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A steel wire of length 4.7 m and
cross section \[=43\cdot \text{2 cm}\] stretches by the same amount as a copper
wire of length 3.5 m and cross section \[0\cdot 5\times {{10}^{-2}}c{{m}^{2}}\]under
a given load. What is the ratio of the Young's modulus of steel to that of
copper?
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Fig. l shows the stress-strain curve for a given
material. What are (a) Young's modulus, and (b) approximate yield strength for
this material?
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The stress versus strain graph for two materials A and B
are shown in Fig. 2. The graphs are on the same scale.
(a)
Which material has greater Young?s modulus?
(b)
Which of the two is stronger material?
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Read each of the statements below carefully and state, with
reasons, if it is true or false.
(a)
The modulus of elasticity of rubber is greater than that of steel.
(b) The stretching of a coil is determined by its shear
modulus.
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Two wires of diameter 0.25 cm, one made of steel and the
other made of brass are loaded as shown in Fig. 3. The unloaded length wire is
1.5 m and that of brass wire is 1.0 m. Compute the of the steel and the brass
wires,
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The edge of aluminium cube are 10 cm long. One face of the
cube is firmly fixed to a vertical wall. A If mass of 100 kg 4 & then
attached to the opposite face of the cube. The shear modulus of aluminium is 25
G Pa. What is the vertical deflection of this face? [V=0cdot 32{{m}^{3}};]
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Four identical hollow cylindrical
columns of steel support a big structure of mass 50,000 kg. The inner and outer
radii of each column are 30 cm and 60 cm respectively. Assume the load
distribution to be uniform, calculate the compressional strain of each column.
The Young's modulus of steel is \[{{\text{S}}_{\text{LA}}}\text{,}{{\text{S}}_{\text{SA}}}\text{
and }{{\text{S}}_{\text{SL}}}\]
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A piece of copper having a
rectangular cross section of \[\theta \]mm x \[\theta >{{90}^{o}}i.e.\] mm
is pulled in tension with 44,500 N, force producing only elastic deformation.
Calculate the resulting strain. Shear modulus of elasticity of copper is \[{{\text{S}}_{\text{SA}}}\text{}{{\text{S}}_{\text{SL}}}\text{,}\]
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A steel cable with a radius of \[\text{a
}\!\!\upsilon\!\!\text{ = a}\] cm supports a chairlift at a ski area. If the
maximum stress is not exceed\[\text{P + }\!\!\rho\!\!\text{ gh +
}\frac{\text{1}}{\text{2}}\text{ }\!\!\rho\!\!\text{ }{{\text{ }\!\!\upsilon\!\!\text{
}}^{\text{2}}}\text{ = a}\] what is the maximum load the cable can support?
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A rigid bar of mass 15 kg is supported symmetrically by
three wires each 2 m long. These at each end are of copper and middle one is of
iron. Determine the ratio of their diameters if each is to have same tension. Youngs
modulus of elasticity for copper and steel are \[\frac{1}{200}\] \[\frac{\text{force}}{\text{area}}\text{=}\frac{\text{mg}}{\text{
}\!\!\pi\!\!\text{ }{{\text{r}}^{\text{2}}}}\text{=}\frac{\text{50
}\!\!\times\!\!\text{ 9 }\!\!\times\!\!\text{ 8}}{\left( \text{22/7}
\right)\text{ }\!\!\times\!\!\text{ }{{\left( \text{1/200}
\right)}^{\text{2}}}}\] and \[=6\cdot 24\times {{10}^{6}}N{{m}^{-2}}.\]\[{{m}^{-3}}\]
respectively.
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A 14.5 kg mass, fastened to the
end of a steel wire of unstretched length 1 m, is whirled in a vertical circle
with an angular velocity of 2 rev is at the bottom of the circle. The
cross-sectional area of the wire is \[{{\text{a}}_{\text{1}}}\text{=3
}\!\!\times\!\!\text{ 0 }\!\!\times\!\!\text{
1}{{\text{0}}^{\text{-5}}}{{\text{m}}^{\text{2}}}\text{;}\] Calculate the
elongation of the wire when the mass is at the lowest point of its path\[{{\text{l}}_{\text{1}}}\text{=4
}\!\!\times\!\!\text{ 7m;}\]
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Compute the bulk modulus of water from the following data:
Initial volume \[{{Y}_{2}}\frac{{{F}_{2}}\times {{l}_{2}}}{{{a}_{2}}\times
\vartriangle {{l}_{2}}}=\frac{F}{3\cdot 0\times {{10}^{-5}}}\times \frac{4\cdot
7}{\vartriangle l}\] litre, pressure increase = 100-0 atmosphere. Final volume \[\therefore
\] litre. (1 atmosphere \[\frac{{{Y}_{1}}}{{{Y}_{2}}}=\frac{4\cdot 7\times
4\times {{10}^{-5}}}{3\cdot 5\times 3\cdot 0\times {{10}^{-5}}}=1\cdot 8\]).
Compare the bulk modulus of water with that of air (at constant temperature).
Explain in simple terms why the ratio is so large.
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What is the density of ocean water at a depth, where the
pressure is 80.0 atm., given that its density at the surface is \[\frac{\left(
6\times 9\cdot 8 \right)\times 1\cdot 0\times 7}{22\times {{\left( 0\cdot
125\times {{10}^{-2}} \right)}^{2}}\times \left( 0\cdot 91\times {{10}^{11}}
\right)}\]Compressibility of water\[=1\cdot 3\times {{10}^{-4}}m\] Given 1 atm \[\left(
\text{1pa=1N}{{\text{m}}^{\text{-2}}}
\right)\text{;g=10m/}{{\text{s}}^{\text{2}}}\text{.}\]
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Compute the fractional change in volume of a glass slab,
when subjected to a hydrautic pressure of 10 atmosphere. Bulk modulus of
elasticity of glass \[=\frac{44500}{\left( 15\cdot 2\times 19\cdot 2\times
{{10}^{-6}} \right)\times 42\times {{10}^{9}}}=3\cdot 65\times {{10}^{-3}}\]
and \[1\,atm=1.03\times {{10}^{5}}\,pa\].
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Determine the volume contraction of a solid copper cube, 10
cm on an edge when subjected to a hydraulic pressure of \[{{D}^{2}}\alpha 1/Y\]
Bulk modulus of copper = 140 G Pa.
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How much should the pressure on a litre of water be changed
to compress it by \[14\cdot 5kg;l=r=1m;v=2\]. Bulk modulus of elasticity of
water \[A=0\cdot 065\times {{10}^{-4}}{{m}^{2}}\]
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Anvils made of single crystal of diamond, with the shape as
shown in Fig. 4, are used to investigate behavior of materials under very high
pressures. Flat faces at the narrow end of the to a compressional force of
50,000 N. What is the pressure at the tip of the anvil?
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A rod of length 1.05 m having negligible mass is supported
at its ends by two wires of steel (wire A) and aluminium (wire B) of equal
lengths as shown in Fig. 5. The cross-sectional area of wires A and B are \[1\,m{{m}^{2}}\]and
\[2\,\,m{{m}^{2}}\], respectively . At what point along the rod should a mass m
be suspended in order to produce (a) equal stresses and (b) equal strains in
both steel and aluminum wires. Given,
\[B=\frac{pV}{\vartriangle V}\]
\[\frac{100\times 1\cdot 013\times
{{10}^{5}}\times 100\times {{10}^{-3}}}{0\cdot 5\times {{10}^{-3}}}\]
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A mild steel wire of length 1.0 m and cross-sectional area \[{{h}_{1}},\]
is stretched, well within its elastic 'limit, horizontally between two pillars.
A mass of 100 g is suspended from' the mid point of the wire.
Calculate the depression at the mid point.
\[B,{{P}_{B}}=58+{{h}_{1}}\] \[{{P}_{A}}={{P}_{B'}}\]
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Two strips of metal are riveted together at their ends by
four rivets, each of diameter 6 mm. What is the maximum tension that can be
exerted by thick riveted strip if the shearing stress on the rivet is not to
exceed\[\text{ }{{\text{ }\!\!\upsilon\!\!\text{ }}_{\text{2}}}\text{=234
km/h}\]? Assume that each rivet is to carry one quarter of the load.
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The Marina Trench is located in the Pacific Ocean, and at
one place it is nearly eleven km beneath the surface of water. The water
pressure at the bottom of the Trench is about \[\frac{1\times \left[
{{65}^{2}}-{{50}^{2}} \right]\times 50}{2\times 9\cdot 8}\] A steel ball of
initial volume\[=4\cdot 4\times {{10}^{3}}kg.\]is dropped into the ocean and
falls to the bottom of the Trench. What is the change in the volume of the ball
when it reaches to the bottom? Bulk modulus for steel\[2\cdot 0\times
{{10}^{-5}}\]
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question_answer22)
Modulus
of rigidity of ideal liquids is
(a)
infinity.
(b)
zero.
(c)
unity.
(d)
some finite small non-zero constant value.
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question_answer23)
The
maximum load a wire can withstand without breaking, when its length is reduced
to half of its original length, will
(a)
be double (b) be half
(c)
be four times (d) remain same.
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question_answer24)
The
temperature of a wire is doubled. The Young's modulus of elasticity
(a)
will also double
(b)
will become four times
(c)
will remain same
(d)
will decrease
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question_answer25)
A
spring is stretched by applying a load to its free end. The strain produced in
the spring is
(a)
volumetric
(b)
shear
(c)
longitudinal and shear
(d)
longitudinal
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question_answer26)
A
rigid bar of mass M is supported symmetrically by three wires each of length \[l\].
Those at each end are of copper and the middle one is of iron. The ratio of
their diameters, if each is to have the same tension, is equal to
(a)
\[{{Y}_{copper}}/{{Y}_{iron}}\] (b) \[\sqrt{\frac{{{Y}_{iron}}}{{{Y}_{copper}}}}\]
(c)
\[\frac{Y_{iron}^{2}}{Y_{copper}^{2}}\] (d) \[\frac{Y_{iron}^{{}}}{Y_{copper}^{{}}}\]
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question_answer27)
A
mild steel wire of length 2L and cross-sectional area A is stretched, well
within elastic limit, horizontally between two pillars. A mass m is suspended
from the mid point of the wire. Strain in the wire is
(a) \[\frac{{{x}^{2}}}{2{{L}^{2}}}\] (b) \[\frac{x}{L}\]
(c)
\[\frac{{{x}^{2}}}{L}\] (d) \[\frac{{{x}^{2}}}{2L}\]
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question_answer28)
A
rectangular frame is to be suspended symmetrically by two strings of equal
length on two supports. It can be done in one of the following three ways:
The
tension in the strings will be
(a) the same in all
cases.
(b) least in (a)
(c) least in (b)
(d) least in (c).
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question_answer29)
Consider
two cylindrical rods of identical dimensions, one of rubber and the other of
steel. Both the rods are fixed rigidly at one end of the roof. A mass M is attached
to each of the free ends at the centre of the rods.
(a)
Both the rods will clongate but there shall be no perceptible change in shape.
(c)
The steel rod will elongate and change shape but the rubber rod will only
elongate.
(c)
The steel rod will elongate without any perceptible change in shape, but the
rubber rod will elongate and the shape of the bottom edge will change to an
ellipse.
(d)
The steel rod will elongate, without any perceptible change in shape, but the
rubber rod will elongate with the shape of the bottom edge tapered to a tip at
the centre.
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question_answer30)
The
strees-strain graphs for two materials are shown in fig. (assume same scale).
(a) Material (ii)
is more elastic than material (i) and hence material (ii) is more brittle.
(b) Material (i)
and (ii) have the same elasticity and the same brittleness.
(c) Material (ii)
is elastic over a larger region of strain as compared to (i).
(d) Material (ii)
is more brittle than material (i).
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question_answer31)
A
wire is suspended from the ceiling and stretched under the action of a weight F
suspended from its other end. The force exerted by the ceiling on it is equal
and opposite to the weight.
(a)
Tensile stress at any cross-section A of the wire is F/A.
(b)
Tensile stress at any cross-section is zero.
(c)
Tensile stress at any cross-section A of the wire is 2F/A.
(d)
Tension at any cross-section A of the wire is F.
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question_answer32)
A
rod of length I and negligible mass is suspended at its two ends by two wires
of steel (wire A) and aluminium (wire B) of equal lengths. The cross-sectional
areas of wires A and B are 1.0 mm2 and 2.0 mm respectively.
\[({{Y}_{Al}}70\times
{{10}^{9}}N{{m}^{-2}}\,\,\text{and }\,{{Y}_{steel}}=\text{ }200\times
{{10}^{9}}N{{m}^{2}})\]
(a)
Mass m should be suspended close to wire A to have equal stresses in both the
wires.
(b)
Mass m should be suspended close to B to have equal stresses in both the wires.
(c)
Mass m should be suspended at the middle of the wires to have equal stresses in
both the wires.
(d)
Mass m should be suspended close to wire A to have equal strain in bolt wires.
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question_answer33)
For
an ideal liquid
(a)
the bulk modulus is infinite.
(b)
the bulk modulus is zero.
(c)
the shear modulus is infinite.
(d)
the shear modulus is zero.
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question_answer34)
A
copper and a steel wire of the same diameter are connected end to end. A
deforming force F is applied to this composite wire which causes a total
elongation of 1 cm. The two wires will have
(a)
the same stress (b) different stress
(c)
the same strain (d) different strain
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question_answer35)
The
Young's modulus for steel is much more than that for rubber. For the same
longitudinal strain, which one will have greater tensile stress?
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question_answer36)
Is
stress a vector quantity?
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question_answer37)
Identical
springs of steel and copper are equally stretched. On which, more work will
have to be done?
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question_answer38)
What
is the Young's modulus for a perfect rigid body?
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question_answer39)
What
is the Bulk modulus for a perfect rigid body?
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question_answer40)
A
wire of length L and radius r is clamped rigidly at one end. When the other end
of the wire is pulled by a force \[f\] its length increases by \[l\]. Another wire
of the same material of length 2L and radius 2r, is pulled by a force \[2f\].
Find the increase in length of this wire.
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question_answer41)
A
steel rod \[(Y=2.0\times {{10}^{11}}\,N{{m}^{-2}}\,and\,\,\alpha
={{10}^{-50}}\,{{C}^{-1}})\]of length, 1 in and area of cross-section \[1\,\,c{{m}^{2}}\]
is heated
from \[0{}^\circ \text{C}\] to \[\text{2}00{}^\circ \text{C}\], without being
allowed to extend of bend. What is the tension produced in the rod?
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question_answer42)
To
what depth must a robber ball be taken in deep sea so that its volume is
decreased by 0.1%. (The bulk modulus of rubber is\[9.8\times
{{10}^{8}}N{{m}^{2}},\] and
the density of sea water is\[{{10}^{3}}kg\text{ }{{m}^{3}}\]).
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question_answer43)
A truck is pulling a
car out of a ditch by means of a steel cable that is 9.1 m long and has a
radius of 5 mm. When the car just begins to move, the tension in the cable is
800 N. How much has a cube stretched? (Young?s modulus for steel is \[2\times
{{10}^{11}}N{{m}^{2}}\].)
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question_answer44)
Two identical solid
balls, one of ivory and the other of wet-clay, are dropped from the same height
on the floor. Which one will rise to a greater height after striking the floor
and why?
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question_answer45)
Consider
a long steel bar under a tensile stress due to forces F acting at the edges
along the length of the bar. Consider a plane making an angle \[\theta \] with
the length. What are the tensile and shearing stresses oil this plane?
(a) For what angle
is the tensile strees a maximum?
(b) For what angle
is the shearing strees a maximum?
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question_answer46)
(a)
A steel wire of mass \[\mu \] per unit length with a circular cross-section has
a radius of 0.1 cm. The wire is of length 10 m when measured lying horizontal,
and hangs from a hook on the wall. A mass of 25 kg is hung from the free end of
the wire. Assuming the wire to be uniform and lateral strains <<
longitudinal strains, find the extension in the length of the wire. The density
of steel is \[7860\,\,kg\,\,{{m}^{-3}}\] (Young?s modulus \[Y=2\times
{{10}^{11}}N{{m}^{2}}\]).
(b)
If the yield strength of steel is \[2.5\times {{10}^{8}}N{{m}^{2}},\] what is the maximum
weight that can be hung at the lower end of the wire?
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question_answer47)
A
steel rod of length 2l, cross-sectional area A and mass M is set rotating in a
horizontal plane about an axis passing through the centre. If Y is the Young?s
modulus for steel, find the extension in the length of the rod. (Assume the rod
in uniform).
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question_answer48)
An
equilateral triangle ABC is formed by two Cu rods AB and BC and one Al rod. It
is heated in such a way that temperature of each rod increases by DT. Find change in the angle ABC.
[Coeff. Of linear expansion for Cu is \[{{\alpha }_{1}}\], Coeff. of linear
expansion for 2 Al is \[{{\alpha }_{2}}\]]
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question_answer49)
In
nature, the failure of structural members usually result from large torque
because of twisting or bending rather than due to tensile or compressive
strains. This process of structural breakdown is called buckling and in cases
of tall cylindrical structures like trees, the torque is caused by its own
weight bending the structure. Thus the vertical through the centre of gravity
does not fall within the base. The elastic torque caused because of this
bending about the central axis of the tree is given by \[\frac{Y\pi
{{r}^{4}}}{4R}\,Y\] is
the Young?s modulus, r is the radius of the trunk and R is the radius of
curvature of the bent surface along the height of the tree containing the centre
of gravity (the natural surface). Estimate the critical height of a tree for a
given radius of the trunk.
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question_answer50)
A
stone of mass m is tied to an elastic string of negligble mass and spring
constant k. The unstretched length of the string is L and has negligible mass.
The other end of the string is fixed to a nail at a point P. Initially the
stone is at the same level as the point P. The stone is dropped vertically from
point P.
(a)
Find the distance y from die top when the mass comes to rest for an instant,
for the first time.
(b)
What is the maximum velocity attained by the stone in this drop?
(c)
What shall be the nature of the motion after the stone has reached its lowest
point?
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