Lecture 20 Static fluids
Today s Topics: Density Pressure, Depth and Pressure Gauges Pascal s Principle Archimedes Principle
Solids and Fluids Solids Maintain their shape. Generally don t flow Limited compressibility Resistant to shear Fluids Can Flow. Adapt to the shape of their container Little resistance to shear Liquids. Limited compressibility Gases. Larger compressibility Fills the container
Mass Density The mass density of a substance is the mass of a substance divided by its volume: r = m V SI Unit of Mass Density: kg/m 3 (g/cm 3 ) Ice has a lower density than liquid water: Lakes (water) freeze from the top down!
Example: Mass Density The average density of the material in intergalactic space is approximately 2.5 10 27 kg/m 3. What is the volume of a gold sample, ρ = 19,300 kg/m 3, that has the same mass as 8.0 10 24 m 3 of intergalactic space? r = M V Þ V = M r The mass of 8.0 10 24 m 3 of intergalactic space is : M M So, V gold = rv = 0.02 kg = = (2.5 10-27 kg/m M 0.02 kg = r 19,300 kg/m 3 3 )(8.0 10 = 1.0 10-6 24 m m 3 3 )
Pressure Pressure is uniform in all directions! P = F A SI Unit of Pressure: 1 N/m 2 = 1Pa Pascal Gauge pressure vs. absolute pressure
ACT: Too much pressure If you push both a pin and the blunt end of a pen against your skin with the same force, what will determine whether your skin will be punctured? a) the pressure on your skin b) the net applied force on your skin c) both pressure and net applied force are equivalent d) neither pressure nor net applied force are relevant here The net force is the same in both cases. For the pin, that force is concentrated over a much smaller area of contact with the skin, such that the pressure is much greater. Because the force per unit area (i.e., pressure) is greater, the pin is more likely to puncture.
ACT: On a Frozen Lake You are walking out on a frozen lake and you begin to hear the ice cracking beneath you. What is your best strategy for getting off the ice safely? a) stand absolutely still and don t move a muscle b) jump up and down to lessen your contact time with the ice c) try to leap in one bound to the bank of the lake d) shuffle your feet (without lifting them) to move toward shore e) lie down flat on the ice and crawl toward shore As long as you are on the ice, your weight is pushing down. What is important is not the net force on the ice, but the force exerted on a given small area of ice (i.e., the pressure!). By lying down flat, you distribute your weight over the widest possible area, thus reducing the force per unit area.
Atmospheric Pressure at Sea Level: 1.013 x 10 5 Pa = 1 atmosphere Arises from the weight of the air (fluid) above us Area of a hand A ~ 200 cm 2 = 0.02 m 2 F = p A~ 2000 N on your hand due to air! atm DEMO: Piston and weight
Pressure and Depth å F y = P A 2 - P1 A- mg = 0 P A PA+ 2 = 1 mg P A = 1 m = Vr PA+ r Vg 2 V = Ah P 2 A = 1 PA+ r Ahg P 2 = P 1 + r hg
ACT: Bubbling Up While swimming near the bottom of a pool, you let out a small bubble of air. As the bubble rises toward the surface, what happens to its diameter? a) bubble diameter decreases b) bubble diameter stays the same c) bubble diameter increases As the bubble rises, its depth decreases, so the water pressure surrounding the bubble also decreases. This allows the air in the bubble to expand (due to the decreased pressure outside) and so the bubble diameter will increase.
ACT: Three Containers Three 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? a) container 1 b) container 2 c) container 3 d) all three are equal 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 because the base is also the same for all containers, the total force is the same.
ACT: The Straw When you drink liquid through a straw, which of the items listed below is primarily responsible for this to work? a) water pressure b) gravity c) inertia d) atmospheric pressure e) mass 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?
Example: Pressure A gas sample is confined within a chamber with a movable piston. A small load is placed on the piston; and the system is allowed to reach equilibrium. If the total weight of the piston and load is 70.0 N and the piston has an area of 5.0 x 10 4 m 2, what is the pressure exerted on the piston by the gas? Note: atmospheric pressure is 1.013 x 10 5 Pa. load The pressure exerted by the gas on the piston will be equal to the pressure exerted by the piston on the gas! piston gas sample P = W Piston+load + P A atm 70.0 N = 5.0 10 4 m 2 +1.013 105 N/m 2 = 2.4 10 5 Pa Don t forget atmospheric pressure!
Example: Pressure and Depth A column of water of height 70.0 cm supports a column of an unknown liquid. Determine the density of the unknown liquid. 70.0 cm 27.0 cm At point A (or any point at that height): P water = P unknown A ρ water gh water + P atm = ρ unknown gh unknown + P atm ρ water h water = ρ unknown h unknown ρ unknown = ρ water h water h unknown = (1.0 103 kg/m 3 )(70.0 cm) 27.0 cm = 2.60 10 3 kg/m 3
Pressure Gauges (Barometer) P 2 = P 1 + r gh P atm = r gh P 2 (A) = P 2 (B) h = Patm r g = 5 ( 1.01 10 Pa) 3 3 2 ( 13.6 10 kg m )( 9.80 m s ) = 0.760 m = 760 mm
ACT: Thermometers Thermometers often use mercury or alcohol in a thin glass tube, but barometers never use alcohol. Why? a) mercury is less flammable than alcohol b) mercury s color is easier to see than alcohol c) mercury is less toxic than alcohol d) mercury is denser than alcohol e) mercury is cheaper than alcohol Mercury is very dense, so the height of the column that supports atmospheric pressure is only 760 mm. A water barometer would require a height of about 10 m, which would be inconvenient. Alcohol is less dense than water, so that would be even worse!
Pascal s Principle Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls. P 2 = P1 + r g ( 0 m) F 2 = A 2 F A 1 1 F 2 = æ F ç 1 è A A 2 1 ö ø
Are we getting something for nothing? Let s calculate the work done on the input and output sides:
Buoyant Force F ( B = P 2 A- PA 1 = P 2 - P 1 )A P 2 - P 1 = r gh F r B = gha V = ha F B =! r V g massof displaced fluid
Archimedes Principle Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: F B! = W disp! Magnitude of buoyant force Weight of displaced fluid Note, if an object floats, it s weight is equal to the buoyant force. W obj = W disp m obj g = ρ fluid V sub g ρ obj V obj g = ρ fluid V sub g ρ obj V obj = ρ fluid V sub
ACT : Archimedes An object floats in water with ¾ of its volume submerged. What is the ratio of the density of the object to that of water? a) ¼ b) 1 / 3 c) 4 / 3 d) ¾ e) 1 / 2 Remember that we have: V disp V object = ρ object ρ fluid so if the ratio of the volume of the displaced water to the volume of the object is 3, the object has 3 the 4 density of water. 4
ACT: Wood in Water Two beakers are filled to the brim with water. A wooden block is placed in beaker 2 so it floats. (Some of the water will overflow the beaker). Both beakers are then weighed. Which scale reads a larger weight? The block in 2 displaces an amount of water equal to its weight, because it is floating. That means that the weight of the overflowed water is equal to the weight of the block, and so the beaker in 2 has the same weight as that in 1. same for both
ACT: Lake water level A boat carrying a large chunk of steel is floating on a lake. The chunk is then thrown overboard and sinks. What happens to the water level in the lake (with respect to the shore)? a) rises b) drops c) remains the same d) depends on the size of the steel Initially the chunk of steel floats by sitting in the boat. The buoyant force is equal to the weight of the steel, and this will require a lot of displaced water to equal the weight of the steel. When thrown overboard, the steel sinks and only displaces its volume in water. This is much less water and so the water level in the lake will drop.
Example: Archimedes Principle A sculpture made of brass (r brass = 8470 kg/m 3 ) is believed to have a secret central cavity. The weight of the sculpture in air is 15.76 N. When it is submerged in water, the apparent weight is 13.86 N. What is the volume of the secret cavity?