C : MATERIALS SCIENCE
D: SOLID MECHANICS
Q. 1 – Q. 9 carry one mark each.
Q.1 A steel wire of diameter 5 mm is bent around a cylindrical drum of radius 0.5 m. The steel wire has modulus of elasticity of 200 GPa. Find the bending moment in the wire in N-m.
Q.2 A compressed air tank having an inner diameter of 480 mm and a wall thickness of 8 mm is formed by welding two steel hemispheres. If the allowable shear stress in the steel is 40 MPa, find the maximum permissible pressure (in MPa) inside the tank.
Q.4 A point in a body is subjected to a bi-axial state of stress, equal in magnitude but opposite in nature.
On a plane inclined at an angle 45° with respect to x-axis (passing through the point), the (A) shear and normal stresses are zero
(B) normal stress is maximum and shear stress is zero (C) shear stress is maximum and normal stress is zero (D) shear stress is maximum and normal stress is non-zero
Q.3 The Euler’s buckling load of a column fixed at both the ends is P. If one of the ends is made free, the buckling load shall change to
(A) P/16 (B) P/8
(C) P/4 (D) P/2
R = 0.5 m d = 5 mm
Q.6 For the pin jointed truss, find the axial force (in kN) in the member 2-5.
Q.5 A weightless beam subjected to two point loads is shown in the figure below.
The shear force diagram of the beam is
(A)
(B)
(C)
(D)
10 kN
10 kN 2 m
1 m 1 m
10 kN
10 kN
10 kN 10 kN
5 kN 5 kN
5 kN
10 kN
Q.7 The supporting structure of a water tank is made of reinforced concrete (RC) with a tubular cross section of inner diameter di, outer diameter do, height l, and Young’s modulus E. The mass of the tank is m. If mass of the supporting structure is neglected, then the natural frequency of the water tank in transverse direction is
(A)
�
3����� −4 ��4�64�3� (B)
�
����� −4 ��4�8�3�
(C) )
�
384����� −4 ��4�360�3� (D)
�
����� −4 ��4�64�3�
Q.8 A mass is attached to a spring and placed horizontally in a frictionless surface. A simple pendulum has been pivoted to the mass. The degree of freedom of this system is
(A) 1 (B) 2 (C) 3 (D) 4
X
y
x l
Q.9 Consider the following two statements
Statement 1: A body of weight W falls from a height h and strikes the ground. If the body starts from rest, the velocity with which it strikes the ground is �2�ℎ , where g is the acceleration due to gravity.
Statement 2: If the same body (initially at rest) slides without friction along an inclined plane PQ (angle of inclination α) starting from an elevation h above point Q, then its velocity at point Q is
�2�ℎ
The correct option is
(A) Both statements 1 and 2 are true (B) Statement 1 is true and 2 is false (C) Statement 1 is false and 2 is true (D) Both statements 1 and 2 are false
W
h
h P
Q W
α
Q. 10 – Q. 22 carry two marks each.
Q.10 A composite bar of length ‘L’ is made of a centrally placed steel plate (50 mm wide x 10 mm thick) with two copper plates (each 30 mm wide x 5 mm thick) connected rigidly on each side. If the temperature of the composite bar is raised by 50°C, find the stress developed in each copper plate in MPa.
(For Steel: Es = 2x105 MPa and αs= 12x10-6 /°C; For Copper: Ec = 1x105 MPa and αc= 17x10-6 /°C)
Q.11 The vertical deflection at the free end of the cantilever beam as shown in figure is
(A) 1400/EI (B) 1400/3EI (C) 200/EI (D) 100/EI
Q.12 A hollow shaft and a solid shaft have the same length and the same outer radius R. The inner radius of the hollow shaft is 0.6 R. Assuming that both the shafts are made of same material and are subjected to the same torque, find the ratio of shear stress in hollow shaft to that in solid shaft.
Q.13 A beam with overhangs carries one point load acting downwards and the other upward. The clockwise moment Pb is applied at each support. The bending moment at the midpoint of the beam is
(A) 0 (B) PL/2 (C) PL (D) PbL
L Steel plate
5 mm 5 mm
Copper plate
Copper plate
10 mm
100 kN
EI = flexural rigidity 2 m 1 m
L P P
b b
Pb Pb
Q.14 A cantilever beam of length L supports a concentrated load P at the free end. The cross section of the beam is rectangular with constant width b and varying depth h. The depth h of this idealized cantilever beam varies in such a way that the maximum normal stress at every cross section remains equal to the allowable bending stress. Considering only the bending stresses, the depth hx of the fully stressed beam at any distance x from the free end shall vary
(A) with square of x (B) with square root of x
(C) linearly with x (D) with cube of x
Q.15 A cantilever beam is subjected to following three different loading conditions:
(a) a concentrated load P at its free end, (b) a couple Mo at its free end and (c) both loads acting simultaneously
(a)
(b)
(c)
The flexural rigidity of the beam may be assumed as EI. The strain energy due to bending when both loads act simultaneously
(A) can be determined by applying the principle of superposition and the strain energy is �
2�3
6��
+
��2�2��
(B) can be determined by applying the principle of superposition and the strain energy is �2�2
6��
+
���32��
(C) cannot be determined by applying the principle of superposition and the strain energy is �
Q.16 A tapered rod has diameter d1 at one end which reduces uniformly to a diameter d2 over the length (L). If the modulus of elasticity of the material is E, the change in the length of the rod due to the application of axial force (P) is
(A) 4��
���1�2 (B) 4��
����12−�22� (C) ��
���1�2 (D) 2��
����12−�22�
Q.17 For a point in a body subjected to a plane stress condition (σx= 100 MPa, σy= 50 MPa and τxy = τyx = 25 MPa), the maximum principal stress in MPa is_____________
Q.18 An isotropic body is subjected to a state of stress given by: σx= 10 MPa and τxy= τyx = −20 MPa.
Assuming G = 0.4E, the volumetric strain is
(A) 5/E (B) 7.5/E (C) 10/E (D) 15/E
Q.19 A block of weight Q rests on an inclined plane and it is attached to a string which runs over a frictionless pulley to carry a block of weight P at its other end. The coefficient of friction between the block of weight Q and the inclined plane is µ. Consider the following cases:
Case I: weight Q starts moving down the inclined plane Case II: weight P starts falling down
The limiting values of ratio P/Q for Case I and Case II respectively are (A) (sin� − � cos �) and (sin � + � cos �)
(B) (�sin � − cos �) and (�sin � + cos �) (C) (sin� + � cos �) and (sin � − � cos �) (D) (�sin � + cos �) and �(sin � − cos �)
Q.20 To unload an item from a truck a crane boom is raised with a constant angular velocity of 1 rad/s relative to the cab and then the cab is rotated about a vertical axis with constant angular velocity of 0.5 rad/s.
If the length of the boom (OP) is 10 m, the velocity of the tip (P) of the boom in m/s is
(A) √2
5 �−2� � − �̂ + 2��� (B) √2
5 �−� � − 2�̂ + ���
(C) 5
√2�−2� � − �̂ + 2��� (D) 5
√2�−� � − 2 �̂ + 2���
Q.21 A block of mass 5 kg moves up on a smooth inclined plane with a velocity of 10 m/s in the direction shown. A bullet of mass 60 g travelling at 500 m/s strikes the block centrally and gets embedded in it. The velocity of the block and embedded bullet in m/s immediately after the impact is
(A) 12.54 at 30° (B) 13.84 at 51.78° (C) 13.84 at 30° (D) 15.62 at 51.78°
Z
X
Y Cab
Boom
45° O
P
30° x y
500 m/s 10 m/s
Q.22 A balloon with ballast (weight) inside it has a gross weight of 500 N. It is falling vertically with a constant acceleration of 2 m/s2. If air resistance is negligible, find the weight of ballast (in N) that must be thrown out in order to give the balloon an upward acceleration of 2 m/s2. (Acceleration due to gravity, g = 9.81 m/s2)
END OF THE QUESTION PAPER
500 N
E : THERMODYNAMICS
Notations used:
P-pressure, V-volume, T-temperature, S-entropy, H-enthalpy, U-internal energy, cp-specific heat at constant pressure, cv-specific heat at constant volume; specific properties are designated by lower case symbols.
Subscripts: R-reduced, C-critical, f-saturated liquid, g-saturated vapor,
Properties of air: cp = 1.005 kJ/(kg.K), specific heat ratio γ = 1.4, Gas constant = 0.287 kJ/(kg.K), Molecular weight = 29 gm/mol.
Universal gas constant = 8.314 kJ/(kmol.K).
Q. 1 – Q. 9 carry one mark each.
Q.1 Entropy is a
(A) Path function (B) Point function
(C) Property independent function (D) Neither path nor point function
Q.2 A small container has gas at high pressure. It is placed in an evacuated space. If the container is punctured, work done by the gas is
(A) Positive (B) Negative (C) Zero (D) ∞
Q.3 The molecular weight of a mixture is 38.4 gm/mol. The mixture is composed of methane and carbon-dioxide gases. The atomic weights of the elements C, H, and O are 12, 1, and 16 gm/mol, respectively. The mole fraction of methane (Xmethane) is ____________ and that of carbon-dioxide (X carbon-dioxide) is _______.
(A) Xmethane = 0.2; X carbon-dioxide = 0.8 (B) Xmethane = 0.8; X carbon-dioxide = 0.2 (C) Xmethane = 0.3; X carbon-dioxide = 0.7 (D) Xmethane= 0.7; X carbon-dioxide = 0.3
Q.4 A system undergoes a change from state 1 to state 2. During this process, the change in the internal energy is ∆U. The change in internal energy of the system when executing the cycle 1-2-1 is equal to
(A) ∆U (B) 2∆U (C) Zero (D) −2∆U
Q.5 Which among the following plots represents a line joining two states with the same dew point temperature on a standard psychrometric chart, with the dry bulb temperature on the X-axis and the humidity ratio on the Y-axis?
(A) (B)
(C) (D)
Q.6 The efficiency of a reversible engine operating between two temperatures is 40 % . The COP of a reversible refrigerator operating between the same temperatures is
(A) 1.5 (B) 2.5 (C) 0.4 (D) 3.5
Q.7 For a superheated vapor that cannot be approximated as an ideal gas, the expression determining a small change in the specific internal energy is
(A) dv
Q.8 The minimum and maximum volumes in an air standard Otto cycle are 100 and 800 cm3. Its thermal efficiency (%) is
(A) 56.47 (B) 94.55 (C) 54.08 (D) 87.50
Q.9 At a saturation temperature Tsat, the difference between the entropy of saturated vapor and entropy of saturated liquid can be expressed as
(A)
(
hf −hg)
Tsat (B)(
hg −hf)
Tsat(C)
(
ug −uf)
Tsat (D)(
uf −ug)
TsatQ. 10 – Q. 22 carry two marks each.
Q.10 A gas in a closed system is compressed reversibly from an initial volume of 0.2 m3 to 0.1 m3 at a constant pressure of 3 bar. During this process, there was a heat transfer of 50 kJ from the gas. The change in internal energy of the gas during this process in kJ is
(A) 20 (B) −80 (C) 80 (D) −20
Q.11 In a closed rigid vessel, air is initially at a pressure of 0.3 MPa and volume of 0.1 m3 at 300 K. A stirrer supplies 100 kJ of work to the air, while 20 kJ of heat is lost to the atmosphere across the container walls. After these processes, the temperature of air changes to _________K.
(A) 321.9 (B) 702.4 (C) 782.4 (D) 620.2
Q.12 A reversible heat engine (E) operates using three thermal reservoirs with temperatures as shown in the following figure. If Q1=Q2, the efficiency of the engine is ___________.
(A) 0.25 (B) 0.125 (C) 0.625 (D) 0.75
Q.13 A metal block of mass 25 kg at 300 K is immersed in an infinitely large liquid nitrogen bath maintained at 77 K. The system comprising of the block and liquid nitrogen attains thermal
equilibrium. The average specific heat of the metal is 0.45 kJ/(kg.K). The entropy generated during the process is ___________kJ/K.
(A) 17.28 (B) 32.5 (C) 47.8 (D) −47.8
Q.14
For a gas obeying the equation of state given by v RT v
P a
=
+
2 , the values of the critical volume and the critical temperature are 0.004 m3/kg and 100 °C, respectively. If the value of the gas constant is 250 J/(kg.K), then the value of the constant ‘a’ is _____________ (N.m4/kg2). Note that the critical point is the point of inflection on the critical isotherm.(A) 124.3 (B) 0.75 (C) 186.58 (D) 248.67
Reservoir 1
T1= 1200K
Reservoir 2
T2= 600 K
Reservoir 3
T3 = 300 KQ1 Q2
Q3
E W
Q.15 A rigid closed vessel is initially filled with 2 kg of water which is a mixture of saturated liquid and saturated vapor states at 2 bar. The vessel is placed in an oven which heats the mixture to the critical state. Using the saturated and critical property values from the table given below, the heat transferred from the oven to the vessel is __________kJ.
Pressure = 2 bar
∂
is known for the gas. The inversiontemperature, given by the condition,
Q.17 The maximum pressure and temperature in an air standard diesel cycle are 44 bar and 1600 K, respectively. If the minimum pressure and temperature are 1 bar and 300 K, respectively, then the cut-off ratio (the ratio of the volume at the end of the heat addition process to that at the beginning of the heat addition process) is
(A) 1.000 (B) 14.920 (C) 2.809 (D) 1.809
Q.18 The thermal efficiency of an air standard Brayton cycle 0.35. The pressure ratio across the turbine is
(A) 4.516 (B) 5.232 (C) 7.535 (D) 8.234
Q.19 Steam is isentropically expanded in a turbine from 80 bar to 7 bar. At the inlet of the turbine (state 1) h1 is 3246 kJ/kg and s1 is 6.52 kJ/(kg.K).
Pressure = 7 bar
hf (kJ/kg) hg (kJ/kg) sf [kJ/(kg.K)] sg [kJ/(kg.K)]
697 2763 2.0 6.7
The enthalpy of the steam exiting the turbine (state 2) in kJ/kg is
(A) 2683.87 (B) 2657.17 (C) 1986.87 (D) 3354.17
Q.20 A thin insulating membrane separates two tanks initially filled with nitrogen [mean cv = 21.6 J/(mol.K)] and carbon-dioxide [mean cv = 11.6 J/(mol.K)] as shown below.
The membrane is ruptured and the gases are allowed to mix to form a homogeneous mixture at equilibrium. During this process there are no heat or work interactions between the tank contents and the surroundings. The final temperature at the equilibrium state in Kelvin is
(A) 344.1 (B) 306.3 (C) 325.0 (D) 346.1
Q.21 Two moist air streams MAS1 and MAS2 are mixed adiabatically. The details of MAS1 and MAS2 are given below in the table.
MAS1 MAS2
h (kJ/kg of dry air) 42 80 v (m3/kg of dry air) 0.85 0.9
Flow rate (m3/min) 85 90
With pressure remaining same and with no work interactions during the mixing process, the enthalpy of the mixed stream is ________________ kJ/kg of dry air.
(A) 122 (B) 61 (C) 81 (D) 108
Q.22 Consider the steady flow of air through an insulated nozzle. The pressure and temperature at the inlet are 120 kPa and 320 K, respectively. The outlet pressure is 1 bar. The inlet velocity is very small and the air undergoes a reversible adiabatic process. The outlet velocity, in m/s, is
(A) 303.7 (B) 180.7 (C) 5.7 (D) 127.3