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ENGR5002 / Week 2
ENGR217 2017 Q A3Past paperOld spec ENGR2172:125 marks30 min

ENGR217 Summer 2017 Q A3[VALID] official answers

Answer ALL parts (a) - (e).

The ends of a water tank (see Figure A3 below) are of radius 20 m and 36 m long, in the form of circular quadrants. We wish to determine the force exerted by the water on one end of the tank. To calculate the force by direct summation or integration is laborious. It is far simpler to choose a suitable volume of liquid (one pressing against the end of the tank and otherwise bounded by planes) and write down the conditions of equilibrium for it.

Take the quadrant-shaped volume of water. The area of the quadrant is 14πr2\tfrac{1}{4}\pi r^2. The forces on this water consist of its weight W, the force F acting on the vertical plane which forms the left hand boundary, and the force R exerted by the inner surface of the tank wall.

[Data Book: water ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}}.]

Figure A3: water tank whose end is a circular quadrant of radius 20 m, 36 m long.
Figure A3: water tank whose end is a circular quadrant of radius 20 m, 36 m long.
Formulas you may need
  • Weight: W=ρgVW = \rho g V (on the formula sheet, Fg=mgF_g = mg)
  • Force on a plane surface: F=ρgzCGAF = \rho g z_{CG} A (on the formula sheet)
  • Centre of pressure: lCP=lCG+IGA lCGl_{CP} = l_{CG} + \dfrac{I_G}{A\, l_{CG}}, rectangle IG=bh312I_G = \dfrac{b h^3}{12} (on the formula sheet)
  • Quadrant centroid from a straight edge: 4r3π\dfrac{4r}{3\pi} (on the formula sheet, semicircle row of the shape table)
  • Curved surface by equilibrium of a fluid volume: R=W2+F2R = \sqrt{W^2 + F^2}, tan⁡θ=W/F\tan\theta = W/F (learn this)
  • Three non-parallel forces in equilibrium are concurrent (learn this)
  1. (a)
    What is the volume of water in the quadrant shaped end of the tank?
    [3]
  2. (b)
    What is the weight W?
    [2]
  3. (c)
    Find the force acting on the left of this volume of water.
    [8]
  4. (d)
    Draw the triangle of forces and indicate magnitudes and direction.
    [2]
  5. (e)
    Find where the force R acts.
    [10]