A Carnot engine operates between a high$-$temperature reservoir at $600$ K and a low$-$temperature reservoir at $300$ K$.$ If the engine absorbs $1000$ J of heat from the high$-$temperature reservoir, calculate:

a$.$ The efficiency of the engine.

b$.$ The work done by the engine.

c$.$ The heat rejected to the low$-$temperature reservoir.

Fluid Mechanics:

Water flows through a horizontal pipe with a diameter of $0\mathrm{.}15$ m at a velocity of $1\mathrm{.}5$ m$/$s$.$ The pipe then narrows to a diameter of $0\mathrm{.}075$ m$.$ Using the principle of continuity, calculate:

a$.$ The velocity of water in the narrower section of the pipe.

b$.$ The mass flow rate of water through the pipe.

Mechanics of Materials:

A steel rod with a length of $3$ meters and a cross$-$sectional area of $0\mathrm{.}02$ m$^2$ is subjected to a tensile force of $100,000$ N$.$ Calculate:

a$.$ The stress in the rod.

b$.$ The strain in the rod if the Young$$s modulus for steel is $200$ GPa.

c$.$ The elongation of the rod.

Dynamics:

A block of mass $15$ kg is sliding down an inclined plane with an angle of $25$ to the horizontal. The coefficient of friction between the block and the plane is $0\mathrm{.}25.$ Calculate:

a$.$ The acceleration of the block.

b$.$ The net force acting on the block.

c$.$ The distance traveled by the block in $4$ seconds.

Heat Transfer:

A flat plate of area $3$ m$^2$ is exposed to solar radiation. The solar irradiance is $900$ W$/$m$^2.$ If the plate is made of a material with an absorptivity of $0\mathrm{.}7,$ calculate:

a$.$ The total solar energy absorbed by the plate per second.

b$.$ The temperature rise of the plate after $15$ minutes if the specific heat capacity of the material is $800$ J$/$kg$$K and the mass of the plate is $7$ kg$.$

Thermodynamics:

A steam turbine operates with an inlet steam condition of $4$ MPa and $400$C and an exit steam condition of $0\mathrm{.}1$ MPa with a dryness fraction of $0\mathrm{.}9.$ Calculate:

a$.$ The work done by the turbine per kilogram of steam.

b$.$ The change in enthalpy.

c$.$ The change in entropy.

Fluid Mechanics:

Air flows through a converging$-$diverging nozzle with an inlet pressure of $400$ kPa, an inlet temperature of $500$ K$,$ and an inlet velocity of $120$ m$/$s$.$ The throat pressure is $250$ kPa. Calculate:

a$.$ The throat temperature.

b$.$ The throat velocity.

c$.$ The throat area assuming isentropic flow.

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