The radial displacement in a rotating disc is governed by the differential equation $$\,\,{{{d^2}u} \over {d{x^2}}} + {1 \over x}{{du} \over {dx}} - {u \over {{x^2}}} = 8x.\,\,\,$$ where $$u$$ is the displacement and $$x$$ is the radius. If $$u=0$$ at $$x=0$$ and $$u=2$$ at $$x=1,$$ calculate the displacement at $$\,x = {1 \over {2.}}$$

The general solution of the differential equation $${x^2}{{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + y = 0$$ is

The probability that two friends share the same birth-month is

A car moving with uniform acceleration covers 450 m in a 5 second interval, and covers 700 m in the next 5 second interval. The acceleration of the car is

The discharge velocity at the pipe exit in figure is

A cylinder of mass $$10$$ kg and area of cross-section $$0.1$$ $${m^2}$$ is tied down with string in a vessel containing two liquids as shown in Figure. Calculate gauge pressure on the cylinder bottom and the tension in the string. Density of water $$ = 1000\,\,kg/{m^3}.$$ Specific gravity of $$A=0.8$$. Specific gravity of $$B$$ (water) $$=1.0$$

A hot fluid at $${200^ \circ }C$$ enters a heat exchanger at a mass flow rate of $${10^4}\,\,kg/hr.$$ Its specific heat is $$2000$$ $$J/kg$$-$$K.$$ It is to be cooled by another fluid entering at $${25^ \circ }C$$ with a mass flow rate $$2500$$ $$kg/hr$$ and specific heat $$400$$ $$J/kg$$-$$K$$. The overall heat transfer coefficient based on outside area of $$20\,\,{m^2}$$ is $$250\,\,\,W/{m^2}K.$$ Find the exit temperature of the hot fluid when the fluids are in parallel flow.

Hot water flows with a velocity of $$0.1$$ $$m/s$$ in a $$100mm$$ long, $$0.1m$$ diameter pipe. Heat lost from the pipe outer wall is uniform and equal to $$420$$ $$W/{m^2}$$. If the inlet water temperature is $${80^ \circ }C,$$ calculate the water temp at the exit. Neglect effect of pipe wall thickness. $${C_p}$$ (water) $$=4.2$$ $$kJ/kg$$-$$K$$ and density of water $$=1000$$ $$kg/\,{m^3}.$$

Heat transfer coefficients for free convection in gasses, forced convection in gases and vapors, and for boiling water lie, respectively, in the range of

A gas filled tube has $$2$$ $$mm$$ inside diameter and $$25$$ $$cm$$ length. The gas is heated by an electrical wire of diameter $$50$$ microns ($$0.05$$ $$mm$$) located along the axis of the tube. Current and voltage drop across the heating element are $$0.5$$ amps and $$4$$ volts, respectively. If the measured wire and inside tube wall temperatures are $${175^ \circ }C$$ and $${150^ \circ }C$$ respectively, find the thermal conductivity of the gas filling the tube.

The temp variation under steady heat conduction across a composite slab of two materials with thermal conductivies $${K_1}$$ and $${K_2}$$ is shown in fig. then, which one of the following statements holds?

A job shop has $$6$$ orders to be completed by a single turning center. The processing times and due dates are as follows:

Assume that all orders are ready for processing. Give a production schedule that minimizes the average flow time. Compare this schedule with one that minimizes tardiness (lateness).

Which one of the following forecasting techniques is not suited for making forecasts for planning production schedules in the short range?

One of the following statements about $$PRS$$ (Periodic Reordering System) is not true. Identify.

In inventory planning, extra inventory is unnecessarily carried to the end of the planning period when using one of the following lot size decision policies:

The basic load rating of a ball bearing is

Bolts in the flanged end of pressure vessel are usually pre- tensioned. Indicate which of the following statements is not true

Which theory of failure will you use for aluminium components under steady loading is

A cylinder of $$155$$ $$mm$$ is to be reduced $$150$$ $$mm$$ diameter in one turning cut with a feed of $$0.15$$ $$mm$$/ revolution and a cutting speed of $$150$$ $$m/min$$ on a $$NC$$ lathe. What are the programmed spindle speeds programmed feed rate, and the material removal rate?

In orthogonal machining operation, the chip thickness and the uncut chip thickness are equal to $$0.45mm$$. If the tool rake angle is $$0$$ deg. The shear plane angle is

The voltage-arc length characteristics of a $$DC$$ arc is given by $$V=20+4L.$$ Where $$L$$ is in $$mm$$. During a welding operation arc length is expected to vary between $$4$$ and $$6$$ $$mm$$ with the welding current limited between $$450$$ and $$550$$amps.

If the power source has a linear characteristic, the open circuit voltage is.

The voltage-arc length characteristics of a $$DC$$ arc is given by $$V=20+4L.$$ Where $$L$$ is in $$mm$$. During a welding operation arc length is expected to vary between $$4$$ and $$6$$ $$mm$$ with the welding current limited between $$450$$ and $$550$$amps.

The arc power at an arc length of $$5mm$$ is

A strip with sectional area $$150mm\,\, \times \,\,4.5mm$$ is being rolled with $$20\% $$ reduction of area using $$450mm$$ diameter rollers. The angle subtended by the deformation zone at the roll center is (in radians)

Two castings of the same metal have the same surface area one casting is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of cube.

Match the following

A flywheel of moment of inertia $$9.8$$ kg m² fluctuates by $$30$$ rpm for a fluctuation in energy of $$1936$$ Jouls. The mean speed of the flywheel is (in rpm)

A Curtis stage, Rateau stage and a $$50$$% reaction stage in a steam turbine are examples of

A Rankine cycle operates between pressures of $$80$$ bar and $$0.1$$ bar. The maximum cycle temperature is $${600^ \circ }C.$$ If the steam turbine and condensate pump efficiencies are $$0.9$$ and $$0.8$$, respectively, calculate the specific work and thermal efficiency. Relevant steam table extract is given below:

The isentropic heat drop in the nozzle of an impulse steam turbine with a nozzle efficiency $$0.9,$$ blade velocity ratio $$0.5,$$ and mean blade velocity $$150$$ $$m/s$$ in $$kJ/kg$$ is

If $$\,\,{V_N}\,\,$$ and $$\alpha $$ are the nozzle exit velocity and nozzle angle in an impulse turbine, the optimum blade velocity is given by