Buoyant force Archimedes principle The buoyant force on

Buoyant force Archimede’s principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object. The buoyant force is applied to the center of gravity of the fluid (center of the submerged volume of the body).

Equal Volumes Feel Equal Buoyant Forces Suppose you had equal sized balls of cork, aluminum and lead, with respective densities of 0. 2, 2. 7, and 11. 3 times the density of water. If the volume of each is 10 cubic centimeters then their masses are 2, 27, and 113 gram. Each would displace 10 grams of water, yielding apparent masses of -8 (the cork would accelerate upward), 17 and 103 grams respectively (and weights of -0. 08, 0. 17 and 1. 03 N). Apparent mass can be defined as apparent weight divided by the gravitational acceleration, g.

Archimedes principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object. Ship empty. Ship loaded with 50 ton of iron. Ship loaded with 50 ton of styrofoam. Volume of the submerged part of the ship (or any other floating object) is equal to the mass of the ship divided by the density of water: weight of the ship buoyancy force, balancing the weight

Ice cube in a glass of water Water level, h? After the piece of ice melts:

Center of gravity vs. center of buoyancy Gravitational force is applied at the center of gravity. Buoyancy force is applied to the center of submerged part – the center of buoyancy. Center of gravity should be below center of buoyancy for stable equilibrium. Is that a necessary condition of equilibrium? There is something wrong with the picture on the left… What? Center of buoyancy is the center of volume of the submerged part of the boat. It cannot possibly be at or above the water-line!

What about a raft? Is its center of gravity situated below the center of buoyancy? How come, those people are so careless and are not afraid to turn over?

Steady flow in a river. Velocity in each point is shown by a vector with the length proportional to the velocity. Velocity gets higher, where the river gets narrower. Flow represented by streamlines, that are everywhere tangent to flow direction. Higher density of the streamlines corresponds to higher flow velocity. In a steady flow there are no variations in velocity and pattern of flow in time. Nevertheless, the actual fluid elements flowing past any particular point at different times are always different. The fluid elements also get accelerated and decelerated as they move along the streamlines.

Motion of fluids obeys the standard laws of mechanics. Newton’s second law: Becomes Navier-Stokes equation: Newton’s second law is actually a complicated differential equation! Any way to make our life easier? ! Let’s try to use the laws of conservation!!

Motion of fluids obeys the standard laws of mechanics. Conservation of mass: Conservation of momentum: Conservation of energy: Using the laws of conservation means doing appropriate bookkeeping and doing algebra instead of solving differential equations!

Steady flow Flow tube A small tubelike region bounded on its sides by a continuous set of streamlines and on its ends by small areas at right angles to the streamlines. Cross-section areas on the left and right ends are: A 1 and A 2. Densities and velocities are: r 1, r 2 and v 1, v 2

Mass of fluid entering the tube from the left over the time interval Dt Steady flow By mass conservation, over the time interval Dt, the same mass is exiting the tube from the right Therefore everywhere along a flow tube If the fluid is incompressible and its density, r, is constant, we have

Does it work for traffic? Once you pass the spot of accident there are more lanes available (larger A) and the traffic speeds up (higher v). What is the matter? Traffic is highly compressible. You have got to use

How does the total energy of a small fluid element change, as it moves inside the flow tube from cross-section 1 to cross-section 2? Kinetic energy: Potential energy:

How does this change in the total energy become possible? There are external forces originating from pressure of the liquid outside the tube, which do work on the fluid element! Positive work as it enters from the left Negative work as it exits from the right The total energy balance

The total energy balance Incompressible fluids – constant density and volume Bernoulli’s equation