0
? tan ?? -tan ?? +
3
8
=0
? tan ?? =
3
8
+tan ??
7.2 Six rods ???? ,???? ,???? ,???? ,???? and ???? are each of weight ?? and are freely jointed
at their extremities so as to form a hexagon; the rod ???? is fixed in a horizontal
position and the middle points of ???? and ???? are joined by a string. Find the
tension in the string.
(2013 : 15 Marks)
Solution:
The middle points ?? and ?? are connected ?? by the string.
Let ?? be the tension in the string. Also let length of the rods be 2?? .
The weight of all the rods can be assumed to be acting at the middle point ?? .
Let ??????? =??
Also for a hexagon interior angle
=
6×?? -2?? 6
=
4?? 6
=
2?? 3
? ?? =?? -
2?? 3
=
?? 3
Let it be given a small displacement in which the angle changes from ?? to ?? +???? .
The line ???? is fixed and so we measure all distances from it. The length ???? and ????
changes.
???? =2???? =2·2?? sin ?? =4?? sin ??
The depth of ?? above the fixed line =???? =2?? sin ?? .
By principle of virtual work
-???? (4?? sin ?? )+6???? (2?? sin ?? ) =0
-4???? cos ?????? +16???? cos ?????? =0
4?? [-?? +3?? ]cos ?????? =0
? ?? =3??
7.3 Two equal uniform rods ???? and ?? ?? ?? each of length ?? , are freely jointed at ?? and
rest on a smooth fixed verticai circle of radius ?? . If ?? ?? is the angle between the
rods, then find the relation between ?? ,?? and ?? by using the principle of virtual
work.
(2014 : 10 Marks)
Solution:
Let ' ?? ' be the centre of the given fixed circle and ' ?? ' be the weight each of the rods
???? and ???? . If ' ?? ' and ' ?? ' are the middle points of ???? and ???? , then the total weight ' 2??
' of the two rods can be taken as acting at ' ?? '. Middle point of ???? . The line ???? is vertical
we have
??? ?? ?? =??????? =??
Also, ???? =??,???? =??/2. If the rods ???? touches the circle at ?? ; then ??????? =90
°
and
???? =?? the radius of circle.
Give the rods a small symmetrical displacement in which ' ?? ' changes to ?? +???? . The
point ?? , remains fixed and the point ' ?? ' is slightly displaced.
The ??????? remains 90
°
, we have the height of ?? above the fixed point ' ?? '.
???? =???? -???? =???? cosec ?? -???? cos ?? ???? =?? cosec ?? -?? 2
cos ??
Equation of virtual work is
-2?? (???? )
ˆ
=?? ?? =(?? cosec ?? -??/2cos ?? )=0
=(-?? cosec ?? cot ?? +??/2)???? =0
?? cosec ?? cot ?? =??/2sin ??
2?? ·
1
sin ?? ·
cos ?? sin ?? =??sin ??
2?? =cos ?=??sin ??
7.4 A square framework formed of uniform heavy rods of equal weight ?? joined
together, is hung up by one corner. A weight ?? is suspended from each of the
three lower corners, and the shape of the square is preserved by a light rod along
the horizontal diagonal. Find the thrust of the light rod.
(2020 : 10 Marks)
Solution:
Let us give symmetrical displacement of the rod about ?? , so that ?? changes to ?? +???? .
So, by Principle of virtual work, we have
Page 5
Edurev123
7. Principle of Virtual Work
7.1 A solid hemisphere is supported by a string fixed to a point on its rim and to a
point on a smooth vertical wall with which the curved surface of the hemisphere is
in contact. If ?? and ?? are inclinations of the string and the plane base of the
hemisphere to the vertical, prove by using the principle of virtual work that
?????? ?? =
?? ?? +?????? ??
(2010: 20 marks)
Solution:
?? = radius of solid hemisphere
?? = weight of solid hemisphere
?? = length of string
??sin ?? +?? sin ?? =?? ? ??cos ?????? +?? cos ?????? =0
? ??cos ?????? =-?? cos ?????? The weight ?? acts at a distance
3?? 8
from centre.
? By principle of virtual work
???? (??cos ?? +?? cos ?? +
3?? 8
sin ?? )=0
? -??sin ???? ??ˆ-?? sin ?? ???? +
3?? 8
cos ?????? =0
? cos ?????? +tan ?? -?? sin ?????? +
3?? 8
cos ?????? =0
? tan ?? -tan ?? +
3
8
=0
? tan ?? =
3
8
+tan ??
7.2 Six rods ???? ,???? ,???? ,???? ,???? and ???? are each of weight ?? and are freely jointed
at their extremities so as to form a hexagon; the rod ???? is fixed in a horizontal
position and the middle points of ???? and ???? are joined by a string. Find the
tension in the string.
(2013 : 15 Marks)
Solution:
The middle points ?? and ?? are connected ?? by the string.
Let ?? be the tension in the string. Also let length of the rods be 2?? .
The weight of all the rods can be assumed to be acting at the middle point ?? .
Let ??????? =??
Also for a hexagon interior angle
=
6×?? -2?? 6
=
4?? 6
=
2?? 3
? ?? =?? -
2?? 3
=
?? 3
Let it be given a small displacement in which the angle changes from ?? to ?? +???? .
The line ???? is fixed and so we measure all distances from it. The length ???? and ????
changes.
???? =2???? =2·2?? sin ?? =4?? sin ??
The depth of ?? above the fixed line =???? =2?? sin ?? .
By principle of virtual work
-???? (4?? sin ?? )+6???? (2?? sin ?? ) =0
-4???? cos ?????? +16???? cos ?????? =0
4?? [-?? +3?? ]cos ?????? =0
? ?? =3??
7.3 Two equal uniform rods ???? and ?? ?? ?? each of length ?? , are freely jointed at ?? and
rest on a smooth fixed verticai circle of radius ?? . If ?? ?? is the angle between the
rods, then find the relation between ?? ,?? and ?? by using the principle of virtual
work.
(2014 : 10 Marks)
Solution:
Let ' ?? ' be the centre of the given fixed circle and ' ?? ' be the weight each of the rods
???? and ???? . If ' ?? ' and ' ?? ' are the middle points of ???? and ???? , then the total weight ' 2??
' of the two rods can be taken as acting at ' ?? '. Middle point of ???? . The line ???? is vertical
we have
??? ?? ?? =??????? =??
Also, ???? =??,???? =??/2. If the rods ???? touches the circle at ?? ; then ??????? =90
°
and
???? =?? the radius of circle.
Give the rods a small symmetrical displacement in which ' ?? ' changes to ?? +???? . The
point ?? , remains fixed and the point ' ?? ' is slightly displaced.
The ??????? remains 90
°
, we have the height of ?? above the fixed point ' ?? '.
???? =???? -???? =???? cosec ?? -???? cos ?? ???? =?? cosec ?? -?? 2
cos ??
Equation of virtual work is
-2?? (???? )
ˆ
=?? ?? =(?? cosec ?? -??/2cos ?? )=0
=(-?? cosec ?? cot ?? +??/2)???? =0
?? cosec ?? cot ?? =??/2sin ??
2?? ·
1
sin ?? ·
cos ?? sin ?? =??sin ??
2?? =cos ?=??sin ??
7.4 A square framework formed of uniform heavy rods of equal weight ?? joined
together, is hung up by one corner. A weight ?? is suspended from each of the
three lower corners, and the shape of the square is preserved by a light rod along
the horizontal diagonal. Find the thrust of the light rod.
(2020 : 10 Marks)
Solution:
Let us give symmetrical displacement of the rod about ?? , so that ?? changes to ?? +???? .
So, by Principle of virtual work, we have
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