Formula Sheet: Engineering Mechanics | Engineering Mechanics - Civil Engineering (CE) PDF Download

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2 
 
ENGINEERING MECHANICS IMPORTANT FORMULA 
FORCE 
1. SYSTEM OF FORCES 
 
1.1. Coplanar forces 
Forces whose lines of action lies on the same plane 
1.2. Non-coplanar forces 
Forces whose lines of action do not lie on the same plane. 
1.3. Collinear forces 
Forces whose lines of action lie on the same line. 
 
1.4. Concurrent forces 
Forces, whose lines of action meet at one point They may or may not be collinear & 
coplanar 
 
1.5. Parallel forces 
Forces, whose lines of action are parallel to each other. They may or may not be 
coplanar. 
1.6. Non-concurrent & Non-parallel forces 
Forces, whose lines of action do not meet or tend to meet at same point. They are also 
not parallel to each other. 
They may or may not be coplanar. 
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2 
 
ENGINEERING MECHANICS IMPORTANT FORMULA 
FORCE 
1. SYSTEM OF FORCES 
 
1.1. Coplanar forces 
Forces whose lines of action lies on the same plane 
1.2. Non-coplanar forces 
Forces whose lines of action do not lie on the same plane. 
1.3. Collinear forces 
Forces whose lines of action lie on the same line. 
 
1.4. Concurrent forces 
Forces, whose lines of action meet at one point They may or may not be collinear & 
coplanar 
 
1.5. Parallel forces 
Forces, whose lines of action are parallel to each other. They may or may not be 
coplanar. 
1.6. Non-concurrent & Non-parallel forces 
Forces, whose lines of action do not meet or tend to meet at same point. They are also 
not parallel to each other. 
They may or may not be coplanar. 
 
3 
 
 
2. RESOLUTION OF FORCES 
The splitting up the given force into number of components, without changing its effect on the body 
is called resolution of a force. 
 
S H = F1 cos ?1 + F2 cos ?2 
S V = F1 sin ?1 + F2 sin ?2 
Resultant force = ( ) ( )
22
HV ? + ? 
tan
V
H
?
?
=
?
 
 
3. LAWS OF RESULTANT FORCE 
3.1. Triangle law of forces 
If two forces acting simultaneously on a particle, be represented in magnitude and 
direction by the two sides of a triangle, taken in order; their resultant may be represented 
in magnitude and direction by the third side of the triangle, taken in opposite order. 
 
R is the resultant of & AB 
3.2. Parallelogram law of forces 
If two forces, acting simultaneously on a particle, are represented in magnitude & 
direction by the two adjacent sides of a parallelogram; their resultant may be represented 
in magnitude & direction by the diagonal of the parallelogram, passing through their point 
of intersection. 
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2 
 
ENGINEERING MECHANICS IMPORTANT FORMULA 
FORCE 
1. SYSTEM OF FORCES 
 
1.1. Coplanar forces 
Forces whose lines of action lies on the same plane 
1.2. Non-coplanar forces 
Forces whose lines of action do not lie on the same plane. 
1.3. Collinear forces 
Forces whose lines of action lie on the same line. 
 
1.4. Concurrent forces 
Forces, whose lines of action meet at one point They may or may not be collinear & 
coplanar 
 
1.5. Parallel forces 
Forces, whose lines of action are parallel to each other. They may or may not be 
coplanar. 
1.6. Non-concurrent & Non-parallel forces 
Forces, whose lines of action do not meet or tend to meet at same point. They are also 
not parallel to each other. 
They may or may not be coplanar. 
 
3 
 
 
2. RESOLUTION OF FORCES 
The splitting up the given force into number of components, without changing its effect on the body 
is called resolution of a force. 
 
S H = F1 cos ?1 + F2 cos ?2 
S V = F1 sin ?1 + F2 sin ?2 
Resultant force = ( ) ( )
22
HV ? + ? 
tan
V
H
?
?
=
?
 
 
3. LAWS OF RESULTANT FORCE 
3.1. Triangle law of forces 
If two forces acting simultaneously on a particle, be represented in magnitude and 
direction by the two sides of a triangle, taken in order; their resultant may be represented 
in magnitude and direction by the third side of the triangle, taken in opposite order. 
 
R is the resultant of & AB 
3.2. Parallelogram law of forces 
If two forces, acting simultaneously on a particle, are represented in magnitude & 
direction by the two adjacent sides of a parallelogram; their resultant may be represented 
in magnitude & direction by the diagonal of the parallelogram, passing through their point 
of intersection. 
 
4 
 
 
Resultant R is given by R = 
22
1 2 1 2
2 cos P P PP ? ++ 
The angle (a) which the resultant makes with P2 
 = ?????? ?? =
?? 1
???? ?? ?? ?? 2
+ ?? 1
?? ???? ?? 
Special cases: 
(i) When ? = 0°, R = P1 + P2 
(ii) When ? = 90°, R = 
22
12
PP + 
(iii) When ? = 180°, R = P1 – P2 
(iv) When P1 = P2, R = 2 cos
2
P
? ??
??
??
 
3.3. Polygon law of forces 
If number of forces acting simultaneously on a particle, be represented in magnitude & 
direction, by the sides of the polygon taken in order, then the resultant of all these forces 
is represented, in magnitude & direction by the closing side of the polygon, taken in 
opposite order. 
 
R is the resultant of , , & A B C D vectors. 
  
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2 
 
ENGINEERING MECHANICS IMPORTANT FORMULA 
FORCE 
1. SYSTEM OF FORCES 
 
1.1. Coplanar forces 
Forces whose lines of action lies on the same plane 
1.2. Non-coplanar forces 
Forces whose lines of action do not lie on the same plane. 
1.3. Collinear forces 
Forces whose lines of action lie on the same line. 
 
1.4. Concurrent forces 
Forces, whose lines of action meet at one point They may or may not be collinear & 
coplanar 
 
1.5. Parallel forces 
Forces, whose lines of action are parallel to each other. They may or may not be 
coplanar. 
1.6. Non-concurrent & Non-parallel forces 
Forces, whose lines of action do not meet or tend to meet at same point. They are also 
not parallel to each other. 
They may or may not be coplanar. 
 
3 
 
 
2. RESOLUTION OF FORCES 
The splitting up the given force into number of components, without changing its effect on the body 
is called resolution of a force. 
 
S H = F1 cos ?1 + F2 cos ?2 
S V = F1 sin ?1 + F2 sin ?2 
Resultant force = ( ) ( )
22
HV ? + ? 
tan
V
H
?
?
=
?
 
 
3. LAWS OF RESULTANT FORCE 
3.1. Triangle law of forces 
If two forces acting simultaneously on a particle, be represented in magnitude and 
direction by the two sides of a triangle, taken in order; their resultant may be represented 
in magnitude and direction by the third side of the triangle, taken in opposite order. 
 
R is the resultant of & AB 
3.2. Parallelogram law of forces 
If two forces, acting simultaneously on a particle, are represented in magnitude & 
direction by the two adjacent sides of a parallelogram; their resultant may be represented 
in magnitude & direction by the diagonal of the parallelogram, passing through their point 
of intersection. 
 
4 
 
 
Resultant R is given by R = 
22
1 2 1 2
2 cos P P PP ? ++ 
The angle (a) which the resultant makes with P2 
 = ?????? ?? =
?? 1
???? ?? ?? ?? 2
+ ?? 1
?? ???? ?? 
Special cases: 
(i) When ? = 0°, R = P1 + P2 
(ii) When ? = 90°, R = 
22
12
PP + 
(iii) When ? = 180°, R = P1 – P2 
(iv) When P1 = P2, R = 2 cos
2
P
? ??
??
??
 
3.3. Polygon law of forces 
If number of forces acting simultaneously on a particle, be represented in magnitude & 
direction, by the sides of the polygon taken in order, then the resultant of all these forces 
is represented, in magnitude & direction by the closing side of the polygon, taken in 
opposite order. 
 
R is the resultant of , , & A B C D vectors. 
  
 
5 
 
FRICTION 
The friction is a force distribution at the surfaces of contact and acts tangential to the surface of 
contact. 
1. LIMITING FRICTION 
The maximum value of frictional force, which comes into play, when a body just begins to slide 
over the surface of the other body. 
2. LAWS OF FRICTION 
2.1. Laws of static friction 
1. Force of friction always acts in a direction, opposite to that in which the body tends to 
move. 
2. The magnitude of the limiting friction bears a constant ratio to the normal reaction 
between the two surfaces. 
= tan
F
Cons t
R
 
Where, 
F = limiting friction 
R = normal reaction 
3. The force of friction is independent of the area of contact between the two surfaces. 
4. The force of friction depends upon the roughness of the surfaces. 
2.2. Laws of kinetic or dynamic friction 
1. The force of friction always acts in a direction opposite to that in which the body is 
moving. 
2. The magnitude of kinetic friction bears a constant ratio ratio to the normal reaction 
between two surfaces. 
This ratio is slightly less than that in case of limiting friction. 
sk
F
R
?
??
=
?
 
2.3. Angle of Repose ( ?) 
The value of the angle of inclination ? corresponding to impending motion is called the 
angle of repose. 
Page 5


 
2 
 
ENGINEERING MECHANICS IMPORTANT FORMULA 
FORCE 
1. SYSTEM OF FORCES 
 
1.1. Coplanar forces 
Forces whose lines of action lies on the same plane 
1.2. Non-coplanar forces 
Forces whose lines of action do not lie on the same plane. 
1.3. Collinear forces 
Forces whose lines of action lie on the same line. 
 
1.4. Concurrent forces 
Forces, whose lines of action meet at one point They may or may not be collinear & 
coplanar 
 
1.5. Parallel forces 
Forces, whose lines of action are parallel to each other. They may or may not be 
coplanar. 
1.6. Non-concurrent & Non-parallel forces 
Forces, whose lines of action do not meet or tend to meet at same point. They are also 
not parallel to each other. 
They may or may not be coplanar. 
 
3 
 
 
2. RESOLUTION OF FORCES 
The splitting up the given force into number of components, without changing its effect on the body 
is called resolution of a force. 
 
S H = F1 cos ?1 + F2 cos ?2 
S V = F1 sin ?1 + F2 sin ?2 
Resultant force = ( ) ( )
22
HV ? + ? 
tan
V
H
?
?
=
?
 
 
3. LAWS OF RESULTANT FORCE 
3.1. Triangle law of forces 
If two forces acting simultaneously on a particle, be represented in magnitude and 
direction by the two sides of a triangle, taken in order; their resultant may be represented 
in magnitude and direction by the third side of the triangle, taken in opposite order. 
 
R is the resultant of & AB 
3.2. Parallelogram law of forces 
If two forces, acting simultaneously on a particle, are represented in magnitude & 
direction by the two adjacent sides of a parallelogram; their resultant may be represented 
in magnitude & direction by the diagonal of the parallelogram, passing through their point 
of intersection. 
 
4 
 
 
Resultant R is given by R = 
22
1 2 1 2
2 cos P P PP ? ++ 
The angle (a) which the resultant makes with P2 
 = ?????? ?? =
?? 1
???? ?? ?? ?? 2
+ ?? 1
?? ???? ?? 
Special cases: 
(i) When ? = 0°, R = P1 + P2 
(ii) When ? = 90°, R = 
22
12
PP + 
(iii) When ? = 180°, R = P1 – P2 
(iv) When P1 = P2, R = 2 cos
2
P
? ??
??
??
 
3.3. Polygon law of forces 
If number of forces acting simultaneously on a particle, be represented in magnitude & 
direction, by the sides of the polygon taken in order, then the resultant of all these forces 
is represented, in magnitude & direction by the closing side of the polygon, taken in 
opposite order. 
 
R is the resultant of , , & A B C D vectors. 
  
 
5 
 
FRICTION 
The friction is a force distribution at the surfaces of contact and acts tangential to the surface of 
contact. 
1. LIMITING FRICTION 
The maximum value of frictional force, which comes into play, when a body just begins to slide 
over the surface of the other body. 
2. LAWS OF FRICTION 
2.1. Laws of static friction 
1. Force of friction always acts in a direction, opposite to that in which the body tends to 
move. 
2. The magnitude of the limiting friction bears a constant ratio to the normal reaction 
between the two surfaces. 
= tan
F
Cons t
R
 
Where, 
F = limiting friction 
R = normal reaction 
3. The force of friction is independent of the area of contact between the two surfaces. 
4. The force of friction depends upon the roughness of the surfaces. 
2.2. Laws of kinetic or dynamic friction 
1. The force of friction always acts in a direction opposite to that in which the body is 
moving. 
2. The magnitude of kinetic friction bears a constant ratio ratio to the normal reaction 
between two surfaces. 
This ratio is slightly less than that in case of limiting friction. 
sk
F
R
?
??
=
?
 
2.3. Angle of Repose ( ?) 
The value of the angle of inclination ? corresponding to impending motion is called the 
angle of repose. 
 
6 
 
 
Since the block is still in equilibrium, it follows from the free body diagram that  
SFx = µsN – mgsin? = 0 ? µsN = mgsin?  
SFy = N – mgcos? = 0 ? N = mgcos?  
Equating above two equations, we get 
µs = tan?  
and since angle of static friction  
µs = tan ?s 
therefore ? = ?s 
i.e., value of angle of repose has the same value as that of angle of static friction. 
2.4. Belt friction 
Belts are used to transfer the energy from one axis to another by winding over pulley or 
drum. 
A flat belt passing over a drum where T1 and T2 (T2 > T1) are the tensions in the belt 
when belt is about to slide to right.