Buoyancy and Archimedes Principle Lecturer: Professor Stephen T Buoyancy and Archimedes Principle Lecturer: Professor Stephen T. Thornton Lecture 28

Reading Quiz A) diminish B) stop altogether C) go out in a straight line D) curve upwards When a hole is made in the side of a Coke can holding water, water flows out and follows a parabolic trajectory. If the container is dropped in free fall, the water flow will: Coca-Cola

Reading Quiz A) diminish B) stop altogether C) go out in a straight line D) curve upwards When a hole is made in the side of a Coke can holding water, water flows out and follows a parabolic trajectory. If the container is dropped in free fall, the water flow will: Water flows out of the hole because the water pressure inside is larger than the air pressure outside. The water pressure is due to the weight of the water. When the can is in free fall, the water is weightless, so the water pressure is zero, and hence no water is pushed out of the hole! Coca-Cola

Density, fluids Pressure Pressure gauges and barometers Last Time Density, fluids Pressure Pressure gauges and barometers

Today Today we are studying fluid dynamics after Archimedes Principle. More on pressure Pascal’s Principle Buoyancy Archimedes Principle Mass flow – equation of continuity Today we are studying fluid dynamics after Archimedes Principle.

Measurement of Pressure; Gauges and the Barometer There are a number of different types of pressure gauges. This one is an open-tube manometer. The pressure in the open end is atmospheric pressure; the pressure being measured will cause the fluid to rise until the pressures on both sides at the same height are equal. Figure 13-10a.

Tire pressure gauge.

Pascal’s Principle: An external pressure applied to an enclosed fluid is transmitted unchanged to every point within the fluid.

Do Pascal’s vases demo Do Cartesian diver demo Pascal vases has five glass tubes with a common reservoir – all will have fluid at the same height independent of shape and area.

A Hydraulic Lift

In the hydraulic lift (show demo at end), the liquid is enclosed, and the pressure is the same throughout.

Pascal’s Principle Figure 13-9. Applications of Pascal’s principle: (a) hydraulic lift; (b) hydraulic brakes in a car.

Conceptual Quiz 1 2 3 A) container 1 B) container 2 C) container 3 D) all three are equal Three open containers are filled with water to the same height and have the same surface area at the base, but the total weight of water is different for each. Which container has the greatest total force acting on its base? 1 2 3

Conceptual Quiz 1 2 3 A) container 1 B) container 2 C) container 3 D) all three are equal Three open containers are filled with water to the same height and have the same surface area at the base, but the total weight of water is different for each. Which container has the greatest total force acting on its base? The pressure at the bottom of each container depends only on the height of water above it! This is the same for all the containers. The total force is the product of the pressure times the area of the base, but since the base is also the same for all containers, the total force is the same. 1 2 3

Conceptual Quiz A) water pressure B) gravity C) inertia D) atmospheric pressure E) mass When you drink liquid through a straw, which of the items listed is primarily responsible for this to function? Click to add notes

Conceptual Quiz A) water pressure B) gravity C) inertia D) atmospheric pressure E) mass When you drink liquid through a straw, which of the items listed below is primarily responsible for this to function? When you suck on a straw, you expand your lungs, which reduces the air pressure inside your mouth to less than atmospheric pressure. Then the atmospheric pressure pushing on the liquid in the glass provides a net upward force on the liquid in the straw sufficient to push the liquid up the straw. Follow-up: Is it possible to sip liquid through a straw on the Moon?

Conceptual Quiz You put a straw into a glass of water, place your finger over the top so no air can get in or out, and then lift the straw from the liquid. You find that the straw retains some liquid. How does the air pressure P in the upper part compare to atmospheric pressure PA? A) greater than PA B) equal to PA C) less than PA Click to add notes

Conceptual Quiz You put a straw into a glass of water, place your finger over the top so no air can get in or out, and then lift the straw from the liquid. You find that the straw retains some liquid. How does the air pressure P in the upper part compare to atmospheric pressure PA? A) greater than PA B) equal to PA C) less than PA H Consider the forces acting at the bottom of the straw: PA – P – r g H = 0 This point is in equilibrium, so net force is zero. Thus, P = PA – r g H and so we see that the pressure P inside the straw must be less than the outside pressure PA.

Buoyant Force Due to a Fluid

Archimedes’ principle: The buoyant force is due to the difference in force at the bottom and the top of the cube of sides L. Archimedes’ principle: An object completely immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object.

Buoyant force equals the weight of the displaced liquid, not the weight of the object!

Buoyancy and Archimedes’ Principle Archimedes’ principle: The buoyant force on an immersed object equals the weight of displaced fluid. The picture below shows an object in the air, partially submerged, and completely submerged.

Do Archimedes’ Principle I demo. This is Archimedes Principle I demo. Get the students to believe that the inside of the silver cylinder has the same volume as the blue cylinder (put the blue cylinder into the silver cylinder). Hang the cylinders from the scale as shown and record their weight. Lower the blue cylinder into the water and show the change in weight. With the blue cylinder still submerged, fill the silver cylinder with water. Since the silver cylinder holds water equal in volume to the blue cylinder, the reading on the scale should return to the original reading before lowering the blue cylinder into the water.

Start Archimedes’ Principle II demo. This is Archimedes Principle II demo. Place ball in beaker of cold water. The ball has a slight positive buoyancy and will just float. When the water is heated with the Bunsen burner its density decreases and the ball consequently sinks.

Flotation An object floats when it displaces an amount of fluid equal to its weight. The total weight of the block equals the weight of the fluid displaced.

Floating an Object That Is More Dense than Water The wood and water in (a) have the same weight, and the metal block and water in (b) have the same weight. Note in (c) that bowl made of the metal floats, because the bowl is hollow.

is water density

Conceptual Quiz: What happens to the water level when the ice melts Conceptual Quiz: What happens to the water level when the ice melts? A) Water overflows. B) Water level decreases. C) Water level stays the same.

Answer: C Water level stays the same. Because the ice cube is floating, it displaces a volume of water equal to its weight. When it melts it becomes water and displaces the same volume of water it displaced. Remember the density of ice is less than that of water.

Conceptual Quiz: Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it. Which weighs more?   A)    The glass without ice cubes. B)    The glass with ice cubes. C)    The two weigh the same.

Answer: C The ice cubes displace exactly their own weight in water, so the two glasses weigh the same amount. It is essential that the ice sticks out above the level of the water.

Conceptual Quiz: Now a pebble sits on top of the ice, and the water is filled to the brim of the glass. What happens when the ice melts? A) The water overflows. B) The water level decreases. C) The water level stays the same. D) The pebble explodes.

Answer: B The ice makes no difference, but now the pebble also displaces an amount of water equal to the pebble’s weight. When the ice melts, the pebble drops to the bottom and displaces a volume of water equal to its own volume. Because the volume of the pebble is less than the volume of the water it displaced originally, there is less water displaced afterwards. The water level drops.

Turbulent flow Laminar flow, also known as streamline flow.

Consider a mass m passing through a particular point in time t Consider a mass m passing through a particular point in time t. The ratio m/ t must be constant at every point or mass will build up.

Gases are compressible, but liquids are not Gases are compressible, but liquids are not. For liquid flow, the density  is constant. Then for liquids. Useful equation for many applications: for example, water hoses.

When using a water hose, we put our thumb over end to increase water speed. Hose nozzle does the same thing. Decrease area, increase speed. A1v1 = A2v2 small A, large v

Conceptual Quiz Imagine holding two identical bricks in place underwater. Brick 1 is just beneath the surface of the water, and brick 2 is held about 2 feet down. The force needed to hold brick 2 in place is: A) greater B) the same C) smaller 1 2 Answer: 2

Conceptual Quiz Imagine holding two identical bricks in place underwater. Brick 1 is just beneath the surface of the water, and brick 2 is held about 2 feet down. The force needed to hold brick 2 in place is: A) greater B) the same C) smaller The force needed to hold the brick in place underwater is W – FB. According to Archimedes’ Principle, FB is equal to the weight of the fluid displaced. Because each brick displaces the same amount of fluid, then FB is the same in both cases. 1 2