Equilibrium of a System or Particles | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
5. Equilibrium of a system or particles 
5.1 A uniform rod ???? is movable about ?? hinge ?? and rests with one end in 
contact with a smooth vertical wall. If the rod is inclined at an angle ????
°
 with the 
horizontal find the reaction at the hinge in magnitude and direction. 
(2009 : 12 Marks) 
Solution: 
Approach : Balance forces and moments to get direction of the reaction at the hinge. 
Let the reaction in the hinge in vertical and horizontal direction by ?? and ?? respectively 
and the weight along the midpoint be ?? and the reaction at the wall (perpendicular to it) 
be ?? . 
Balancing forces in horizontal and vertical direction 
?? =?? and ?? =?? 
                                                    
Let length of rod =2?? 
Now taking moments about the hinge to negate effect of reaction by hinge. 
?? ·2asin 30
°
=?? ·?? ·cos 30
°
 
where 
2?? =???? 
Page 2


Edurev123 
5. Equilibrium of a system or particles 
5.1 A uniform rod ???? is movable about ?? hinge ?? and rests with one end in 
contact with a smooth vertical wall. If the rod is inclined at an angle ????
°
 with the 
horizontal find the reaction at the hinge in magnitude and direction. 
(2009 : 12 Marks) 
Solution: 
Approach : Balance forces and moments to get direction of the reaction at the hinge. 
Let the reaction in the hinge in vertical and horizontal direction by ?? and ?? respectively 
and the weight along the midpoint be ?? and the reaction at the wall (perpendicular to it) 
be ?? . 
Balancing forces in horizontal and vertical direction 
?? =?? and ?? =?? 
                                                    
Let length of rod =2?? 
Now taking moments about the hinge to negate effect of reaction by hinge. 
?? ·2asin 30
°
=?? ·?? ·cos 30
°
 
where 
2?? =???? 
?                                                                            ?? =
?? 2
cot 30
°
=
v3?? 2
 
?                                                                            ?? =
v3?? 2
,?? =?? 
           Net reaction=v?? 2
+?? 2
=?? v
3
4
+1 
=
v7
2
?? 
Let ?? be angle of resultant from horizontal. 
tan ?? =
v3?? 2
?? =
v3
2
? ?? =tan
-1
 
v3
2
 
 
5.2 A ladder of weight ?? rests with one end against a smooth vertical wall and the 
other end rests on a smooth floor. If the inclination of the ladder to the horizon is 
????
°
, find the horizontal force that must be applied to the lower end to prevent the 
ladder from slipping down. 
(2011 : 20 Marks) 
Solution: 
Let ???? be the ladder. 
Forces acting on the ladder are 
1. Normal reaction R at A perpendicular to ???? . 
2. Normal reaction ?? at ?? perpendicular to ???? . 
3. Weight ?? of rod acting vertically downwards through ?? , the mid-point of ???? . 
Let ?? be the required horizontal force. Resolving horizontally and vertically, we get 
?? =?? (??) 
 and                                                                         ?? =?? 
Again, taking moments about ?? , we get 
?? ·???? =?? ·???? sin 60
°
 
Page 3


Edurev123 
5. Equilibrium of a system or particles 
5.1 A uniform rod ???? is movable about ?? hinge ?? and rests with one end in 
contact with a smooth vertical wall. If the rod is inclined at an angle ????
°
 with the 
horizontal find the reaction at the hinge in magnitude and direction. 
(2009 : 12 Marks) 
Solution: 
Approach : Balance forces and moments to get direction of the reaction at the hinge. 
Let the reaction in the hinge in vertical and horizontal direction by ?? and ?? respectively 
and the weight along the midpoint be ?? and the reaction at the wall (perpendicular to it) 
be ?? . 
Balancing forces in horizontal and vertical direction 
?? =?? and ?? =?? 
                                                    
Let length of rod =2?? 
Now taking moments about the hinge to negate effect of reaction by hinge. 
?? ·2asin 30
°
=?? ·?? ·cos 30
°
 
where 
2?? =???? 
?                                                                            ?? =
?? 2
cot 30
°
=
v3?? 2
 
?                                                                            ?? =
v3?? 2
,?? =?? 
           Net reaction=v?? 2
+?? 2
=?? v
3
4
+1 
=
v7
2
?? 
Let ?? be angle of resultant from horizontal. 
tan ?? =
v3?? 2
?? =
v3
2
? ?? =tan
-1
 
v3
2
 
 
5.2 A ladder of weight ?? rests with one end against a smooth vertical wall and the 
other end rests on a smooth floor. If the inclination of the ladder to the horizon is 
????
°
, find the horizontal force that must be applied to the lower end to prevent the 
ladder from slipping down. 
(2011 : 20 Marks) 
Solution: 
Let ???? be the ladder. 
Forces acting on the ladder are 
1. Normal reaction R at A perpendicular to ???? . 
2. Normal reaction ?? at ?? perpendicular to ???? . 
3. Weight ?? of rod acting vertically downwards through ?? , the mid-point of ???? . 
Let ?? be the required horizontal force. Resolving horizontally and vertically, we get 
?? =?? (??) 
 and                                                                         ?? =?? 
Again, taking moments about ?? , we get 
?? ·???? =?? ·???? sin 60
°
 
?                                              ?? ·???? cos 60
°
=?? ·???? sin 60
°
 
?                                                       ?? ·
????
2
·
1
2
=?? ·???? ·
v3
2
 
?                                                                        ?? =
?? 2v3
 
?                                                                        ?? =?? =
?? 2v3
 
 
5.3 A heavy hemispherical shell of radius a has a particle attached to a point on 
the rim, and rests with the curved surface in contact with a rough sphere of radius 
?? at the highest point. Prove that if 
?? ?? >v?? -?? , the equilibriun is stable, whatever 
be the weight of the particle. 
(2012 : 20 Marks) 
Solution: 
Let ?? '
 be the centre of the base of the hemispherical shell of radius a. Let a weight be 
attached to the rim of the hemispherical shell at ?? . 
The centre of gravity ?? 1
 of the hemispherical shell is on its symmetrical 
radius ???? and ?? ?? 1
=
1
2
?? '
?? =
?? 2
. 
Let ?? be the centre of gravity of the combined body consisting of the hemispherical shell 
and the weight at ?? . Then ?? lies on the line ?? ?? 1
. 
Page 4


Edurev123 
5. Equilibrium of a system or particles 
5.1 A uniform rod ???? is movable about ?? hinge ?? and rests with one end in 
contact with a smooth vertical wall. If the rod is inclined at an angle ????
°
 with the 
horizontal find the reaction at the hinge in magnitude and direction. 
(2009 : 12 Marks) 
Solution: 
Approach : Balance forces and moments to get direction of the reaction at the hinge. 
Let the reaction in the hinge in vertical and horizontal direction by ?? and ?? respectively 
and the weight along the midpoint be ?? and the reaction at the wall (perpendicular to it) 
be ?? . 
Balancing forces in horizontal and vertical direction 
?? =?? and ?? =?? 
                                                    
Let length of rod =2?? 
Now taking moments about the hinge to negate effect of reaction by hinge. 
?? ·2asin 30
°
=?? ·?? ·cos 30
°
 
where 
2?? =???? 
?                                                                            ?? =
?? 2
cot 30
°
=
v3?? 2
 
?                                                                            ?? =
v3?? 2
,?? =?? 
           Net reaction=v?? 2
+?? 2
=?? v
3
4
+1 
=
v7
2
?? 
Let ?? be angle of resultant from horizontal. 
tan ?? =
v3?? 2
?? =
v3
2
? ?? =tan
-1
 
v3
2
 
 
5.2 A ladder of weight ?? rests with one end against a smooth vertical wall and the 
other end rests on a smooth floor. If the inclination of the ladder to the horizon is 
????
°
, find the horizontal force that must be applied to the lower end to prevent the 
ladder from slipping down. 
(2011 : 20 Marks) 
Solution: 
Let ???? be the ladder. 
Forces acting on the ladder are 
1. Normal reaction R at A perpendicular to ???? . 
2. Normal reaction ?? at ?? perpendicular to ???? . 
3. Weight ?? of rod acting vertically downwards through ?? , the mid-point of ???? . 
Let ?? be the required horizontal force. Resolving horizontally and vertically, we get 
?? =?? (??) 
 and                                                                         ?? =?? 
Again, taking moments about ?? , we get 
?? ·???? =?? ·???? sin 60
°
 
?                                              ?? ·???? cos 60
°
=?? ·???? sin 60
°
 
?                                                       ?? ·
????
2
·
1
2
=?? ·???? ·
v3
2
 
?                                                                        ?? =
?? 2v3
 
?                                                                        ?? =?? =
?? 2v3
 
 
5.3 A heavy hemispherical shell of radius a has a particle attached to a point on 
the rim, and rests with the curved surface in contact with a rough sphere of radius 
?? at the highest point. Prove that if 
?? ?? >v?? -?? , the equilibriun is stable, whatever 
be the weight of the particle. 
(2012 : 20 Marks) 
Solution: 
Let ?? '
 be the centre of the base of the hemispherical shell of radius a. Let a weight be 
attached to the rim of the hemispherical shell at ?? . 
The centre of gravity ?? 1
 of the hemispherical shell is on its symmetrical 
radius ???? and ?? ?? 1
=
1
2
?? '
?? =
?? 2
. 
Let ?? be the centre of gravity of the combined body consisting of the hemispherical shell 
and the weight at ?? . Then ?? lies on the line ?? ?? 1
. 
Let ???? =h. Also, here ?? 1
=?? ?? ?? 2
=?? . 
 
The equilibrium will be stable if 
1
h
>
1
?? 1
+
1
?? 2
. Here ?? 1
 is the radius of the upper body, ?? 2
 is 
the radius of the lower body and h is the height of the C.G. of the upper body above the 
point of contact. 
?                                                    
1
h
>
1
?? +
1
?? or h<
????
?? +??                                                               (??) 
The value of h depends on the weight of the particle aiiached at ?? . 
? The equilibrium will be stable whatever be the weight of the particle attached at ?? , if 
the relation (i) holds even tor the maximum value of h. 
Now h will be maximum if ?? '
?? is minimum, i.e., if ???? is perpendicular to ?? ?? 1
. 
Let                                          ??? '
???? =?? 
Then, in right angled ??????? 
Page 5


Edurev123 
5. Equilibrium of a system or particles 
5.1 A uniform rod ???? is movable about ?? hinge ?? and rests with one end in 
contact with a smooth vertical wall. If the rod is inclined at an angle ????
°
 with the 
horizontal find the reaction at the hinge in magnitude and direction. 
(2009 : 12 Marks) 
Solution: 
Approach : Balance forces and moments to get direction of the reaction at the hinge. 
Let the reaction in the hinge in vertical and horizontal direction by ?? and ?? respectively 
and the weight along the midpoint be ?? and the reaction at the wall (perpendicular to it) 
be ?? . 
Balancing forces in horizontal and vertical direction 
?? =?? and ?? =?? 
                                                    
Let length of rod =2?? 
Now taking moments about the hinge to negate effect of reaction by hinge. 
?? ·2asin 30
°
=?? ·?? ·cos 30
°
 
where 
2?? =???? 
?                                                                            ?? =
?? 2
cot 30
°
=
v3?? 2
 
?                                                                            ?? =
v3?? 2
,?? =?? 
           Net reaction=v?? 2
+?? 2
=?? v
3
4
+1 
=
v7
2
?? 
Let ?? be angle of resultant from horizontal. 
tan ?? =
v3?? 2
?? =
v3
2
? ?? =tan
-1
 
v3
2
 
 
5.2 A ladder of weight ?? rests with one end against a smooth vertical wall and the 
other end rests on a smooth floor. If the inclination of the ladder to the horizon is 
????
°
, find the horizontal force that must be applied to the lower end to prevent the 
ladder from slipping down. 
(2011 : 20 Marks) 
Solution: 
Let ???? be the ladder. 
Forces acting on the ladder are 
1. Normal reaction R at A perpendicular to ???? . 
2. Normal reaction ?? at ?? perpendicular to ???? . 
3. Weight ?? of rod acting vertically downwards through ?? , the mid-point of ???? . 
Let ?? be the required horizontal force. Resolving horizontally and vertically, we get 
?? =?? (??) 
 and                                                                         ?? =?? 
Again, taking moments about ?? , we get 
?? ·???? =?? ·???? sin 60
°
 
?                                              ?? ·???? cos 60
°
=?? ·???? sin 60
°
 
?                                                       ?? ·
????
2
·
1
2
=?? ·???? ·
v3
2
 
?                                                                        ?? =
?? 2v3
 
?                                                                        ?? =?? =
?? 2v3
 
 
5.3 A heavy hemispherical shell of radius a has a particle attached to a point on 
the rim, and rests with the curved surface in contact with a rough sphere of radius 
?? at the highest point. Prove that if 
?? ?? >v?? -?? , the equilibriun is stable, whatever 
be the weight of the particle. 
(2012 : 20 Marks) 
Solution: 
Let ?? '
 be the centre of the base of the hemispherical shell of radius a. Let a weight be 
attached to the rim of the hemispherical shell at ?? . 
The centre of gravity ?? 1
 of the hemispherical shell is on its symmetrical 
radius ???? and ?? ?? 1
=
1
2
?? '
?? =
?? 2
. 
Let ?? be the centre of gravity of the combined body consisting of the hemispherical shell 
and the weight at ?? . Then ?? lies on the line ?? ?? 1
. 
Let ???? =h. Also, here ?? 1
=?? ?? ?? 2
=?? . 
 
The equilibrium will be stable if 
1
h
>
1
?? 1
+
1
?? 2
. Here ?? 1
 is the radius of the upper body, ?? 2
 is 
the radius of the lower body and h is the height of the C.G. of the upper body above the 
point of contact. 
?                                                    
1
h
>
1
?? +
1
?? or h<
????
?? +??                                                               (??) 
The value of h depends on the weight of the particle aiiached at ?? . 
? The equilibrium will be stable whatever be the weight of the particle attached at ?? , if 
the relation (i) holds even tor the maximum value of h. 
Now h will be maximum if ?? '
?? is minimum, i.e., if ???? is perpendicular to ?? ?? 1
. 
Let                                          ??? '
???? =?? 
Then, in right angled ??????? 
                                             tan ?? =
?? '
?? 1
?? '
?? =
?? /2
?? =
1
2
 ?                                        sin ?? =
1
v5
 ?        Minimum value of ???? =???? sin ??                                                          =?? ×
1
v5
=
?? v5
 
? The maximum value of h 
 =?? - the minimum value of ????
 =?? -
?? v5
=
?? (v5-1)
v5
 
? from (i), for stable equilibrium 
?? (v5-1)
v5
<
????
?? +??         
 i.e., 
v5-1
v5
<
?? ?? +?? i.e., (v5-1)?? <??         
 i.e., 
?? ?? >v5-1         
 
5.4 The end links of a uniform chain slide along a fixed rough horizontal rod. 
Prove that the ratio of the maximum span to the length of the chain is 
?? ?????? [
?? +v?? +?? ?? ?? ] where ?? is the coefficient of friction. 
(2012: 20 marks) 
Solution: 
Let ???? be the maximum span of the chain. Hence, the end links ?? and ?? are in limiting 
equilibrium, each under three forces, namely the normai reaction ?? perpendicular to ???? 
(upwards), the force of friction ???? along the fixed horizontal rod outwards and the 
tension ?? along the tangent at ?? (or ?? ). 
If ?? is the resuitant of the forces ?? and ???? at ?? , then ?? will make an angle ?? (where 
tan ?? =?? ) with the direction of ?? . 
For the equilibrium of ?? the resultant ?? of ?? and ?? ?? at ?? will be equal and opposite to the 
tension ?? at ?? . But the tension at ?? acts along the tangent to the chain at ?? , therefore the 
tangent to the catenary at ?? makes an angle ?? ?? =
?? 2
-?? to the horizontal. 
Thus, for the point ?? of the catenary, we have 
?? =?? ?? =
?? 2
-??