4 mechanics
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MechanicsTRANSCRIPT
Parallelogram law of forces:
Exercise sheet 1
Q1. The resultant of two forces P and Q acting at a point is R. If Q is doubled R also gets doubled. If Q is reversed R again is doubled. Show the ratio P,Q and R is P:Q:R=2:3:2[RTU 2011]
Sol:
From Parallelogram law of forces: forces P and Q acting at a point and inclined to each other at an angle then the resultant R magnitude
R2=P2+Q2+2.P.Q.CoseqI
Appling condition : Q is doubled R also gets doubled
(2R)2=P2+(2Q)2+2.P. (2Q).Cos
4R2=P2+4Q2+4.P.Q.CoseqII
Appling condition : If Q is reversed R again is doubled
(2R)2=P2+(-Q)2+2.P. (-Q).Cos
4R2=P2+Q2-2.P.Q.CoseqIII
Adding I and III
5R2=2P2+2Q2eqIV
Multiply III by 2 and adding to II
4R2=P2+2Q2
2Q2 = (4R2-P2) eqV
From IV and V
5R2=P2+2Q2
5R2=P2+(4R2-P2)
R=PeqVI
Putting the value R=P in V
4P2=P2+2Q2
eqVII
From VI & VII
P:Q:R=2:3:2
Q2. Two forces P=3kN Q=4kN, Find the greatest and least resultant possible
Sol:
Resultant Is Given From Parallelogram Law Of Forces:
R2=P2+Q2+2.P.Q.CoseqI
Maximum Cos = 1 at =90
Rmax==7kN
Minimum Cos = 0 at =0
Rmin==5kN
Q3. Two coplanar forces 1N and 2N are acting at a point on a rigid body. Their resultant is 3N find the angles between the forces
Sol:
From Parallelogram law of forces: forces P and Q acting at a point and inclined to each other at an angle then the resultant R magnitude
R2=P2+Q2+2.P.Q.CoseqI
=cos-1 = cos-1 = cos-1 (4/4)=0O
Q4. Two coplanar forces 2N and 3N are acting at a point on a rigid body. Their resultant is 4N find all the angles between the forces.
Sol:
From Parallelogram law of forces: forces P and Q acting at a point and inclined to each other at an angle then the resultant R magnitude
R2=P2+Q2+2.P.Q.CoseqI
=cos-1 = cos-1 = cos-1 (1/4)=75.5O
Let be angle between P and R
From Parallelogram law of forces: forces P and Q acting at a point and inclined to each other at an angle then the resultant R direction will be given by
=5.51
=28.8O
Let be angle between Q and R
=-=75.5-28.8=46.7
Q5. Two force A=2F and B=F are acting on a particle. If first is doubled second increased by 12 kN but their direction remains unchanged find F
Sol:
From Parallelogram law of forces : forces A and B acting at a point and inclined to each other at an angle then the resultant R direction will be given by
Appling condition : A=2F and B=F
eqI
Appling condition : A=4F and B=(F+12)
eqII
Equating (I) and (II)
4F+12 Cos+FCos=(F+12)(2+cos)
F=12 kN
Q6. A bolt subjected to two forces as shown. Find the magnitude and direction of the resultant force.
Sol:
Geometric solution: Law of parallelogram:
Magnitude
Angle
Direction from ground= 30.3+20 =50.3 O
Resolution Of A Force Solution:
Q7. The screw eye is subjected to two forces, F1 and F2. Determine the magnitude and direction of the resultant force.
Sol:
Geometric solution: Law of parallelogram:
Magnitude
Angle
Direction of FR measured from the horizontal
Q
.
P
2
2
Q
2
P
2
R
-
-
1
.
2
2
2
1
2
2
2
3
-
-
3
.
2
2
2
3
2
2
2
4
-
-
5
.
3
5
.
75
cos
2
3
5
.
75
sin
2
cos
P
Q
sin
P
tan
1.93
=
+
=
q
+
q
=
a
q
q
a
cos
sin
tan
B
A
B
+
=
q
q
q
q
a
Cos
Sin
F
F
F
+
=
+
=
2
cos
2
sin
tan
q
q
a
cos
)
12
(
4
sin
)
12
(
tan
+
+
+
=
F
F
F
q
q
q
q
cos
)
12
(
4
sin
)
12
(
2
+
+
+
=
+
F
F
F
Cos
Sin
N
Cos
R
Cos
F
F
F
F
R
O
8
.
279
45
.
200
.
100
.
2
200
100
45
20
65
.
.
.
2
2
2
2
1
2
2
2
1
=
+
+
=
=
-
=
+
+
=
q
q
Q
3
.
30
.
.
2
1
2
1
=
+
=
-
q
q
a
Cos
F
F
Sin
F
Tan
N
Cos
R
Cos
F
F
F
F
R
O
213
65
.
150
.
100
.
2
150
100
65
15
10
90
.
.
.
2
2
2
2
1
2
2
2
1
=
+
+
=
=
-
-
=
+
+
=
q
q
Q
N
Cos
Sin
Tan
Cos
F
F
Sin
F
Tan
39
65
.
150
100
65
.
150
.
.
1
2
1
2
1
=
+
=
+
=
-
-
q
q
a
f
f
=
+
=
o
o
o
8
.
54
15
8
.
39
P
Q
2
3
=
4
3
2
4
3
2
2
+
+
0
4
3
2
2
+
+