Stability of Floating Bodies

Stability of Floating Bodies - Ability of a floating  body to return return to its neutral neutral position position after the the external force has been applied and removed.

the bottom. What is the value of righting moment. Relative equilibrium of Fluids a. Moving vessel with constant acceleration (horizontal motion)

Metacenter –  Metacenter  –  a  a point on the vertical neutral axis through which the buoyant force always acts for a small angle of tilt. For stability to exist, the objects center of gravity must be below its metacenter. For partially submerged objects the location of metacenter is found by: AM = ABo + MBo Where: AM = distance from the bottom of the object to the metacenter ABo = distance from the bottom of the object to the center of buoyancy. The location of the buoyancy B o is located at the geometric center of the displacement volume. MBo = Is/V Is = moment of inertia about the largest axis of the area produced if the object were cut at the water line. MBo = B2/12D (1 + tan2 θ/2) MG = MBo –  GB  GBo Where MG = metacenter height 1.

A rectangular scow 9m wide 15m long and 3.6m high has a draft in sea water of 2.4m. Its center of gravity is 2.7m above the  bottom of the the scow. Determine Determine the initial initial metacentric height and final metacentric if the body is tilted until one end is just submerged in water. 2. If the center of gravity of a ship in the upright position is 10m above the center of gravity of the portion under water, the displacement being 1000 metric tons, and the ship is tipped 30 ⁰ causing the center of buoyancy to the shift sidewise by 8m. Find the location of the metacenter from

tan θ = a/g  b. Vertical motion. P1 = γwh ( 1 + a/g) upward P2 = γwh(1 –  h(1 –  a/g)  a/g) downward 1. An open tank 1.90m square weighs 3500N and contains 0.95m of water. It is acted by an unbalanced force of 11000N parallel to a  pair of sides. sides. Find the constant constant acceleration acceleration of the tank and the forces acting on the sides of the tank. 2. An unbalanced vertical force of 300N upward accelerates a volume of 0.050m 3 of water. If the water is 0.9m deep in a cylindrical tank. What is the acceleration of the tank and the force acting on the bottom of the tank? Rotating Vessel a.

Cylindrical Cylindrical vessels with free liquid surfaces  –  if  if an open vessel is partly filled with water or any liquid and is rotated at a certain velocity about its vertical axis, its free surface becomes concave in form

2 2

y = ω r  /2g where: y = height of the paraboloid (m) ω = angular velocity (rad/sec) r = radius (m)

1. An open cylindrical tank one meter in diameter and 2.5m high is 3/5 full of water. If the tank is rotated about its vertical axis, what speed should it have in rpm so that a. The water could just reach the rim of the tank.  b. The depth of water at the center is zero. c. There is no water at the bottom with in 20cm from the vertical axis. 2. An open cylindrical tank is 1.20m in diameter and 2.10m high is 2/3 full of water. a. Find the amount of water spilled out if the vessel will at a constant speed of 90 rpm.  b. What speed in rpm will the vessel rotate without spilling of water? Closed cylindrical tank 1. A closed cylindrical vessel, axis vertical, 2m high and 0.60m in diameter is filled with water, the pressure intensity at the top being 196.2kPa. the metal side is 2.5mm. If the vessel is rotated at 240rpm. Compute the  pressure in the wall and against the top. Find also the hoop tension. Bernoulli’s Equation 1. A liquid of specific gravity 1.60 flows in a 6cm horizontal pipe. The total energy at a certain point in the flow is 80J/N. The elevation of the pipe above a fixed datum is 2.6m. If the pressure at the specified point is 80kPa. Determine the velocity of flow and  power at specified points. 2. In the free siphon shown, compute the following: a. Pressure of the water in the tube at B.  b. Pressure of the water in the tube at A. c. If the vapor pressure of water is 0.1799m of water, how high “h”

above the free surface can point B  before the siphon action breaks down. Assume the atmosphere  pressure is 101kPa. 3. Water enters a pump through a 250mm diameter pipe at 35kPa. It leaves the pump at 140kPa through a 150mm diameter pipe. If the flow rate is 150 liter/sec. find the horsepower delivered to the water by the  pump. Assume the suction and discharge sides of pump are at the same elevation.