Q.1 Discuss the nature of the sequence (x1/n), x 0.
Q.2 (a) Test for the convergence of the series whose general term is .
(b) Test for the convergence or divergence of the series :
Q.3 Developing the Fourier series of , show that
Q.4 For
Q.5 Test the analyticity of the following function :
where is the complex conjugate of z.
Q.6 If f (z) = u + iv is an analytic function, find f (z) if u – v = ex (cos y – sin y).
Q.7 Evaluate : along
(i) the straight line joining (1 – i) to (2 + i);
(ii) the curve x = t + 1, y = 2t2 – 1.
Q.8 Evaluate : where c is the circle | z | = 2.
Q.9 Evaluate : .
Q.10 Discuss the bijectivity of the following mapping : f : defined by f (x) = , for all
a, b, ; R being the set of real numbers.
Q.11 Solve : .
Q.12 Solve : .
Q.13 Solve : .
Q.14 Solve in a power series of the following equation :
Q.15 Solve : .
Q.16 Solve : .
Q.17 Show that : .
Q.18 Solve in series the equation .
Q.19 Solve , given that when x = 0 and z = 0 when y is an odd multiple of .
Q.20 A tightly stretched string with fixed end points x = 0 and x = 1 is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points a velocity , find the displacement of the string at any distance x from one end at any time t.
Q.1 (a) Enumerate the difference between the bulk modulus and compressibility. Also derive a relationship between bulk Modulus (K) and Pressure (P) of a gas for isothermal as well as adiabatic processes.
(b) The dynamic viscosity of oil used for lubrication between a shaft and sleeve is 10 poise. Calculate the power lost in the bearing for a sleeve length of 100 mm. Given the thickness of the oil film as 2.5 mm and shaft diameter 0.1 m. The shaft is rotating at the rate of 1500 rpm.
Q.2 (a) What do you understand by the temperature lapse rate (L) with elevation? Derive an expression for temperature at any given point in a compressible fluid underadiabatic condition.
(b) Calculate the capillary effect (in millimetre) in a glass tube of 5 mm diameter when immersed in (1) water (2) Mercury. The temperature of liquids is 20o C and the values of the surface tension of water and mercury at 20o in contact with air are 0.0736 N/m and 0.5145 N/m, respectively. The angle of contact for water is 0o that for mercury 1.3o. Take the density of water at 20oC as 998 Kg/m3.
Q.3 (a) Define Metacentre. Derive an expression for finding out the metacentric height of a ship which is floating in the Dead Sea. Assume all the required values within permissible limits.
(b) A solid cylinder of diameter 10 m has a height of 15 m. Find the metacentric height of the cylinder when it is floating in water with its axis vertical. The specific gravity of the cylinder is 0.9.
Q.4 (a) The velocity rector in a fluid flows given as
Find the velocity and acceleration of a fluid particle at time t = 1, when the coordinate of the particle are (2, 1, 3).
(b) The following cases represent the two velocity components. Determine the third component of velocity such that they satisfy the continuity equilibrium. Assume the flow to be incompressible.
(i)
(ii)
Q.5 (a) A hydraulic press has a ram of 50 cm diameter and a plunger of 10 cm diameter
(Figure 1). It is used for lifting a weight of 200 kN. Find the force required at the plunger.
Figure 1
(b) A single column manometer is connected to a pipe containing a liquid of specific gravity S1. The pressure in the pipe is 5.21 N/cm2. Area of the reservoir is 100 times the area of the tube for mercury manometer reading as shown (Figure 2). If the specific gravity of mercury is 13.6 then find the specific gravity of the liquid.
Figure 2
Q.6 (a) The atmospheric pressure at sea land is 10.15 N/cm2. Determine the pressure at a height of 10,000 m assuming the pressure variation follows : (i) Hydrostatic Law and
(ii) Isothermal Law. The density of the air is given as 1.3 kg/m3.
(b) A rectangular gate 5 m x 2 m is hinged at its base and inclined at 45o to the horizontal as shown in Figure 3. To keep the gate in stable position, a counter weight of 4000 kgf is attached at the upper end of the gate as shown in the figure. What will be the depth of water at which the gate begins to fall? The friction of the hinge and pulley and the weight of the gate are negligible.
Figure 3
Q.7 (a) Discuss in brief the various types of flows encountered in our daily life. What do you understand by rotational flow? Derive a general equation for the rotational flow.
(b) The velocity potential function is given by :
f = 5 (x2 – y2)
Calculate the velocity components at points (3, 4), (– 4, 6), (7, 8) and (– 9, – 12).
Q.8 (a) Prove that the actual discharge in a Venture can be measured by :
where various terms have their usual meaning.
(b) A vertical pipe is carrying water in vertical upward direction, if it is fitted with a venturi of 30 cm x 15 cm size (of inlet and outlet). A differential manometer connected to the inlet and throat gives a reading 20 cm. calculate the discharge rate. Given Cd = 0.98.
Q.9 (a) Water is filled in a hemispherical tank of d = 4 m, the water level is 1.5 m. An orifice of
d = 50 mm is provided at the bottom. Find the time required by water (i) to fall from 1.5 m to 1.0 m and (ii) for completely emptying of the tank, take Cd = 0.60.
(b) A hemispherical tank of radius R is fitted with an orifice of area ‘a’ at its bottom
(Figure 4). The tank contains a liquid where initial length is H1 and in time t the height of the liquid falls to H2. Derive the relation between t, H, a, and R.
Figure 4
Q.10 (a) A pipe line of 60 cm diameter bifurcates at a Y divide into two branches of diameters equal to 40 and 30 cm each. If the rate of flow in the main pipe is 1.5 m and the mean velocity of flow is 7.5 m/s in 30 cm pipe, determine the rate of flow in the other branch pipe.
(b) A jet of water of diameter 50 mm strikes a 40 x 40 cm2 plate of uniform thickness with a velocity of 10 m/s at the centre plate which is suspended vertically by a hinge on its top horizontal edge (Figure 5). The weight of the plate is 98.1 N. The jet strikes normal to the plate. What force must be applied at the lower edge of the plate so that the plate is kept vertical? If the plate is allowed to deflect freely, what will be the inclination of the plate with vertical due to the force exerted by jet of water?
Figure 5
Q.1 (a) Define an isolated system. Distinguish between the term ‘Change of state’, ‘Path’ and ‘Process’ with the help of suitable example.
(b) What are intensive and extensive properties? Explain what is meant by Thermodynamic equilibrium.
(15)
Q.2 (a) A vacuum gauge mounted on a condenser reads 740 mm Hg. What is the absolute pressure in the condenser in kPa when the atmospheric pressure is 760 mm of Hg.
(b) A vessel of cylindrical shape is 50 cm in diameter and 75 cm high. It contains 4 Kg of a gas. The pressure measured with manometer indicates 620 mm of Hg above atmosphere when barometer reads 760 mm of Hg. Determine.
(i) The absolute pressure of the gas in the vessel in bar, and
(ii) Specific volume and density of the gas.
(15)
Q.3 (a) A man of gas is compressed in a quasi-static process form 80 KPa, 0.1 m3 to 0.4 MPa, 0.03 m3. Assuming that the pressure and volume are related by PVn = constant, find the work done by the gas system.
(b) Define a thermodynamic system. Differentiate between open system, closed system and an isolated system with the help of suitable examples.
(15)
Q.4 (a) State the first law of thermodynamics for closed system under going a cycle. Explain zeroth law of thermodynamics.
(b) During are cycle the working fluid in an engine engages in two work interactions; 15 KJ to the fluid and 44 KJ from the fluid, and three heat interaction, two of which are known; 75 KJ to the fluid and 40 KJ from the fluid. Evaluate the magnitude and direction of the third heat transfer.
(15)
Q.5 Estimate the pressure which would be exerted by 3.7 kg of CO in a 0.030 m3 container at
215 k employing :
(i) the ideal gas equation of state, and
(ii) Van der Waal’s equation.
(10)
Q.6 (a) Define Internal Energy and prove that it is a property of a system.
(b) A cylinder contains 0.45 m3 of a gas at 1 x 105 N/m2 and 80oC. The gas is companied to a volume of 0.13 m3, the final pressure being 5 x 10 N/m2. Determine :
(i) The man of gas;
(ii) The value of index ‘n’ for companion;
(iii) The increase in internal energy of the gas; and
(iv) The heat rejected or received by the gas during compression.
Take r = 1.4, R = 294.2 J/KgoC.
(10)
Q.7 A steam boiler has a total volume of 3 m3. Initially it contains 2 m3 of saturated steam and 1 m3 of saturated liquid at 30 bar. Calculate the mass of vapour, mass of liquid, quality of the steam, its specific internal energy and specific enthalpy.
(10)
Q.8 In an ideal air cycle refrigeration system, air enters the compressor at 2 bar, 7oC and is compressed to 5 bar. The air then cooled at constant pressure to 55oC and then expanded in a turbine to 1 bar. The cooling capacity of the system is 12 kW. Assume air behaves as a perfect gas with Cp = 1.005 kJ/kg.K and Cv = 0.718 kJ/kg.K. Find COP, mass flow rate of air and the power required by the system.
(10)
Q.9 An air compressor has a volumetric efficiency of 80%. When tested, the discharge state being 600 kPa, 170oC and thee inlet state 200 kPa, 20oC. If the clearance is 6%, predict the new volumetric efficiency when the discharge pressure is increased to 800 kPa. Assume that the ratio of real to ideal volumetric efficiency and the exponent n remain constant.
(10)
Q.10 8 kg of nitrogen is cooled in a rigid tank from 300oC to 37oC. The initial pressure is 30 bar. Calculate the changes in entropy, internal energy and enthalpy. Assume nitrogen to be an ideal gas with Cp = 1.024 kJ/kg.K and Cv = 0.745 kJ/kg.K.
(10)
Q.1 (a) A gravity dam (specific gravity of the material = 2.4) as shown retains water behind it. Find the resultant of the forces in magnitude and direction, and locate its point of application on base B1 B2.
Figure 1
(b) Find the value of h, and H if b = 50 m, and F. B = 1.2 m (as before), for the resultant R to fall within the middle third of b.
Q.2 A body A is acted upon by seven forces (coplanar) as shown in Figure 2. Determine the sum of forces acting along x-x, and the sum along y-y. Evaluate the resultant, R, (in magnitude and direction with respect to x-x) of the forces.
Figure 2
Q.3 A regular hexagon ABCDEF carries forces as shown in Figure 3 (AB = 2.0 m). Find the total moment of the forces about A, D and T respectively.
Figure 3
Q.4 (a) Two cantilevers (fixed at one end) are differently loaded as shown in Figure 4. Determine the sections (force and moment) at the fixed ends in each case.
(a) (b)
Figure 4
(b) A block A sets on a rigid plane surface (m = 0.53). Check whether the block shall move. What should be the max value of m for the block just to be on the point of moving if P1 is increased by 50 N
Figure 5
Q.5 (a) The position vector of a particle at any time t with respect to 3 coordinate axes is given by :