DEPARTMENT OF MECHANICAL ENGINEERING
KANPUR INSTITUTE OF TECHNOLOGY, KANPUR
ENGINEERING MECHANICS (EME
Discuss various laws and principle of engineering mechanics.
Explain various force systems.
e of transmissibility of a force.
What are the necessary and sufficient conditions of equilibrium of a system of
coplanar force system?
Define moment of force about a point. Also explain varignons’s
Explain free body diagram with suita
Explain the following
a. Polygon law of forces
b. Parallelogram law of forces
c. Triangle law of forces
Explain laws of static and dynamic friction.
Write notes on:
a. Cone of friction
b. Coeff. of friction
c. Angle of friction
d. Angle of repose e. Limiting friction
Find expressions for the following due to belt friction in a pulley drive:
a. Tension in the belt on tight and slack side (T
b. Torque transmitted.
c. Reaction on the be
aring of pulley.
What is friction? Give some useful application of friction.
Define a beam and classify different types of beams on the basis of:
a. Support conditions.
What do you understand by shear force and bending moment and w
importance in beam design?
How shearing force and bending moment diagrams are drawn for a beam?
Explain rules for shear force and bending moment diagram.
What do you understand by the term ‘Point of contraflexure’?
Define a t
russ. What is the difference between a frame and truss?
How the trusses are classified? Define the following trusses.
a. Perfect truss
b. Imperfect truss.
c. Deficient Truss
d. Redundant truss.
What is simple truss? What assumptions ar
e made in the analysis of a simple truss?
What are various methods of analysis of truss? What is basically found when analysis of truss is done? What is
ntage of method of section over
the method of joint?
What is the difference between a s
imply supported truss and a cantilever truss? Discuss the method of finding out
reactions in both the cases. Is it essential to find out the reactions in a cantilever truss before analyzing it?
Explain the following elastic constants and establish rela
tion between them:
a. Modulus of elasticity
b. Modulus of rigidity
c. Bulk modulus
d. Poisson’s ratios.
strain diagram for structural steel and C.I.
Explain the following:
a. Complementary shear stress
b. Shear strain energ
What are the assumptions made in the theory of pure bending?
Explain the following:
a. Pure Torsion
b. Section modulus
c. Polar section modulus
What is the moment of a couple? Find the magnitude of a couple of two forces F1 and F2 both
equal in magnitude
co opposite in direction acting at d1 and d2 distance from a fixed point in a body.
State Lami’s theorem.
Prove that moment of inertia of a triangular section about the base of the section
is given by bh
section and h
height of the section)
Prove that moment of inertia of a disc of radius R about Z
axis perpendicular to
its plane is Izz = MR
Explain the parallel axis theorem and perpendicular axis theorem with an example.
What do you m
a. Centre of gravity
c. Principle of moment of inertia
d. Composite bodies.
Define radius of gyration.
How do you determine centroid of composite area?
The moment of inertia of a circular disc of radius ‘r’ and mass ‘m
’ about any
diameter is mr
/4. Calculate the
moment of inertia about an axis passing through
the centre and perpendicular to its plane.
Show that moment of inertia of a rectangular section of bxd is (bd
/12) and (bd
/13) about horizontal axis passing
C.G. and about horizontal axis
passing through base respectively.
Determine the centroid for the following cases by first principal.
a. Centroid of a triangle
b. Centroid of a semicircle.
Show that the moment of inertia of a circula
r area about the centroidal axes Ixx =
/64 .Where D is the
Find the mass moment of inertia of a sphere of radius ‘r’ and mass ‘m’ will it be
different along x,y and z axis.
Explain polar modulus. Calculate
it for circular section (Torsion of shaft)
Show that the total strain energy U =
/4G X volume.
Stating the assumptions drive torsion equation T / I p =
/ r = G
a. pure torsion
b. Compound shaft
c. Polar moment of inertia.
What do you by simple bending. What are the assumptions taken in the theory of pure bending.
Drive the relation E = 2G (I +
for Aluminum and cast iron.
What is strain energy ? Show U = 0.5
/ E be the strain energy per unit volume due to direct stress
Define Poisson’s ratio, resilience, modulus of rigidity, shear strain
and proof stress.
Explain ductile and brittle materials.
Explain the principle of super position and show the stresses in bars of
Show that the elongation of a rectangular body due to self w
elongation of a circular
body due to self w
E = 3K (1
Radius of ball 1=100mm and ball 2=50mm ; Weight of ball 1=2000N and ball 2
=800N Determine reactions at A, B, C, D
Three uniform, homogeneous and smooth spheres A, B & C weighing 300N, 600N & 300N respectively and having diameters
800mm, 1200mm & 800mm respectively are placed in a trench as shown in Fig. above. Dete
rmine the reactions at the contact points
P, Q, R and S
A rigid circular roller of weight 5000N rest on a smooth inclined plane and is held in position by a chord AC as shown in Fig
Find the tension in the chord if there is a horizontal force o
f magnitude 1000N acting at C.
A Cylinder 1m diameter and 10kg mass is lodged between cross pieces that makes an angle of 60
other As shown in Fig. above. Determine the Tension in the horizontal rope DE
Find support reactions at A,
Find x to keep the bar AB in equilibrium.
Determine the magnitude and direction of the resultant of the following set of forces acting on a body.
(i) 200N inclined 30 degree with east towards north
(ii) 250N towards the north
(iii) 300N towards North West
(iv) 350N inclined at 40 degree with west towards south.
What will be the equilibrant of the force system
Find the resultant of forces 2, 3, 4, 5, 6 N that act at an angular point of a regular hexagon towards the other a
points taken in order.
A uniform wheel of 50cm diameter and 1kN weight rest against a rigid rectangular block of thickness 20 cm as shown in
Fig.. Considering all surfaces smooth, determine
(a) Least Pull to be applied through the centre of whe
el to just turn it over the corner of the block.
(b) Reaction of the block
Fig below shows a sphere resting in a smooth V shaped groove and subjected to a spring force. The spring is compressed
to a length of 100mm from its free Length of 150mm.
If the stiffness of spring is 2KN/mm. Determine the contact
reactions at A and B.
Determine the resultant of the forces acting tangential to the circle of radius 3m as shown in Fig given below. What will
be its location with respect to the centre
of the circle?
A rigid bar is subjected to a system of parallel forces as shown in Fig given below.
Reduce this system to
(i) A single force (ii) a single force
moment system at A
(iii) A single force
moment system at B
Forces equal to P, 2P
, 3P and 4P act along the sides AB,BC,CD and DA of a square ABCD. Find the magnitude, direction
and line of action of the resultant.
The frictional less pulley A shown in Fig is supported by two bars AB and AC which are hinged at B and C to a
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 shown in Fig. Determine the forces in AB and AC. Neglect the size of pulley.
One end of a split horizontal b
eam ACB is fixed into wall and the other end B rests on a roller support. A hinge is at point
A.A crane of weight 50KN is mounted on the beam and is lifting a load of 10KN at the end L. The C.G of the crane acts
along the vertical line CD and KL=4m.Neglect
the weight of the beam, find the reaction / moments at A &B.
section of a block is an equilateral triangle. It is hinged at A and rests on a roller at B. It is pulled by means of a
string attached at C. If the weight of the block is M
g and the string is horizontal, determine the force P which should be
applied through string to just lift the block off the roller.
A 12m boom AB weighs 1KN, the distance of the centre of gravity G being 6m from A. For the position shown, determine
e tension T in the cable and the reaction at B.
A body of weight 100N rest on a rough horizontal surface (µ=03) and is acted upon by a force applied at an angle
of 30 degree to the horizontal.
(a)What force is required to just cause the
body to slide over the surface?
(b) Proceed to determine the inclination and magnitude of minimum force required to set the block in to impending
Q20. wooden block of weight 50N rest on a horizontal plane. Determine the force required to just (a) p
ull it and (b) push
it. Take µ=04 between the mating surface. Comment on the result.
Q21. A body resting on a rough horizontal plane required a pull of 24N inclined at 30 degree to the plane just to move it.
It was also found that a push of 30N at 30 degre
e to the plane just enough to cause motion to impend. Make calculation
for the weight of the body and the co
efficient of friction.
Q22. Two blocks A & B of weight 4kN and 2 kN Respectively are in equilibrium as shown in Fig.. Presuming that co
f friction between the blocks as well as between block and floor is 0.25, make calculation for the force P
required to move the block A.
Calculate the force P required to move the lower block B and tension in the cable. Take coefficient of fri
ction at all the
contact surfaces to be 0.3.
Q24. State whether B is stationary with respect to ground and A or B is stationary with respect to A. Determine the
minimum value of weight W in the pan so that motion starts .coefficient of friction between gro
und and block is 0.1 and
between block A and B is 0.28.
What should be the value of è in Fig given below which will make the motion of 900N block down the plane to impend?
Coefficient of friction all contact surfaces is 1/3.
What is the value
of P in the system as shown in Fig to cause the motion to impend? Assume the pulley is smooth and
coefficient of friction between the other contact surfaces is 0.2.
Find whether block A is moving up or down the plane in Fig. for the data given be
low. Weight of block A and B are 300N
and 600N respectively. Coefficient of friction between plane AB and block A is 0.2.Coefficient of friction between plane
BC and block AB is 0.25.Assume the pulley is smooth.
Two identical blocks A and B are connec
ted by a rod and they rest against vertical and horizontal planes respectively as
shown in Fig. If sliding impends when θ=45 degree, determine the coefficient of friction, assuming it to be the same for
both floor and wall.
What is the least value
of P to cause motion to impend? Assume the coefficient of friction to be 0.20.
Block A of mass 12kg and block B of mass 6kg are connected by a string passing over a smooth pulley. If µ =0.12 at all
surfaces of contact find smallest value of P force t
o maintain equilibrium.
Two block having weights W
are connected by A string and rest on horizontal plane as shown in Fig.If the angle
of friction for each block is
φ, find magnitude and direction of least force P.
Find whether a cylinder of 800N will slip or not under the action of a tangential force of 200N as shown in Fig. Take
µ=0.5 at all contact surfaces.
Two blocks connected by a horizontal l
ink AB are supported on two rough planes as shown in Fig. The coefficient of
friction for the block on the horizontal plane is 0.4. the limiting angle of friction for block B on the inclined plane is 20
degree. What is the smallest weight W of the block A
for which equilibrium of the system can exist if weight of block B is
Two blocks A and B each weighing 1500N are connected by a uniform horizontal bar which weighs 1000N.If the angle of
limiting friction under each block is 15 degree, find the fo
rce P directed parallel to the 60 degree plane that will cause
motion impending to the right.
The mass of A is 23 kg and the mass of B is 36kg.The coefficient of friction are 0.6 between A & B,0.2 between B and the
plane and 0.3 between the rop
e and fixed drum. Determine the maximum mass of M before motion impends.
Determine the force P to cause motion to impend.µ=0.25 at all contact contact surfaces. The pulley is frictionless.
Determine the coefficient of friction between the rop
e and pulley if the coefficient of friction between the block A and
the plane is 0.28.
A uniform ladder, 5 m long weighs 180 N. it is placed against a wall making an angle of 60
with floor. The co
efficient of friction between the wall and the ladder
is 0.25 and between the floor and ladder is 0.35. The ladder
has to support a man 900N at its top. Calculate
The horizontal force P to be applied to the ladder at the floor level to prevent slipping.
If the force P is not applied, what should be
the minimum inclination of the ladder with the horizontal so
that there is no slipping of it with the man at its top?
A uniform ladder of length 10m and weighing 20N is placed against a smooth vertical wall with its lower end
8m from the wall. In thi
s position the ladder is just to slip. Determine
(a) The co
efficient of friction between the ladder and floor.
(b) Frictional force acting on the ladder at the point of contact between ladder and floor.
A ladder of length “L” rests against a wall,
the angle of inclination being 45
. If the co
efficient of friction
between the ladder and the ground and that between the ladder and the wall be 0.5 each. What will be the
maximum distance along the ladder to which a man whose weight is 2 times the weight
of ladder may ascend
before the ladder begins to slip?
DRAW SHEAR FORCE AND BENDING MOMENT DIAGRAM
OF THE FOLLOWING LOAD DIAGRAM.
OVER HANGING BEAM
SIMPLY SUPPORTED BEAM
SIMPLE STRESS & STRAIN
Q.1 A steel bar 2.4m long and 30mm square is elongated by a load of 500N.If Poisson ratio is 0.25, find the
increase in volume .Given E=0.2×106 N/mm2
Q.2. The piston of a steam engine is 300mm in diamet
er and the piston rod is of 50mm diameter. The steam
pressure is 1N/mm2. Find the stress in the piston rod and elongation in a length of 800mm. Take E=200GPa.
Q.3. Three bars of equal length and having cross
sectional areas in the ratio 1:2:4 are all subje
cted to equal
load. Compare their strain energy.
Q.4. A metallic rectangular rod 1.5m long and 40mm wide and 25mm thick is subjected to an axial tensile
of 120KN. The elongation of the rod is 0.9mm. Calculate the stress, strain and modulus of elastici
Q.5. One meter long steel rod of rectangular section 80mm x 40mm is subjected to an axial tensile load of
KN. Find the strain energy and maximum stress produced in it when the load is applied gradually. Take
Q.6. A circular rod of 10
0mm diameter and 500mm long is subjected to a tensile force of 1000KN.
the modulus of rigidity, bulk modulus and change in volume if poison’s ratio is 0.3 and E=2x10
Q.7.The piston of a steam engine is 300mm in diameter and the piston rod
is of 50mm diameter. The steam
pressure is 1 N/mm2.Find the stress in the piston rod and elongation in a
length of 800mm.Take:E=200GPa.
Q.8.A bar of 25mm diameter is subjected to a pull of 60kN.The measured extension over a gauge length of
250mm is 0.15m
m and change in diameter is 0.004mm. Calculate the modulus of elasticity, modulus of
and Poisson’s ratio.
Q.9.A steel rod of diameter 50mm and 2.5m long is subjected to a pull of 100kN. To what length the rod
be bored centrally so that the
total extension will increase by 15% under the same pull, the bore being
diameter? Take E=200GN/m
Q.10.The bar shown in Fig is subjected to a tensile load of 50kN. Find the diameter of the middle portion if
stress is limited to 130MN/ m
. Find a
lso the length of the middle portion if the total elongation of the bar
0.15mm .Take E=200GN/m
Find stresses in all three parts of the bar if cross
sectional area of bar is 1000mm
1.A wooden beam of rectangular section is s
ubjected to a bending moment of 5kNm.If the depth of the
is to be twice the breadth and stress in the wood is not to exceed 60N/m2, find the dimension of the
tion of the beam
2.A rectangular beam with depth 150mm and width 100mm is subjec
ted to a maximum bending moment of
kNm, find the maximum
stress in the beam.
3.A rectangular beam of 200mm in width and 400mm in depth is simply supported over a span of 4m and
carried a udl of 10K=kN/m. determine the maximum bending stress in the bea
4.A rectangular beam of cross
is simply supported over a span of 5m.What uniformly
distributed load the beam may carry
(i)when the height is 300mm (ii) when the height is 200mm.The bending
stress is not to exceed 130N/ mm
made of C.I. having a circular section of 50mm external diameter and 25mm internal diameter is
supported at two points 4m apart. The beam carries a concentrated load of 100N at the centre. Find the
maximum bending s
tress in the beam?
6.A hollow circular
bar having outside diameter twice the inside diameter is used as a beam subjected to a
bending moment of 50kNm.Determine the inside diameter of the bar if allowable bending stress is limited to
7. Find the dimension of the strongest rectangular
beam that can be cut from a log of 250mm
8. Three beams have the same length, same stress and same bending moment. The cross
sections of the
beams are a square, a rectangle with depth twice the width and a circle. Determine the ratios of weight o
circular and rectangular beams with respect to that of square beam.
9.For a given stress compare the moment of resistance of a beam of square section when placed (i) with two
sides horizontal and(ii) with its diagonal horizontal.
10.A water mains 500mm
external diameter and 25mm thick is full of water and is freely supported for 20m
span. Determine the maximum bending stress induced in the pipe metal if the weight of water and that of
is taken in to account. The specific weight of water and steel as
11.A cantilever with a constant breadth of 100mm has a span of 2.5m.It carries a uniformly distributed load
20kN/m. Determine the depth of the section at the middle of the length of the cantilever and also at the
end if the
stress remains the same throughout and is equal to 120 MN/ m
12.A simply supported beam 1m long and
section fails when a central load of 600N is
applied to it. What intensity of UDL would cause failure of a cantile
ver beam 2m long and
deep made of same material.
13.Deterrmine the longest span of a simply supported beam carry a UDL of 6kN/m without exceeding a
bending stress of 120MN/ m
.The depth and moment of inertia of the symmetrical I
section are 20cm and
14.. A simply supported beam 200mm wide and 250mm deep carries a UDL of intensity 800N/mm
entire span of 4m find maximum stress developed in the beam.
15. For I section given flange = 250mm wide and 25mm thick, web = 15mm thick, over
all depth =
beam has span of 10m and carries a UDL of intensity 50KN/m for the entire span.
Find the stress produced
due to bending.
Q.1. Compare the strength between hollow and a solid circular shaft for same material, same length
Q.2. The diameter of a shaft is 20cm. Find the safe maximum torque which can be transmitted by
the shaft, if
the permissible shear stress in the shaft material is 4000N/cm
and permissible angle of
twist is 0.2º per meter
length. Take G= 8x10
. if the shaft rotates at 320rpm what maximum
power can be transmitted by the
Q.3.If the maximum torque transmitted by a solid shaft exceeds the mean by 30% in each
revolution, find a
suitable shaft diameter to transmit 75 kW power at 200rpm
.Take allowable shear
stress as 70 N/ mm
Q.4.What external and internal diameter is required for a hollow shaft to transmit 50kW of power at
the shear stress is limited to 100MN/ m
.Take outside diameter to be twice of inside
culate the diameter of a circular shaft to transmit 75 kW at
200rpm.Allowable shear stress is
to 50MN/ m
and twist 1o in 2 m shaft length .Take G=400GPa
Q.6.Compare the weight of a solid and hollow shaft of same material, same length, same tor
que and same
stress. The internal diameter of hollow shaft is 2/3 of its outer diameter.
Q.7.A hollow shaft of 3m length is subjected to a torque such that the maximum shear stress produced is
75MPa.The external and internal diameters of the shaft are 150m
m and 100mm respectively. Find the shear
stress at the inside surface.Take:G=75MPa
Q.8.A solid circular shaft is required to transmit 200kW power at 100rpm.Determine the diameter of the
permissible shear stress is 60 N/mm
.calculate the energy sto
red per meter length of the shaft.
Q.9.A solid circular shaft is to transmit power 160kW at 180rpm.What will be the suitable diameter of this
if the permissible stress in the shaft material should not exceed 2MPa and twist per meter
CENTRE OF GRAVITY & MOMENT OF INERTIA
Determine the centroid of an arc of radius
from first principle.
Distinguish between centroid and centre of gravity.
Determine the centroid of a tria
ngle of base width ‘
’ and height ‘
’ by the method of integration.
Locate the centroid of a semicircle from its diametral axis using the method of integration.
Explain the terms moment of inertia and radius of gyration of a plane figure.
) Perpendicular axis theorem
) Parallel axis theorem of moment of inertia.
Determine the moment of inertia of the areas specified below by first principle:
) Triangle of base width
about its centroid axis parallel
to the base.
) A semicircle about its centroidal axis parallel to the diametral axis.
State and explain theorems of Pappus
Locate the centre of gravity of the right circular cone of base radius
Determine the centr
e of gravity of a solid hemisphere of radius
from its diametral axis.
Determine the centroid of the built
up section in Figure. Express the coordinates of centroid with respect to
Find the coordinates of the ce
ntroid of the shaded area with respect to the axes shown in Figure
With respect to the coordinate axes
locate the centriod of the shaded area shown in Figure
Locate the centroid of the plane area shown in Figure
The strength of a 400 mm deep and 200 mm wide
beam of uniform thickness 10 mm, is increased by welding a 250
mm wide and 20 mm thick plate to its upper flanges as shown in Figure. Determine the moment of inertia and the radii
of gyration of the
composite section with respect to cetroidal axes parallel to and perpendicular to the bottom edge
A plate girder is made up of a web plate of size 400 mm × 10 mm, four angles of size 100 mm × 100 mm × 10 mm and
cover plates of size 300 mm ×
10 mm as shown in Figure. Determine the moment of inertia about horizontal and vertical
section of a plain concrete culvert is as shown in Figure. Determine the moment of inertia about the horizontal
Determine the centroid of the built
up section shown in
Figure and find the moment of inertia and radius of
about the horizontal centroidal axis.
Find the moment of inertia about the horizontal centroidal axis and about th
base A B
Find the centroid of the plain lamina shown
Find the moment of inertia about the horizontal centroidal axis. As shown in the Figure
A square prism of cross section 200mm
200mm and height 400mm stands ve
rtically and centrally over a cylinder of
diameter 300mm and height 500mm. Calculate the mass moment of inertia of the composite solids about the vertical
axis of symmetry if the mass density of the material is 2000
Find out the Centre of G
ravity of a uniform lamina shown in f
consists of a rectangle
a circle and a
ow triangular section shown in f
is symmetrical about its vertical axis.
A hollow semicircular section has its outer and inne
r diameter of 200 mm and 120 mm respectively as shown in figure
Find the moment of inertia of a hollow rectangular section about its centre of gravity, if the external dimensions are 40
mm deep and 30 mm wide and internal dimensions are 25 mm
deep and 15 mm wide
Find the moment of inertia of a T
section with flange as 150 mm × 50 mm and web as 150 mm × 50 mm about X
X and Y
Y axes through the centre of gravity of the section.
Find the centre of gravity of I
wn in figure below
Fid the centre of Gravity of I Section in figure below
Find the moment of inertia of the area shaded in figure below
Find out the centre of gravity with respect to the coordinate axes as shown in f
Determine the centre of gravity of the uniform plane lamina shown in figure below ( The symmetrical about Y
A body consist of solid hemisphere of radius 4 cm and a right circular solid cone of height of 12 cm.
The hemisphere &
cone have common base are made of of same material. Locate the position of centre of gravity of the composite body.
Locate the centroid of the are Shown in figure with respect to the axes indicated in the figure
A triangular plate in the form of an isosceles triangle ABC has base BC = 10cm and altitude 12 cm. From this plate, a
portion in the shape of an isosceles triangle OBC is removed. If O is the mid
point of an altitude of triangle ABC, then
distance of CG of the reminder section from the base.
From a rectangular plate whose cross
section is 8 X 6cm, a circular disc of 3cm
in area is removed. If the CG of the
remainder is 1mm from CG off the rectangular plate, work out the distan
ce of the centre of disc from the centre of
A rectangular lamina ABCD 20 X 25 cm has a rectangular hole of 5 X6 cm as shown in figure. Locate the centroid of the
From a circular lamina of diameter d, a square hole ha
s been punched out. If one diagonal of the square coincides with
the radius of the circle, determine the distance of the centre of remainder from the centre of the circle.
Determine the centoid of the lamina shown in figure below.
ocate the centroid of the T
section as shown in figure
Locate the position of the centroid of the plane shaded area depicted in the figure below
An isosceles triangle is to be cut from one edge of a square plate of side 1m such th
at the remaining part of the plate
remains in equilibrium in any position when suspended from the apex of the triangle. Calculate the area of the triangle
to be removed.
The frustum of a right circular cone has a bottom radius 5
cm & top radiu
s 3 cm. and height 8 cm. A coaxial cylindrical
hole of 4cm. is made throughout the frustum. Locate the position of centre of gravity of the remaining solid.
A hemisphere and a cone have their bases joined together; two bases being same size. Wh
at should be the greatest
height of the cone in terms of radius of the base so that the combined body of the cone and hemisphere may stand
Locate the centroid of the Z
sewction as shown in figure.
A rod rectangular hole is
made in a triangular section as shown in figure below. Determine the moment of inertia of the
i) About base AB
ii) About the centroidal axis parallel to the base AB.
For the figure shown below, calculate the moment of inertia about
axis. What will be the corresponding radius of
Detremine the Moment of Inertia of the Z
section as shown in figure about the centroidal axis x
x and y
Find the moment of inertia of lamina with a circular hole of 1
5mm diameter about the axes AB &CD as shown in figure.
Determiine the distance d at which the two 2 X 6 cm rectangles shown in figure be spaced so that I
Figure shows the cross section of a cast iron beam. Determine the moment of inertia of the section about the horizontal
and vertical axes passing through the centroid of the section.