EN09 105 ENGINEERING
MECHANICS
Tutorial
sheet - I
PARALLELOGRAM
LAW OF FORCES
1. The angle between the resultant
(35N) of two forces and one of the forces (15N) is 38.210.Find the
other force and its inclination with the resultant. (Ans: 25N,600) [ apply triangular law]
2. The greatest & least
resultants of two forces acting on a particle are 35kN & 5kN respectively.
If 25KN is the magnitude of the resultant for the given system of forces F1
and F2,prove that the forces are at right angles.( hints: When R is
max, θ= 0,R is min θ= 1800)
3. Two forces equal to 2F and F act
on a particle. If the first be doubled and the second increased by 15N, the
direction of the resultant remains unaltered. Find the value of F.( Ans:15N )
4. Two forces P and Q of magnitude
25N and 10N are acting at a point. The force P and Q makes angles 15° and 45°
measured counterclockwise with horizontal. Determine the resultant in magnitude
and direction. ( Ans: 34.03N,α=8.45°)
5. A boat is moved by pulling with
forces 200N and 240N acting at an angle 60° as shown in the figure. Find the
resultant in magnitude and angle made by resultant with 200N and 240N.(Refer
Figure 1)
(Ans:
R= 381.5 N, α=27°, β=33°)
6. A boat is moved uniformly by
pulling with forces P = 240N and Q = 200 N .What must be the inclination of the
resultant force with P and Q to have R=400N as shown in the figure.(Refer
Figure 2) (Ans: α=22.33°,
β=27.13°)
7. The resultant of two forces when
they act an angle of 60° is 14N.When the same forces act at right angles R = 12
N .Determine magnitude of two forces.(Ans: P=11.03N,Q=4.71N)
8. What force P combined with a
vertical force Q=12 N will give a horizontal resultant of 16N. (Ans:
P= 20 N, α=36.86°)
9. A telephone cable is clamped to
the pole as shown in the figure, knowing that the tension in the left hand
portion of the cable is T1= 800N.Determine the Tension T2
in the right hand side portion if the resultant of forces T1 and T2
exerted by cable at A is to be vertical. Also find the magnitude of the
resultant. .(Refer Figure 3) (Ans: T2 = 852.3 N, R=567.3N)
LAMI’S THEOREM
10. An electric light fixture
weighing 15N hangs from a point C by two strings Ac and BC.The strings AC and
BC.The string AC is inclined at 60° the horizontal and string BC at
45° to the horizontal as shown in the figure. Using Lami’s theorem
or otherwise determine the forces in the strings AC and BC. .(Refer Figure 4) (Ans:
TBC = 7.76N, TAC =10.98N)
11. A system of connected flexible
cables are shown in figure supporting two vertical forces 200N and 250N at
points B and D. Determine the forces in the various segments of the cable. .
(Refer Figure 5) (Ans: TBD =
183.01N, TDE =224.14N, TBC = 336.6N, TAB =326.79N)
12. A wire rope is fixed at two
points A and D as shown in the figure. Two weights 20kN and 30kNare attached to
it at B and C respectively. The weights rests with portions AB and BC inclined
at angles 30° and 50° respectively to the vertical. Find
the tension in the wire segments AB, BC and CD.Also find the inclination of
segment CD to the vertical.
(Refer Figure 6) (Ans: TCD =25.05kN,
TBC = 29.24kN, TAB =44.79kN, θ=63.42°)
13. A wire is fixed at two points A
and D as shown in the figure. Two weights 20kN and 25kN are supported at B and
C respectively. When equilibrium is reached it is found that the inclination of
AB is 30° and of CD is 60° to the vertical. Determine the
tension in the segments AB, BC and CD of the rope and also find the inclination
of BC to the vertical.
(Refer Figure 7) (Ans: TCD =38.97kN,
TBC = 23.84kN, TAB =44.79kN, θ=54.8°)
14. A fine light string ABCDE whose
extremity A is fixed, has weights W and W1 attached to it at B and C
and passes around a smooth peg at D,carrying a weight of 40N at free end E.In
the position of equilibrium, BC is horizontal and AB,CD makes angles 150° and
120° respectively with BC.Find
i.
Tension
in AB, BC, CD, DE.
ii.
Values
of W and W1.
iii.
Pressure on peg at D.
(Ans:
TBC =20N, TCD =40N, TDE = 40N, TAB =23.09N,
W=11.55N, W1 =34.64N)
COMPOSITION
OF CONCURRENT FORCES BY METHOD OF RESOLUTION
15. Determine the magnitude and
direction of the resultant of three forces acting on a hook as shown in figure.
.(Refer Figure 8) (Ans: R= 161.48N, θ=18.18°)
16. The forces 20,30,40,50 and ‘10
+X’ N are acting at one of the angular points of the regular hexagon towards
the other five angular points taken in order. Find the magnitude and direction
of the resultant.(replace ‘X’ with your roll number) (Ans: R= 159.9N, θ=76.64°)
17. The following forces act at a
point
20N,
inclined at 30° North of East
25N
towards North
30N
towards North West
35N
inclined at 40° South of West
Find
the magnitude and direction of the resultant forces in analytical method and
graphical method. (Ans: R= 45.6N, θ=47.68°)
18. A particle is acted upon by three
forces of magnitude 50N, 100N and 130N along three sides of an equilateral
triangle taken in order. Find the magnitude and direction of the resultant
force graphically.
(Ans:
R= 70N, θ=222°)
19. ABCDEF is a regular
hexagon,Forces4,X,Y,6,2 acts along AB,CA,AE,AD,FA respectively. Find the
magnitude of X and Y in order that the system may be in equilibrium. (Ans:
X= 9.24N, Y= 1.15N)
20. Four coplanar forces are acting
at a point as shown in the figure. Determine the magnitude and direction of the
resultant force graphically and analytically. .(Refer Figure 9) (Ans:
R= 260.25N, θ=17.4°)
21. Five strings are tied at a point
are pulled in all direction, equally spaced from one another. If the magnitude
of pull on three consecutive strings is 50N, 70N and 60N respectively. Find
analytically and graphically, the magnitude of pulls on the other two strings.
(Ans:
X= 56.17N, X= 72.35N)
22. A sphere of weight 100N is tied
to a smooth wall as shown in the figure. Find the tension in the string and
reaction of the wall. (Refer Figure 10) (Ans: TAB
=103.52N , RC =26.79N)
23. Determine the horizontal force P
to be applied to a block of weight 1500N to be held it in position on smooth
inclined plane AB which makes angle of 30° with the horizontal.
(Refer Figure 11) (Ans: R =1732.05N, P =866.025N)
24. A roller of weight 10kN rest on a smooth
horizontal floor and is connected to the floor by a bar AC. Determine the force
in the bar AC and the reaction from the floor if roller subjected to a
horizontal force of kN and inclined force of 7kN. (Refer Figure 12). (Ans:
TAC =0.058kN, RB=14.98kN)
25. Two smooth spheres each of radius
100mm and weight 100N rests on a horizontal channel having vertical wall,
distance between which is 360mm.Find the reactions at points of contact.
(Refer Figure 13) (Ans: RA = RD=133.34kN,
RB=166.67kN, RC= 200kN)
26. Find the reactions on the contact
points if
a. If
the weight of sphere is 120N (Ans: RA =60N, RC =103.92N)
b. If
the weight of two spheres are 1000N.(Refer Figure 14a & 14b)(Ans: RA
= 866.03N, RB=1443.37N, RC=1154.69N, RD=
500N)
27. Two cylinders A of weight 4000N
and of weight 2000N rest on a smooth inclined plane as shown in the figure.
They are connected by a bar of negligible weight hinged to each cylinder at its
geometric centre by smooth pins. Find the force P to be applied as shown in the
figure such that it will hold the system in the given position. (Refer
Figure 15) (Ans: RD=5378.93N, RC=5464.1N, RB=
-4898.98N,P=1071.8N)
28. Two cylinder of diameter 1m and
one cylinder of diameter 2m are placed symmetrically in a pit 3m wide(2m
cylinder on top.).Find all the contact forces if the surfaces are smooth.
Weight of 1m cylinder is 100N and weight of 2m cylinder is 200N. (Ans: RA =
RB=134.16N, RC=RD= 89.44N, RE=RF
= 200N)
29. The system in figure is in equilibrium
for the conditions of loading shown. Subsequently, the load Q is increased by
200N.How far must the load P be moved to the right to preserve equilibrium. (Refer
Figure 16) (Ans: L’ =8 m)
30. A roller of radius 40m weighing
3000N is to be pulled over a rectangular block of height 20cm as shown in the
figure by a horizontal force applied at the end of a string wound round
circumference of the roller. Find the magnitude of the horizontal force which
will just turn the roller over the corner of rectangular block. Also determine
the magnitude and direction of reaction at A and B. All surfaces may be taken
as smooth. (Refer Figure 17) (Ans: R B=3464.1N,P=1732.05N)
31. If
in the above problem, force P is applied horizontally at the centre of the roller,
what would be the magnitude of force? Also determine the least force and its
line of action at roller centre for turning roller around the rectangular
block. (Refer Figure 18) (Ans: R B=6000N,P=5196N, Pmin=2598N)
32. Two equal lengths of tubing, of
weight 2W each, are placed on two racks so that each rack supports half the
weight of tubing. Neglecting friction at all surfaces, determine reactions
exerted by the racks at A, B and C when α is 45°. Find the
least value of α in which equilibrium is possible.
(Refer Figure 19)(Ans: RA = 2.46W N, RB=0.55W
N, RC=4.89W N, α = 40.9°)
33. A 600 N cylinder is supported by
the frame BCD as shown in the figure. The frame is hinged at D. Determine the
reactions at A, B, C and D. (Refer Figure 20) )(Ans: RA =
RB= RC= 200N, RD= 632.5N)
34. Three bars are hinged at A and D
and pinned at B and C as shown in the figure to form a four bar mechanism.
Determine the value of P which will prevent motion.
(Refer Figure 21) (Ans: P =3047.2N)
35. Cords are looped around a small
spacer separating two cylinders each weighing 400N and passes over frictionless
pulleys to weights of 200N and 600N.Determine the angle θ and the normal
reaction r between the cylinders and the smooth surface inclined 15° to the
horizontal as shown in the figure. (Refer Figure 22) )(Ans: RA =
274N,θ =63.2°)
36. Three cylinders are piled in a
rectangular ditch as shown in the figure. Neglecting friction find the reaction
between cylinder A and vertical wall. Weights and radii of each cylinder are
given below.
(Refer Figure 23)(Ans: R = 226.86N)
Cylinder
|
Weight(N)
|
Radius(mm)
|
A
|
75
|
100
|
B
|
200
|
150
|
C
|
100
|
125
|
37. The frictionless pulley A is
shown in the figure is supported by two bars AB and AC which are hinged at B
and C to a vertical wall. The flexible cable DG hinged at D goes over the
pulley and supports a load of 20kN at G. The angles between various members are
as shown in the figure. Determine the forces in the bar AB and AC. Neglect the
size of the pulley.
(Refer Figure 24)(Ans: FAB = 0, FAC=34.64N)
38. A 500 N cylinder, 1m in diameter
is loaded between the cross pieces which makes an angle of 60° with each other
and are pinned at C. Determine the tension in horizontal rope DE assuming a
smooth floor. (Refer Figure 25)(Ans: T = 374N)
39. Three spheres A, B and C are
placed in a trench with smooth side walls and floor as shown in the figure. The
centre to centre distance of spheres A and b is 600mm.Find the reactions at P, Q,
R and S. Weights and diameters of each cylinder are given below.
(Refer Figure 26)(Ans: RP = 2.15kN, RQ=7.44kN,
RR=7.03kN, RS= 2.29kN)
Cylinder
|
Weight( kN)
|
Diameters(mm)
|
A
|
4
|
500
|
B
|
4
|
500
|
C
|
8
|
800
|
40. Three uniform, homogeneous and
smooth spheres A, B and C are placed in a trench as shown in the figure.
Determine the reactions at the contact points P, Q, R and S. Weights and
diameters of each cylinder are given below. (Refer Figure 27)(Ans: RP =
61.24N, RQ=631.58N, RR=1095.21N, RS= 290.43N)
Cylinder
|
Weight(N)
|
Diameters(mm)
|
A
|
300
|
800
|
B
|
600
|
1200
|
C
|
300
|
800
|
PARALLEL
FORCE SYSTEM / NON PARALLEL NON CONCURRENT FORCE SYSTEM
41. Find the resultant of the
following parallel force systems(Refer Figure 28a & 28b)
a.
(Ans
: R= 600N,x from A =45cm)
b.
(Ans
: R= 125N,x from A =3.06m)
42. A system of parallel forces is
acting on a rigid body as shown in the figure. Reduce the system to a
a.
Single
force.
b.
Single
force and couple at A.
c.
Single
force and couple at B. (Refer Figure 29a & 29b)
43. A system of loads acting on a
beam as shown in the figure. Determine the resultant of the loads in magnitude
and direction. (Refer Figure 30)
(Ans: E
= 68.06kN,α=81.55°,x from A = 3.326m)
44. Find the resultant of the force
system shown in the figure acting on a lamina of equilateral triangle shape.
Also make the magnitude and direction of the equilibrant. (Refer Figure 31)
(Ans: E
= 91.19N,α=35.84°,x from A = 317.06mm)
45. ABCD is a square each side
being 20mm long and E is the middle
point of AB. Forces 7,8,12,5,9 and 6 N acts on the body along the lines
AB,EC,BC,BD,CA and DE respectively. Find the magnitude, direction and position
with respect to ABCD, of the single force required to keep the body in
equilibrium. (Ans: E = 11.46kN,α=73°,x from B = 9.98mm)
46. The figure shows a tong used for
lifting a weight. Find the pressure applied at points A and B of the weight of
100N. Take a =100mm,b=200mm,c=200mm and
d=400mm. (Refer Figure 32)
(Ans: HB=80.8N,VB
= 50N,R = 95.02N)
47. Determine the magnitude,
direction and line of action of the equilibrant of the set of forces as shown
in the figure, which will keep the plane body ABCDEFGH in equilibrium. (Refer
Figure 33)
(Ans:
E = 23.65kN, α=24.37°, x from A = 1.041m).
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