Constrained Motion | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
3. Constrained Motion 
3.1 Find the length of an endless chain which will hang over a circular pulley of 
radius a so as to be in contact with three-fourth of the circumference of the pulley. 
(2009 : 15 Marks) 
Solution: 
Since the chain is in contact with 3/4 th of the pulley 
                ANB  =
3
4
× Circumference of pulley 
 =
3
4
×2???? =
3????
2
??????? =
1
4
×2?? =
?? 2
?                                                          ??????? =??????? =
?? 4
 
 
Let ???? be the directrix and ???? the axis of the caternary of the part of the chain hanging 
from the pulley. Let C be the parameter. 
Then,                                                             ???? =?? 
Tangent at ?? is perpendicular to radius. 
So, the tangential angle at ?? , i.e., ?? ?? =
?? 4
 
From ??? ?? '
?? 
??? =???? sin 
?? 4
=
?? v2
 
? At ?? 
Page 2


Edurev123 
3. Constrained Motion 
3.1 Find the length of an endless chain which will hang over a circular pulley of 
radius a so as to be in contact with three-fourth of the circumference of the pulley. 
(2009 : 15 Marks) 
Solution: 
Since the chain is in contact with 3/4 th of the pulley 
                ANB  =
3
4
× Circumference of pulley 
 =
3
4
×2???? =
3????
2
??????? =
1
4
×2?? =
?? 2
?                                                          ??????? =??????? =
?? 4
 
 
Let ???? be the directrix and ???? the axis of the caternary of the part of the chain hanging 
from the pulley. Let C be the parameter. 
Then,                                                             ???? =?? 
Tangent at ?? is perpendicular to radius. 
So, the tangential angle at ?? , i.e., ?? ?? =
?? 4
 
From ??? ?? '
?? 
??? =???? sin 
?? 4
=
?? v2
 
? At ?? 
?? =?? log (tan ?? +sec ?? ) 
?                                                         
?? v2
=?? log (tan 
?? 4
+sec 
?? 4
) 
 =?? log (1+v2)
?                                                            ?? =
?? v2log (1+v2)
?? =?? tan ?? ??? ?? =
?? v2log (1+v2)
 
?                   
 Arc ???? =
2?? v2log (1+v2)
 Total length of chain  =
v2?? log (1+v2)
+
3????
2
 
3.2 A beam ???? rests on two supports ?? and ?? , where ???? =???? =???? . It is found 
that the beam will tilt when a weight of ?? ???? is hung from ?? or when a weight of 
?? ???? is hung from ?? . Find the weight of the beam. 
(2020: 15 marks) 
Solution: 
(i) Let ???? =???? =???? =?? , and ?? be centroid, w, weight of beam when weight is being 
from ?? and Beam is about to tilt, then reaction at ?? =0 
 
Balancing torque about ?? : 
???? (?? )=?? ?? (??) 
(ii) When, weight is hung from ?? , reaction at ?? would be zero. 
Balancing torque about ?? : 
Page 3


Edurev123 
3. Constrained Motion 
3.1 Find the length of an endless chain which will hang over a circular pulley of 
radius a so as to be in contact with three-fourth of the circumference of the pulley. 
(2009 : 15 Marks) 
Solution: 
Since the chain is in contact with 3/4 th of the pulley 
                ANB  =
3
4
× Circumference of pulley 
 =
3
4
×2???? =
3????
2
??????? =
1
4
×2?? =
?? 2
?                                                          ??????? =??????? =
?? 4
 
 
Let ???? be the directrix and ???? the axis of the caternary of the part of the chain hanging 
from the pulley. Let C be the parameter. 
Then,                                                             ???? =?? 
Tangent at ?? is perpendicular to radius. 
So, the tangential angle at ?? , i.e., ?? ?? =
?? 4
 
From ??? ?? '
?? 
??? =???? sin 
?? 4
=
?? v2
 
? At ?? 
?? =?? log (tan ?? +sec ?? ) 
?                                                         
?? v2
=?? log (tan 
?? 4
+sec 
?? 4
) 
 =?? log (1+v2)
?                                                            ?? =
?? v2log (1+v2)
?? =?? tan ?? ??? ?? =
?? v2log (1+v2)
 
?                   
 Arc ???? =
2?? v2log (1+v2)
 Total length of chain  =
v2?? log (1+v2)
+
3????
2
 
3.2 A beam ???? rests on two supports ?? and ?? , where ???? =???? =???? . It is found 
that the beam will tilt when a weight of ?? ???? is hung from ?? or when a weight of 
?? ???? is hung from ?? . Find the weight of the beam. 
(2020: 15 marks) 
Solution: 
(i) Let ???? =???? =???? =?? , and ?? be centroid, w, weight of beam when weight is being 
from ?? and Beam is about to tilt, then reaction at ?? =0 
 
Balancing torque about ?? : 
???? (?? )=?? ?? (??) 
(ii) When, weight is hung from ?? , reaction at ?? would be zero. 
Balancing torque about ?? : 
 
?? (?? -?? )=?????? (???? ) 
From (1) and (2) 
?? (?? ) =(?? +?? )????
?? =(???? )?? 
3.3 If a planet, which revolves around the Sun in a circular orbit, is suddenly 
stopped in its orbit, then find the time in which it would fall into the Sun. Also, find 
the ratio of its falling time to the period of revolution of the planet. 
[2021 : 10 marks] 
Solution: 
Let a planet describing a circular path of radius a and centre ?? (the sun) be stopped at 
the point ?? of its path. Then, it will begin to move towards ?? along the straight line PS-
under the acceleration ?? /( distance )
2
. If ?? is the position of the planet at time ?? such 
that ???? =?? , then the acceleration at ?? is 
?? ?? 2
 directed towards ?? . 
 
 
? The equation of motion of the planet at ?? is 
Page 4


Edurev123 
3. Constrained Motion 
3.1 Find the length of an endless chain which will hang over a circular pulley of 
radius a so as to be in contact with three-fourth of the circumference of the pulley. 
(2009 : 15 Marks) 
Solution: 
Since the chain is in contact with 3/4 th of the pulley 
                ANB  =
3
4
× Circumference of pulley 
 =
3
4
×2???? =
3????
2
??????? =
1
4
×2?? =
?? 2
?                                                          ??????? =??????? =
?? 4
 
 
Let ???? be the directrix and ???? the axis of the caternary of the part of the chain hanging 
from the pulley. Let C be the parameter. 
Then,                                                             ???? =?? 
Tangent at ?? is perpendicular to radius. 
So, the tangential angle at ?? , i.e., ?? ?? =
?? 4
 
From ??? ?? '
?? 
??? =???? sin 
?? 4
=
?? v2
 
? At ?? 
?? =?? log (tan ?? +sec ?? ) 
?                                                         
?? v2
=?? log (tan 
?? 4
+sec 
?? 4
) 
 =?? log (1+v2)
?                                                            ?? =
?? v2log (1+v2)
?? =?? tan ?? ??? ?? =
?? v2log (1+v2)
 
?                   
 Arc ???? =
2?? v2log (1+v2)
 Total length of chain  =
v2?? log (1+v2)
+
3????
2
 
3.2 A beam ???? rests on two supports ?? and ?? , where ???? =???? =???? . It is found 
that the beam will tilt when a weight of ?? ???? is hung from ?? or when a weight of 
?? ???? is hung from ?? . Find the weight of the beam. 
(2020: 15 marks) 
Solution: 
(i) Let ???? =???? =???? =?? , and ?? be centroid, w, weight of beam when weight is being 
from ?? and Beam is about to tilt, then reaction at ?? =0 
 
Balancing torque about ?? : 
???? (?? )=?? ?? (??) 
(ii) When, weight is hung from ?? , reaction at ?? would be zero. 
Balancing torque about ?? : 
 
?? (?? -?? )=?????? (???? ) 
From (1) and (2) 
?? (?? ) =(?? +?? )????
?? =(???? )?? 
3.3 If a planet, which revolves around the Sun in a circular orbit, is suddenly 
stopped in its orbit, then find the time in which it would fall into the Sun. Also, find 
the ratio of its falling time to the period of revolution of the planet. 
[2021 : 10 marks] 
Solution: 
Let a planet describing a circular path of radius a and centre ?? (the sun) be stopped at 
the point ?? of its path. Then, it will begin to move towards ?? along the straight line PS-
under the acceleration ?? /( distance )
2
. If ?? is the position of the planet at time ?? such 
that ???? =?? , then the acceleration at ?? is 
?? ?? 2
 directed towards ?? . 
 
 
? The equation of motion of the planet at ?? is 
?? ????
????
=-
?? ?? 2
 
(-ve sign is taken as the acceleration at ?? is in the direction of ?? decreasing) 
?                                                    ?? ????
????
=-
?? ?? 2
???? 
                                 integrating, 
?? 2
2
=
?? ?? +?? , where ?? is a constant. 
But at ?? , 
?? =?? ?? =?? and ?? =0 
 ?                                                           0=
?? ?? +?? ??? =-
?? ?? ?                                                        
?? 2
2
=
?? ?? -
?? ?? =
?? (?? -?? )
????
 
[Note that the planet begins to move along PS with zero velocity at P] 
?                                                            ?? =
????
????
=-
v
(
2?? ?? )×v(
?? -?? ?? ) 
(-ve sign is taken because ?? decreases as tincreases) 
? ???? =-v(
?? 2?? )×v(
?? ?? -?? )???? (??) 
If ?? , is the time taken by the planet from P to S, then integrating (i), we have 
                       ? ?
?? 1
0
????? =-v(
?? 2?? )×? ?
0
?? =?? ?v(
?? ?? -?? )????
 ?                           ?? 1
=-v(
?? 2?? )×? ?
?? /2
0
?v(
?? cos
2
 ?? ?? -?? cos
2
 ?? )×2?? cos ?? ×sin ?????? 
Putting ?? =?? cos
2
 ?? , so that ???? =-2?? cos ?? sin ?????? 
 =?? v(
?? 2?? )
?? /2
? ?
2
0
?cos
2
 ?????? =?? v(
?? 2?? )? ?
?? /2
0
?(?? +cos 2?? )????
 =?? v(
?? 2?? )[?? +
1
2
sin 2?? ]
0
?? /2
=
?? ?? 3/2
2v(2?? )
 
Page 5


Edurev123 
3. Constrained Motion 
3.1 Find the length of an endless chain which will hang over a circular pulley of 
radius a so as to be in contact with three-fourth of the circumference of the pulley. 
(2009 : 15 Marks) 
Solution: 
Since the chain is in contact with 3/4 th of the pulley 
                ANB  =
3
4
× Circumference of pulley 
 =
3
4
×2???? =
3????
2
??????? =
1
4
×2?? =
?? 2
?                                                          ??????? =??????? =
?? 4
 
 
Let ???? be the directrix and ???? the axis of the caternary of the part of the chain hanging 
from the pulley. Let C be the parameter. 
Then,                                                             ???? =?? 
Tangent at ?? is perpendicular to radius. 
So, the tangential angle at ?? , i.e., ?? ?? =
?? 4
 
From ??? ?? '
?? 
??? =???? sin 
?? 4
=
?? v2
 
? At ?? 
?? =?? log (tan ?? +sec ?? ) 
?                                                         
?? v2
=?? log (tan 
?? 4
+sec 
?? 4
) 
 =?? log (1+v2)
?                                                            ?? =
?? v2log (1+v2)
?? =?? tan ?? ??? ?? =
?? v2log (1+v2)
 
?                   
 Arc ???? =
2?? v2log (1+v2)
 Total length of chain  =
v2?? log (1+v2)
+
3????
2
 
3.2 A beam ???? rests on two supports ?? and ?? , where ???? =???? =???? . It is found 
that the beam will tilt when a weight of ?? ???? is hung from ?? or when a weight of 
?? ???? is hung from ?? . Find the weight of the beam. 
(2020: 15 marks) 
Solution: 
(i) Let ???? =???? =???? =?? , and ?? be centroid, w, weight of beam when weight is being 
from ?? and Beam is about to tilt, then reaction at ?? =0 
 
Balancing torque about ?? : 
???? (?? )=?? ?? (??) 
(ii) When, weight is hung from ?? , reaction at ?? would be zero. 
Balancing torque about ?? : 
 
?? (?? -?? )=?????? (???? ) 
From (1) and (2) 
?? (?? ) =(?? +?? )????
?? =(???? )?? 
3.3 If a planet, which revolves around the Sun in a circular orbit, is suddenly 
stopped in its orbit, then find the time in which it would fall into the Sun. Also, find 
the ratio of its falling time to the period of revolution of the planet. 
[2021 : 10 marks] 
Solution: 
Let a planet describing a circular path of radius a and centre ?? (the sun) be stopped at 
the point ?? of its path. Then, it will begin to move towards ?? along the straight line PS-
under the acceleration ?? /( distance )
2
. If ?? is the position of the planet at time ?? such 
that ???? =?? , then the acceleration at ?? is 
?? ?? 2
 directed towards ?? . 
 
 
? The equation of motion of the planet at ?? is 
?? ????
????
=-
?? ?? 2
 
(-ve sign is taken as the acceleration at ?? is in the direction of ?? decreasing) 
?                                                    ?? ????
????
=-
?? ?? 2
???? 
                                 integrating, 
?? 2
2
=
?? ?? +?? , where ?? is a constant. 
But at ?? , 
?? =?? ?? =?? and ?? =0 
 ?                                                           0=
?? ?? +?? ??? =-
?? ?? ?                                                        
?? 2
2
=
?? ?? -
?? ?? =
?? (?? -?? )
????
 
[Note that the planet begins to move along PS with zero velocity at P] 
?                                                            ?? =
????
????
=-
v
(
2?? ?? )×v(
?? -?? ?? ) 
(-ve sign is taken because ?? decreases as tincreases) 
? ???? =-v(
?? 2?? )×v(
?? ?? -?? )???? (??) 
If ?? , is the time taken by the planet from P to S, then integrating (i), we have 
                       ? ?
?? 1
0
????? =-v(
?? 2?? )×? ?
0
?? =?? ?v(
?? ?? -?? )????
 ?                           ?? 1
=-v(
?? 2?? )×? ?
?? /2
0
?v(
?? cos
2
 ?? ?? -?? cos
2
 ?? )×2?? cos ?? ×sin ?????? 
Putting ?? =?? cos
2
 ?? , so that ???? =-2?? cos ?? sin ?????? 
 =?? v(
?? 2?? )
?? /2
? ?
2
0
?cos
2
 ?????? =?? v(
?? 2?? )? ?
?? /2
0
?(?? +cos 2?? )????
 =?? v(
?? 2?? )[?? +
1
2
sin 2?? ]
0
?? /2
=
?? ?? 3/2
2v(2?? )
 
But the time period ?? of the planets revolution is given by, ?? =
2?? ?? 3/2
v
?? 
?                          
?? 1
?? =
1
4v2
=
v2
8
??? 1
=(
v2
8
)?? 
i.e. the time taken by the planet from ?? to ?? . is 
v2
8
 times the period of the planiet's 
revolution.