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ENGR5002 / Week 1
ENGR217/266 2023 Q A1Past paperCurrent spec2:125 marks30 min

ENGR217/266 Summer 2023 Q A1[VALID] official answers

The U-tube manometer shown in Figure A1-1 has the two ends exposed to gauge pressures pleftp_{left} and prightp_{right}. The blue area is occupied by water (take water density to be ρw=1000 kg m−3\rho_w = 1000\ \mathrm{kg\,m^{-3}}), while the grey one is filled with a fluid of unknown properties.

Figure A1-1: U-tube manometer (14 cm, 3.5 cm, 8.5 cm, 12.5 cm levels). Figure A1-2: gate hinged at A, inclined at 60 deg, held by a horizontal cable (1.8 m wetted, 2.4 m long).
Figure A1-1: U-tube manometer (14 cm, 3.5 cm, 8.5 cm, 12.5 cm levels). Figure A1-2: gate hinged at A, inclined at 60 deg, held by a horizontal cable (1.8 m wetted, 2.4 m long).
Formulas you may need
  • Hydrostatic pressure: p(z)=p0+ρgzp(z) = p_0 + \rho g z; equal pressure at equal levels in a continuous static fluid (on the formula sheet)
  • Gauge pressure: pg=p−patmp_g = p - p_{atm} (on the formula sheet)
  • Force on a plane surface: F=ρgzCGAF = \rho g z_{CG} A (on the formula sheet)
  • Centre of pressure along the plane: lCP=lCG+IGA lCGl_{CP} = l_{CG} + \dfrac{I_G}{A\, l_{CG}} (on the formula sheet)
  • Rectangle: Ixc=ba312I_{xc} = \dfrac{b a^3}{12} (also given in the question) (on the formula sheet)
  • Depth of a point on an inclined plane: z=lsin⁡θz = l \sin\theta (learn this)
  • Moment equilibrium about the hinge: ∑MA=0\sum M_A = 0 (learn this)
  1. (a)
    Supposing pleftp_{left} and prightp_{right} to be equal to the atmospheric pressure, determine the density of the unknown fluid.
    [3]
  2. (b)
    Assuming the left end to be open (pleftp_{left} is the atmospheric pressure, i.e., pleft,gauge=0p_{left,gauge} = 0) and pright=15p_{right} = 15 Pa (gauge), show that the unknown fluid has the same density as the water.
    [2]
  3. (c)
    The rectangular gate shown in Figure A1-2 weighs 350 N. The gate, which has b=1.2b = 1.2 m wide and a=2.4a = 2.4 m long, is hinged in A and held in place by a horizontal cable. Taking water density to be ρw=1000 kg m−3\rho_w = 1000\ \mathrm{kg\,m^{-3}}, determine the hydrostatic force exerted by the water on the gate.
    [4]
  4. (d)
    The location of the centre of pressure on the gate (i.e. the moment of inertia of a rectangle is Ixc=ba3/12I_{xc} = b a^3/12).
    [4]
  5. (e)
    Sketch the free body diagram.
    [4]
  6. (f)
    Write down the momentum equilibrium about the hinge.
    [4]
  7. (g)
    Determine the force exerted by the cable on the gate.
    [4]