Question

1- Comment on the effects of changing the position of G on the position of

metacentre.

2- Explain how unstable equilibrium might be achieved?

(Fluid Subject)

Answer 1

• When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the center of buoyancy moves relative to the body.

• As a result of above observation stable equlibrium can be achieved, under certain condition, even when G is above B. Figure illustrates a floating body (a) boat, for example, in its equilibrium position.

• Important points to note here are
  a) The force of buoyancy F_B is equal to the weight of the body W
  b) Center of gravity G is above the center of buoyancy in the same vertical line.
  c) Figure shows the situation after the body has undergone a small angular displacement theta with respect to the vertical axis.
  d) The center of gravity G remains unchanged relative to the body.
  e) During the movement, the volume immersed on the right hand side increases while that on the left hand side decreases. Therefore the center of buoyancy moves towards the right to its new position B'.

• Let the new line of action of the buoyant force (which is always vertical) through B' intersects the axis BG (the old vertical line containing the center of gravity G and the old center of buoyancy B) at M. For small values of theta () the point M is practically constant in position and is known as metacenter. For the body shown in Fig, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position. If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable. When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium. Therefore, for a floating body, the stability is determined not simply by the relative position of B and G, rather by the relative position of M and G. The distance of metacenter above G along the line BG is known as metacentric height GM which can be written as

• GM = BM - BG

• Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows:
  
  GM > 0 (M is above G)                                      Stable equilibrium
  GM = 0 (M coinciding with G)                          Neutral equilibrium
  GM < 0 (M is below G)                                      Unstable equilibrium