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Page 1
Edurev123
Mechanics and Fluid Dynamics
1. Generalised coordinate
1.1 A perfectly rough sphere of mass ?? and radius ?? , rests on the lowest point of
a fixed spherical cavity of radius ?? . To the highest point of the movable sphere is
attached a particle of mass ?? '
and the system is disturbed. Show that the
oscillations are the same as those of a simple pendulum of length
(?? -?? )
?? ?? '
+
?? ?? ?? ?? +?? '
(?? -
?? ?? )
(2009: 30 marks)
Solution:
Now,
?? ?? ''
?=?? ?? ''
???? ?=?? (?? +?? )
(?? -?? )?? ?=??
(?? -?? )???
?=?? ?? ?
Now, total kinetic energy, ?? can be written as
?? =
1
2
?? '
?? ?? 2
+
1
2
?? ?? ????
2
+
1
2
?? ????
?? 2
Now,
Page 2
Edurev123
Mechanics and Fluid Dynamics
1. Generalised coordinate
1.1 A perfectly rough sphere of mass ?? and radius ?? , rests on the lowest point of
a fixed spherical cavity of radius ?? . To the highest point of the movable sphere is
attached a particle of mass ?? '
and the system is disturbed. Show that the
oscillations are the same as those of a simple pendulum of length
(?? -?? )
?? ?? '
+
?? ?? ?? ?? +?? '
(?? -
?? ?? )
(2009: 30 marks)
Solution:
Now,
?? ?? ''
?=?? ?? ''
???? ?=?? (?? +?? )
(?? -?? )?? ?=??
(?? -?? )???
?=?? ?? ?
Now, total kinetic energy, ?? can be written as
?? =
1
2
?? '
?? ?? 2
+
1
2
?? ?? ????
2
+
1
2
?? ????
?? 2
Now,
?? ?? ?=(?? -?? )sin??? ????
?? =(?? -?? )cos??? ·???
?? ?? ?=(?? -?? )cos??? ????
?? =(?? -?? )-sin??? ·???
?? ?? 2
?=???
?? 2
+???
?? 2
=(?? -?? )
2
???
2
?? ????
?=
2
5
?? ?? 2
and ?? =?? ?
?? ?? ?=(?? -?? )sin??? +?? sin??? ???
?? ?=(?? -?? )cos??? ·???
+?? cos??? ·?? ?
?? ?? ?=(?? -?? )cos??? -?? cos??? ???
?? ?=(?? -?? )(-sin??? )·???
+?? sin??? ·?? ?
???
?? 2
+???
?? 2
?=?? ?? 2
=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
?? (cos??? cos??? -sin??? sin??? )
?=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
(?? ·†
?
-???? )? (ignoring the term ???? as it is very small)
?˜(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
Now potential energy,
Now lagrange,
?? =-???? (?? -?? )cos??? -?? '
?? {(?? -?? )cos??? -?? cos??? }
?? =?? -?? ?? =
1
2
?? '
{(?? -?? )
2
???
2
+?? 2
???
2
+2(?? -?? )?? ???
?? ?
}+
Take this point as zero potential
As
1
2
?? (?? -?? )
2
???
2
+
1
2
×
2
5
?? ?? 2
?? ?
2
+???? (?? -?? )cos??? +
?? '
?? {(?? -?? )cos??? -?? cos??? }
So, the generalised coordinate will only be one
Convert ?? to ?? .
Also
(?? -?? )???
=?? ?? ?
So,
?? =
1
2
?? '
{?? ?? ?
2
+?? 2
?? ?
2
+2?? 2
???
2
}+
1
2
?? ?? 2
?? ?
2
+
?? ?? 2
5
?? ?
2
+???? (?? -?? )cos??? ?+?? '
?? (?? -?? )cos??? -?? '
???? cos???
Page 3
Edurev123
Mechanics and Fluid Dynamics
1. Generalised coordinate
1.1 A perfectly rough sphere of mass ?? and radius ?? , rests on the lowest point of
a fixed spherical cavity of radius ?? . To the highest point of the movable sphere is
attached a particle of mass ?? '
and the system is disturbed. Show that the
oscillations are the same as those of a simple pendulum of length
(?? -?? )
?? ?? '
+
?? ?? ?? ?? +?? '
(?? -
?? ?? )
(2009: 30 marks)
Solution:
Now,
?? ?? ''
?=?? ?? ''
???? ?=?? (?? +?? )
(?? -?? )?? ?=??
(?? -?? )???
?=?? ?? ?
Now, total kinetic energy, ?? can be written as
?? =
1
2
?? '
?? ?? 2
+
1
2
?? ?? ????
2
+
1
2
?? ????
?? 2
Now,
?? ?? ?=(?? -?? )sin??? ????
?? =(?? -?? )cos??? ·???
?? ?? ?=(?? -?? )cos??? ????
?? =(?? -?? )-sin??? ·???
?? ?? 2
?=???
?? 2
+???
?? 2
=(?? -?? )
2
???
2
?? ????
?=
2
5
?? ?? 2
and ?? =?? ?
?? ?? ?=(?? -?? )sin??? +?? sin??? ???
?? ?=(?? -?? )cos??? ·???
+?? cos??? ·?? ?
?? ?? ?=(?? -?? )cos??? -?? cos??? ???
?? ?=(?? -?? )(-sin??? )·???
+?? sin??? ·?? ?
???
?? 2
+???
?? 2
?=?? ?? 2
=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
?? (cos??? cos??? -sin??? sin??? )
?=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
(?? ·†
?
-???? )? (ignoring the term ???? as it is very small)
?˜(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
Now potential energy,
Now lagrange,
?? =-???? (?? -?? )cos??? -?? '
?? {(?? -?? )cos??? -?? cos??? }
?? =?? -?? ?? =
1
2
?? '
{(?? -?? )
2
???
2
+?? 2
???
2
+2(?? -?? )?? ???
?? ?
}+
Take this point as zero potential
As
1
2
?? (?? -?? )
2
???
2
+
1
2
×
2
5
?? ?? 2
?? ?
2
+???? (?? -?? )cos??? +
?? '
?? {(?? -?? )cos??? -?? cos??? }
So, the generalised coordinate will only be one
Convert ?? to ?? .
Also
(?? -?? )???
=?? ?? ?
So,
?? =
1
2
?? '
{?? ?? ?
2
+?? 2
?? ?
2
+2?? 2
???
2
}+
1
2
?? ?? 2
?? ?
2
+
?? ?? 2
5
?? ?
2
+???? (?? -?? )cos??? ?+?? '
?? (?? -?? )cos??? -?? '
???? cos???
?? ?=?? ?
2
(2?? '
+
7?? 10
)?? 2
+(?? +?? '
)?? (?? -?? )cos?
????
?? -?? -?? '
???? cos??? ??? ??? ?
?=2?? ?
(2?? '
+
7?? 10
)?? 2
??? ??? ?
?=
?? (?? -?? )
(?? +?? '
)?? (?? -?? )(-1)sin?
????
(?? -?? )
+?? '
???? sin??? ?=-?? (?? +?? '
)?? sin?
????
(?? -?? )
+?? '
???? sin???
But
sin?
????
?? -?? ˜
????
?? -?? and sin??? ˜??
So,
??? ??? =-?? (?? +?? '
)?? ????
(?? -?? )
+?? '
??????
Now, using Lagrange's theorem
or
?? ????
(
??? ????
)-
??? ??? =0
?? ¨
(4?? '
+
7?? 5
)?? 2
+?? (?? +?? '
)
?????? ?? -?? -?? ?? '
????
?? ¨
(4?? '
+
7?? 5
)?? 2
+?????? [
?? (?? +?? '
)-?? '
(?? -?? )
?? -?? ]=0
?? ¨
+
?? ?? (4?? '
+
7?? 5
)?? 2
(
???(2?? -?? )+????
(?? -?? )
)?? =0
This is S.H.M. equation.
or
?? ¨
+???
2
?? =0
So,
?? 2
?=
?? {?? '
(2-
?? ?? )+?? }
(4?? '
+
7?? 5
)(?? -?? )
=
?? ?? ?? ?=
(4?? '
+
7?? 5
)(?? -?? )
?? +?? '
(2-
?? ?? )
Page 4
Edurev123
Mechanics and Fluid Dynamics
1. Generalised coordinate
1.1 A perfectly rough sphere of mass ?? and radius ?? , rests on the lowest point of
a fixed spherical cavity of radius ?? . To the highest point of the movable sphere is
attached a particle of mass ?? '
and the system is disturbed. Show that the
oscillations are the same as those of a simple pendulum of length
(?? -?? )
?? ?? '
+
?? ?? ?? ?? +?? '
(?? -
?? ?? )
(2009: 30 marks)
Solution:
Now,
?? ?? ''
?=?? ?? ''
???? ?=?? (?? +?? )
(?? -?? )?? ?=??
(?? -?? )???
?=?? ?? ?
Now, total kinetic energy, ?? can be written as
?? =
1
2
?? '
?? ?? 2
+
1
2
?? ?? ????
2
+
1
2
?? ????
?? 2
Now,
?? ?? ?=(?? -?? )sin??? ????
?? =(?? -?? )cos??? ·???
?? ?? ?=(?? -?? )cos??? ????
?? =(?? -?? )-sin??? ·???
?? ?? 2
?=???
?? 2
+???
?? 2
=(?? -?? )
2
???
2
?? ????
?=
2
5
?? ?? 2
and ?? =?? ?
?? ?? ?=(?? -?? )sin??? +?? sin??? ???
?? ?=(?? -?? )cos??? ·???
+?? cos??? ·?? ?
?? ?? ?=(?? -?? )cos??? -?? cos??? ???
?? ?=(?? -?? )(-sin??? )·???
+?? sin??? ·?? ?
???
?? 2
+???
?? 2
?=?? ?? 2
=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
?? (cos??? cos??? -sin??? sin??? )
?=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
(?? ·†
?
-???? )? (ignoring the term ???? as it is very small)
?˜(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
Now potential energy,
Now lagrange,
?? =-???? (?? -?? )cos??? -?? '
?? {(?? -?? )cos??? -?? cos??? }
?? =?? -?? ?? =
1
2
?? '
{(?? -?? )
2
???
2
+?? 2
???
2
+2(?? -?? )?? ???
?? ?
}+
Take this point as zero potential
As
1
2
?? (?? -?? )
2
???
2
+
1
2
×
2
5
?? ?? 2
?? ?
2
+???? (?? -?? )cos??? +
?? '
?? {(?? -?? )cos??? -?? cos??? }
So, the generalised coordinate will only be one
Convert ?? to ?? .
Also
(?? -?? )???
=?? ?? ?
So,
?? =
1
2
?? '
{?? ?? ?
2
+?? 2
?? ?
2
+2?? 2
???
2
}+
1
2
?? ?? 2
?? ?
2
+
?? ?? 2
5
?? ?
2
+???? (?? -?? )cos??? ?+?? '
?? (?? -?? )cos??? -?? '
???? cos???
?? ?=?? ?
2
(2?? '
+
7?? 10
)?? 2
+(?? +?? '
)?? (?? -?? )cos?
????
?? -?? -?? '
???? cos??? ??? ??? ?
?=2?? ?
(2?? '
+
7?? 10
)?? 2
??? ??? ?
?=
?? (?? -?? )
(?? +?? '
)?? (?? -?? )(-1)sin?
????
(?? -?? )
+?? '
???? sin??? ?=-?? (?? +?? '
)?? sin?
????
(?? -?? )
+?? '
???? sin???
But
sin?
????
?? -?? ˜
????
?? -?? and sin??? ˜??
So,
??? ??? =-?? (?? +?? '
)?? ????
(?? -?? )
+?? '
??????
Now, using Lagrange's theorem
or
?? ????
(
??? ????
)-
??? ??? =0
?? ¨
(4?? '
+
7?? 5
)?? 2
+?? (?? +?? '
)
?????? ?? -?? -?? ?? '
????
?? ¨
(4?? '
+
7?? 5
)?? 2
+?????? [
?? (?? +?? '
)-?? '
(?? -?? )
?? -?? ]=0
?? ¨
+
?? ?? (4?? '
+
7?? 5
)?? 2
(
???(2?? -?? )+????
(?? -?? )
)?? =0
This is S.H.M. equation.
or
?? ¨
+???
2
?? =0
So,
?? 2
?=
?? {?? '
(2-
?? ?? )+?? }
(4?? '
+
7?? 5
)(?? -?? )
=
?? ?? ?? ?=
(4?? '
+
7?? 5
)(?? -?? )
?? +?? '
(2-
?? ?? )
or,
So, oscillations are the same as those of a simple pendulum of above length
?? =
(4?? '
+
7?? 5
)(?? -?? )
?? +?? '
(2-
?? ?? )
1.2 A sphere of radius a and mass ?? rolls down a rough plane inclined at an angle
?? to the horizontal. If ?? be the distance of the point of contact of sphere from a
fixed point on the plane, find the acceleration by using Hamilton's equations.
(2010 : 30 Marks)
Solution:
Figure-1 below depicts the situation given in problem.
?? ? Radius of gyration of given sphere
Kinetic energy,
?? ?=
1
2
?? (???
2
+?? 2
???
2
)
?=
1
2
?? (???
2
+
2
5
?? 2
???
2
)
?=
1
2
?? (???
2
+
2
5
???
2
)=
7
10
?? ???
2
Figure-1
(?? =
2
?? ?? 2
for sphere )
(In pure rolling, ??? =?? ???
)
Potential energy,
?? =-?????? sin
Page 5
Edurev123
Mechanics and Fluid Dynamics
1. Generalised coordinate
1.1 A perfectly rough sphere of mass ?? and radius ?? , rests on the lowest point of
a fixed spherical cavity of radius ?? . To the highest point of the movable sphere is
attached a particle of mass ?? '
and the system is disturbed. Show that the
oscillations are the same as those of a simple pendulum of length
(?? -?? )
?? ?? '
+
?? ?? ?? ?? +?? '
(?? -
?? ?? )
(2009: 30 marks)
Solution:
Now,
?? ?? ''
?=?? ?? ''
???? ?=?? (?? +?? )
(?? -?? )?? ?=??
(?? -?? )???
?=?? ?? ?
Now, total kinetic energy, ?? can be written as
?? =
1
2
?? '
?? ?? 2
+
1
2
?? ?? ????
2
+
1
2
?? ????
?? 2
Now,
?? ?? ?=(?? -?? )sin??? ????
?? =(?? -?? )cos??? ·???
?? ?? ?=(?? -?? )cos??? ????
?? =(?? -?? )-sin??? ·???
?? ?? 2
?=???
?? 2
+???
?? 2
=(?? -?? )
2
???
2
?? ????
?=
2
5
?? ?? 2
and ?? =?? ?
?? ?? ?=(?? -?? )sin??? +?? sin??? ???
?? ?=(?? -?? )cos??? ·???
+?? cos??? ·?? ?
?? ?? ?=(?? -?? )cos??? -?? cos??? ???
?? ?=(?? -?? )(-sin??? )·???
+?? sin??? ·?? ?
???
?? 2
+???
?? 2
?=?? ?? 2
=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
?? (cos??? cos??? -sin??? sin??? )
?=(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
(?? ·†
?
-???? )? (ignoring the term ???? as it is very small)
?˜(?? -?? )
2
???
2
+?? 2
?? ?
2
+2(?? -?? )???
?? ?
Now potential energy,
Now lagrange,
?? =-???? (?? -?? )cos??? -?? '
?? {(?? -?? )cos??? -?? cos??? }
?? =?? -?? ?? =
1
2
?? '
{(?? -?? )
2
???
2
+?? 2
???
2
+2(?? -?? )?? ???
?? ?
}+
Take this point as zero potential
As
1
2
?? (?? -?? )
2
???
2
+
1
2
×
2
5
?? ?? 2
?? ?
2
+???? (?? -?? )cos??? +
?? '
?? {(?? -?? )cos??? -?? cos??? }
So, the generalised coordinate will only be one
Convert ?? to ?? .
Also
(?? -?? )???
=?? ?? ?
So,
?? =
1
2
?? '
{?? ?? ?
2
+?? 2
?? ?
2
+2?? 2
???
2
}+
1
2
?? ?? 2
?? ?
2
+
?? ?? 2
5
?? ?
2
+???? (?? -?? )cos??? ?+?? '
?? (?? -?? )cos??? -?? '
???? cos???
?? ?=?? ?
2
(2?? '
+
7?? 10
)?? 2
+(?? +?? '
)?? (?? -?? )cos?
????
?? -?? -?? '
???? cos??? ??? ??? ?
?=2?? ?
(2?? '
+
7?? 10
)?? 2
??? ??? ?
?=
?? (?? -?? )
(?? +?? '
)?? (?? -?? )(-1)sin?
????
(?? -?? )
+?? '
???? sin??? ?=-?? (?? +?? '
)?? sin?
????
(?? -?? )
+?? '
???? sin???
But
sin?
????
?? -?? ˜
????
?? -?? and sin??? ˜??
So,
??? ??? =-?? (?? +?? '
)?? ????
(?? -?? )
+?? '
??????
Now, using Lagrange's theorem
or
?? ????
(
??? ????
)-
??? ??? =0
?? ¨
(4?? '
+
7?? 5
)?? 2
+?? (?? +?? '
)
?????? ?? -?? -?? ?? '
????
?? ¨
(4?? '
+
7?? 5
)?? 2
+?????? [
?? (?? +?? '
)-?? '
(?? -?? )
?? -?? ]=0
?? ¨
+
?? ?? (4?? '
+
7?? 5
)?? 2
(
???(2?? -?? )+????
(?? -?? )
)?? =0
This is S.H.M. equation.
or
?? ¨
+???
2
?? =0
So,
?? 2
?=
?? {?? '
(2-
?? ?? )+?? }
(4?? '
+
7?? 5
)(?? -?? )
=
?? ?? ?? ?=
(4?? '
+
7?? 5
)(?? -?? )
?? +?? '
(2-
?? ?? )
or,
So, oscillations are the same as those of a simple pendulum of above length
?? =
(4?? '
+
7?? 5
)(?? -?? )
?? +?? '
(2-
?? ?? )
1.2 A sphere of radius a and mass ?? rolls down a rough plane inclined at an angle
?? to the horizontal. If ?? be the distance of the point of contact of sphere from a
fixed point on the plane, find the acceleration by using Hamilton's equations.
(2010 : 30 Marks)
Solution:
Figure-1 below depicts the situation given in problem.
?? ? Radius of gyration of given sphere
Kinetic energy,
?? ?=
1
2
?? (???
2
+?? 2
???
2
)
?=
1
2
?? (???
2
+
2
5
?? 2
???
2
)
?=
1
2
?? (???
2
+
2
5
???
2
)=
7
10
?? ???
2
Figure-1
(?? =
2
?? ?? 2
for sphere )
(In pure rolling, ??? =?? ???
)
Potential energy,
?? =-?????? sin
? ?? =?? -?? =
7
10
?? ???
2
+???? ?? sin??? Now, ?? ?? =
??? ????
=
7
10
?? ×2???=
7
5
?? ???
? ??? =
5?? ?? 7?? ? Hamiltonian, ?? =-?? +?? ?? ·??? =-
7
10
?? ???
2
-?????? sin??? +?? ?? ·
5?? ?? 7?? ? ?? =
-5?? ?? 2
14?? -?????? sin??? +
5?? ?? 2
7?? ? ?? =
5
14?? ?? ?? 2
-???? ?? sin??? ( Putting ???=
5?? ?? 7?? )
? One of the Hamiltonian's equation gives
???
?? ?=
-??? ??? =+???? sin??? ??
7
5
?? ??¨?=???? sin??? ????¨?=
5
7
?? sin???
? Acceleration of sphere is
5
7
?? sin??? .
1.3 Obtain the equations governing the motion of a spherical pendulum.
(2012 : 12 Marks)
Solution:
Let ' ?? ' be the mass of the bob of spherical pendulum, which can swing in any direction,
traces out a sphere of constant length ?? .
Using polar co-ordinates ?? and ?? , the kinetic energy,
and potential energy,
?? ?=
1
2
?? (???
2
+?? 2
???
2
+?? 2
sin
2
??? ?? ?
2
)
?? ?=-?????? cos???
For spherical pendulum, the length ' ?? ' is constant.
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Generalised coordinate Notes
Generalised coordinate Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Generalised coordinate.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
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Generalised coordinate UPSC Questions
The "Generalised coordinate UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
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The content of Generalised coordinate is prepared as per the latest UPSC syllabus.
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