An ideal Rankine cycle operates with a high$-$pressure steam at $6$ MPa and $500$C$,$ and it condenses at $0\mathrm{.}1$ MPa. Calculate:

a$.$ The thermal efficiency of the cycle.

b$.$ The specific work output.

c$.$ The heat added in the boiler per kilogram of steam.

Fluid Mechanics:

A horizontal Venturi meter with a throat diameter of $0\mathrm{.}1$ m is used to measure the flow rate of water. The inlet diameter is $0\mathrm{.}2$ m$,$ and the pressure difference between the inlet and the throat is $5000$ Pa$.$ Calculate:

a$.$ The flow rate of water through the meter.

b$.$ The velocity of water at the throat.

c$.$ The velocity of water at the inlet.

Mechanics of Materials:

A beam with a rectangular cross$-$section $($width $=0\mathrm{.}2$ m$,$ height $=0\mathrm{.}4$ m$)$ is simply supported at both ends and subjected to a uniformly distributed load of $10$ kN$/$m over its entire length of $5$ meters. Calculate:

a$.$ The maximum bending moment in the beam.

b$.$ The maximum shear force.

c$.$ The maximum bending stress.

Dynamics:

A flywheel with a moment of inertia of $10$ kg$$m$^2$ is subjected to a constant torque of $50$ Nm$.$ Starting from rest, calculate:

a$.$ The angular acceleration.

b$.$ The angular velocity after $10$ seconds.

c$.$ The rotational kinetic energy after $10$ seconds.

Heat Transfer:

A cylindrical rod of length $0\mathrm{.}5$ m and diameter $0\mathrm{.}02$ m is made of copper $($thermal conductivity $=400$ W$/$m$$K$).$ One end of the rod is maintained at $300$C$,$ while the other end is kept at $100$C$.$ Calculate:

a$.$ The steady$-$state heat transfer rate through the rod.

b$.$ The temperature gradient along the rod.

c$.$ The heat transfer per unit area at the hot end of the rod.

Thermodynamics:

A piston$-$cylinder device contains $2$ kg of air at $300$ K and $500$ kPa. The air undergoes an isothermal expansion to a final pressure of $100$ kPa. Calculate:

a$.$ The work done by the air.

b$.$ The heat transfer during the process.

c$.$ The final volume of the air.

Fluid Mechanics:

Oil $($density $=850$ kg$/$m$^3,$ viscosity $=0\mathrm{.}1$ Pa$$s$)$ flows through a $0\mathrm{.}1$ m diameter horizontal pipe with a velocity of $1$ m$/$s$.$ Calculate:

a$.$ The Reynolds number.

b$.$ The friction factor if the flow is laminar.

c$.$ The pressure drop over a length of $50$ meters.

Mechanics of Materials:

A steel column with a length of $3$ meters and a circular cross$-$section of diameter $0\mathrm{.}1$ meters is subjected to a compressive load of $200$ kN$.$ Calculate:

a$.$ The critical buckling load using Euler's formula $($E $=200$ GPa$).$

b$.$ The safety factor against buckling.

c$.$ The maximum compressive stress in the column.

Dynamics:

A projectile is launched with an initial velocity of $50$ m$/$s at an angle of $30$ to the horizontal. Calculate:

a$.$ The maximum height reached by the projectile.

b$.$ The time of flight.

c$.$ The horizontal range of the projectile.

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