Introduction

The metacentric height (GM) is calculated as the distance between the center of gravity(G) of a floating body and its metacenter(M). It is the measure of the static stability of the floating body. GM = BM - BG

Theory

A point about which a floating body tries to oscillate is called the metacenter of that body. The distance between the metacenter (M) and the center of gravity (G) of the floating body is known as metacentric height (GM). When a small angular displacement is given to a body floating in a liquid, it starts oscillation about some point M. This point about which the body starts oscillating is called the metacenter.

The following formula is used for the calculation of the Metacentric height of a floating body:

GM = (w1.Y)/(W + w1). tan θ

Metacenter (M) may be defined as the point of intersection of the axis of the body passing through the center of gravity (G) and original center of buoyancy (B) and a vertical line passing through the new center of buoyancy (B) of the titled position of the body. Buoyancy is the tendency of the fluid to lift a submerged body. The resultant upward force or thrust exerted by a fluid on the submerged body is known as Force of Buoyancy. According to Archimedes Principle, FB = weight of the volume of liquid displaced by the body. The Center of Gravity(G) is the point where is the weight of the body is acting. The Center of Buoyancy (B) is the point to which the force of Buoyancy is acting. The Center of Buoyancy is the center of gravity of the volume of liquid displaced by the body.



The relation between the center of gravity and metacenter in different three types of equilibrium:

  1. Neutral equilibrium

    If GM = 0 (M coinciding with G)

  2. Stable equilibrium

    If GM > 0 (M is above G)

  3. Unstable equilibrium

    If GM < 0 (M is below G)


Objective

To calculate the Metacentric height of a floating body.

Procedure

The procedure to find the metacentric height of a floating body is as follows:

  1. Take an empty tank and fill it with water up, the height of the water level is (Z1).
  2. Now place the floating body having weight W in the tank,the water level rises upto Z2.
  3. Adjust the floating ship.
  4. After adjusting, add load of weight (w) to the horizontal beam of the floating body at centre of the beam.
  5. Now displaced the load at a known distance Y from the centre of the beam.
  6. Now the floating body will tilt at some angle on one side and observe the tilt angle.
  7. Finally, calculate the metacentric height using the given formula:
    8. Metacentric Height (GM) = (w*Y)/((W+w)*tan(θ)

Observations

  • Weight of floating body = W
  • Additional weight added on the horizontal beam of the floating body = w1
  • The distance of w1 from the center = Y
  • Tilt angle = θ