# what

Topics: Buoyancy, Metacentric height, Density Pages: 6 (915 words) Published: March 8, 2014
﻿57:020 Mechanics of Fluids & Transfer Processes
Laboratory Experiment #2

STABILITY OF A FLOATING BODY

Principle
A floating body is said to be stable at its position, if it returns to that position following a small disturbance.

Introduction
Laboratory experiment 2 is an exercise in hydrostatics. It is designed to demonstrate the stability of a floating cylinder and to familiarize the student with the concept of buoyancy, metacenter, and metacentric height. It is also an experimental verification of the theory presented in the textbook. The center of the buoyancy (C, the centroid of the displaced volume of fluid) of a floating body depends on the shape of the body and on the position in which it is floating. If the body is disturbed by a small angle of heel, the center of buoyancy changes because the shape of the submerged volume is changed. The point of intersection of the lines of action of the buoyancy force before and after heel is called the metacenter (M) and the distance between the center of gravity (G) and M, is called the metacentric height (GM, see Fig. 1).

The expression for the metacentric height GM is

GM =
Ioo
V -CG

where Ioo is the moment of inertia of the waterline area about the axis of disturbance, and V is volume of the displaced liquid. For stability the metacentric height GM must be positive. Stability (restoring force) increases with increasing GM. The objective of this experiment is to find the metacentric height and asses the stability of the several floating bodies.

Apparatus
There are three components of the experimental set-up:
1. Large vertically-standing cylinder containing fresh water 2. Small cylinder with a detachable cap at one end
3. Sand and a metric balance to weigh the cylinders

Procedures
1. Weigh the small cylinder and its cap together (mc)
2. Place the small empty cylinder into the large vertical cylinder containing fresh water and observe that it is unstable. 3. Pour a small amount of sand into the small cylinder (to give some ballast) and note if it is still unstable. Estimate how much of the cylinder is submerged, h, and then measure the height of sand, hb.

4. Continue adding sand until the small cylinder remains vertical, i.e., stable. Measure the amount of cylinder that is submerged. Make sure that the open end of the small cylinder is capped when stability is reached. 5. Remove the small cylinder and measure its weight. Also measure the height of sand and record your results on the data sheet.

Measurements
According to procedures described above, measure the following quantities:

Case
Mass (gm)
h (cm)
hb (cm)
Fresh( unstable)

Fresh (stable)

Data Analysis
Figure 2 shows a slender, hollow circular cylinder which has been made stable in the vertical position by means of ballast placed inside. The cylinder floats in a liquid of density, ρ, with a depth of immersion, h. The center of buoyancy is C, and Gc and Gb

are the centers of gravity of the cylinder and ballast, respectively. The center of gravity of the h
cylinder/ballast is G and its
metacenter is M. Other dimensions are shown in the figure. The masses of the cylinder and ballast, mc and mb, t
will be measured during the
experiment.

M Gc

C
h L Gb h/2

O r R

Verify that:
Fig. 2. Hollow cylinder with ballast.
1. The formula for h in terms of the cylinder radius, the density of the liquid ρ , and the masses, mc and mb,
h = (mc + mb )
πR2 ρ
2. The formula for OG in terms of cylinder/ballast dimensions and the masses, mc and mb is
OG m
+ OG m
m L / 2 + m (h
/ 2 + t )
OG = c c b b = c b b (mc + mb )
3. The expression for the metacentric height, GM, is
mc + mb

GM...