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a) i) State any FOUR assumptions made in the theory of torsion.

ii) Show that the strain energy ‘u’ stored in a solid shaft of diameter ‘d’, length ‘l’ and modulus of rigidity G, when subjected to pure torque T, is given by the expression:

u=T2 x Volume of the shaft

4G

where T is the maximum shear stress. (7 marks)

b) A composite shaft is used to transmit 380KW at a speed of 750 rev/min. The composite shaft is made by passing a solid cylindrical shaft of 65mm diameter and 1.8m long, 65mm and 75mm internal and external diameters respectively. They are then rigidly joined together at their ends. The solid shaft is made of steel and the hollow shaft is made of brass.

Determine:

The maximum and minimum stresses in the two shafts.

The angle of twist.

The total strain stored:

Take G for brass=35GN/m2

and G for steel=78 GN/m2 (13 marks)

SECTION C: FLUID MECHANICS

Answer any ONE question from this section.

a) Explain the following terms:

Geometrical similarity

Dynamic similarity (3 marks)

b) Show from the first principles the requirements for dynamic similarity between two fluid motions when considering:

Viscous resistance

Wave resistance (6 marks)

The air resistance R of a supersonic plane during flight, is a function of its length L, velocity V, air dynamic velocity µ, air density ?, and bulk modulus K. Show that the air resistance R is given by:

R=pl^2 V^2Ø{µ/pvl ,k/(pv^2 )}

where Ø means a “function of”. (11 marks)

a) i) Define the following terms with reference to fluid flow:

Critical velocity

Reynold’s number

ii) Show that the flow rate Q, of a fluid of dynamic viscosity ?, flowing under laminar conditions through a horizontal circular pipe of diameter ‘d’, length ‘L’, with a mean velocity V, when the pressure difference between the ends is P, is given by the expression:

Q=(ppd^4)/128?L

Hence show that the pressure difference p can be given by the expression

P=32?lv/d^2 (121/2 marks)

b) Oil is pumped through a pipe 120mm diameter and 900m long. The pressure difference between the ends is 420KN/m2. The dynamic viscosity of the oil is 1.42 N-S/m2 and the relative density is 0.9.

Show that the flow is viscous if Reynold’s number is 2100.

Calculate the electric power of the motor required if the mechanical efficiency between the pump and the electric motor is 80%.

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