OAT: Fluid Statics
principle stating that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by that object
Archimedes' principle
Example: How would you find approximately how much pressure must be exerted on a given volume of water in order to compress it to 99.8% of its volume? (B for water = 2.2 x 10⁹ N/m²)
100-99.8=0.2%, which is 0.002, and P=B(∆V/V) so plug in known values to get 2.2x10⁹(0.002/1)
What is the density of water in kg/m³?
1000
Example: The image represents a section through a horizontal pipe of varying diameters into which open four vertical pipes. If water is allowed to flow through the pipe in the direction indicated, in which of the vertical pipes will the water level be lowest? Why?
2 because it has the smallest area, lowest pressure, and highest velocity
Example: Bars A and B are attached to a wall on the left and pulled with equal forces to the right. Bar B, with twice the radius, is stretched half as far as Bar A, Which bar has the larger value of Young's modulus?
A
Example: Two wooden balls of equal volume but different density are held beneath the surface of a container of water. Ball A has a density of 0.5 gm/cm³ and ball B has a density of 0.7 gm/cm³. When the balls are released, they will accelerate upward to the surface. Which has the greater acceleration? Why?
A because it has the same volume but a smaller mass
What is the volume equilibrium equation for a hydraulic lever?
A1d1=A2d2
What is the formula for stress?
F/A
What is the pressure equilibrium equation for a hydraulic lever?
F1/A1=F2/A2
What is the work equilibrium equation for a hydraulic lever?
F1d1=F2d2
In the equation F1/A1=F2/A2, how do you find F? How do you find A?
F=mg, A=πr²
What is the formula for force required to move something through a viscous fluid using the coefficient of viscosity (η), area, velocity, and distance?
F=ηA(v/d)
What is the formula for buoyant force (Fb) using gravitational acceleration, density of the fluid, and volume of the submerged object?
Fb=pgV
What is the SI unit of viscosity?
Ns/m²
What is the general formula for pressure involving force and area?
P=F/A
What is the formula for the absolute pressure (P) in a fluid due to gravity on an object somewhere below the surface (pressure at surface=P₀)?
P=P₀+pgh
What is the formula for hydrostatic pressure?
P=pgh
principle stating that if an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount
Pascal's principle
What is the formula for gauge pressure (Pg) involving absolute and atmospheric pressure?
Pg=P-P(atm)
If pressure at the surface of a liquid = 1 atm, then how do you find gauge pressure (Pg)?
Pg=pgh
What is the formula for the weight of an object that includes density?
W=pVg
What is the formula for Young's modulus (Y) that includes force, area, original length, and change in length?
Y=(F/A)/(∆L/L)
attractive force that a molecule of liquid feels toward the molecules of some other substance Example: water droplets sticking to the windshield of a car
adhesion
attractive force that a molecule of liquid feels toward the other molecules of the same liquid Example: water droplets sticking to each other
cohesion
According to Bernoulli's equation, when height is constant, does increased velocity mean increased or decreased pressure?
decreased
Example: How would you find the specific gravity of benzene, with p = 879 kg/m³?
density of benzene/density of water=879/1000=0.879
True or false: Pressure depends on the density of an object in a liquid.
false
Example: A diver in the ocean is 20 m below the surface of sea water with a density of 1025 kg/m³. How would you find the absolute pressure and gauge pressure?
find absolute pressure by using P(atm)-pgh, which is 1.013x10⁵-(1025)(9.8)(20), and find gauge pressure by using the previous result-1.013x10⁵
According to Bernoulli's principle, if a fluid has low velocity, does it have low or high pressure?
high
Example: If the cork in the image is removed, water flows out. Assume the downward velocity of the water at the top of the tank is negligible. How does the velocity of the water as it leaves the pipe depend on the air pressure at the surface of the water?
higher pressure, higher velocity
Example: A cup in the shape of a cylinder is filled with soda. If a hole is cut in the side of the cup so the soda shoots out, what is the direction of the soda flow the instant it leaves the cup? Why?
horizontal because pressure of a liquid is always perpendicular to the walls of its container
According to Bernoulli's equation, when height is constant, does decreased velocity mean increased or decreased pressure?
increased
What are the standard units for density?
kg/m³
Pressures increases ____ with depth below the surface of a liquid. (i.e. What kind of relationship?)
linearly
According to Bernoulli's principle, if a fluid has high velocity, does it have low or high pressure?
low
The elasticity of a solid is characterized by a factor called a what?
modulus
Is water flow faster in narrow or wider passages?
narrow
What is the formula for density (p, actually rho)?
p=m/V
Example: How would you find the density of a wooden block floating in the ocean (1024 kg/m³) with half its volume submerged?
set the weight of the block (b) equal to the buoyant force/weight of the water (w) so you have p(b)V(b)g=1/2p(w)V(b)g, then simplify to p(b)=1/2p(w) and do 1024/2
While both fluids and solids can exert forces perpendicular to their surfaces, only which one can withstand shear (tangential) forces?
solids
ratio of the density of a substance to the density of water (no units)
specific gravity
The larger the Young's modulus, the weaker (flexible) or stronger (rigid) the material?
stronger
True or false: Pressure depends on the density of a liquid.
true
What is the continuity equation involving area and velocity?
v1A1=v2A2
measure of the internal friction of a fluid; because of this, a force must be exerted to cause one layer of fluid to slide past another
viscosity
Pressure in a fluid is created by the ____ of the fluid above the submerged object pressing down on that object.
weight
Example: An aluminum cube is suspended by a cable in a tank of water such that it is completely submerged but remains at a fixed depth. Is there a net vertical force on the cube due only to the water pressure? If so, what direction? Why or why not?
yes, upward because the areas of the top and bottom faces are the same but water pressure increases with depth
the point beyond which a material will not return to its original dimensions once the force is removed; if more stress is applied, eventually the ultimate strength is reached, and beyond that point, rupture occurs
yield strength
What is the formula for strain?
∆L/L
What is the formula for the bulk modulus (B) that includes change in pressure, original volume, and decrease in volume?
B=∆P/(∆V/V)
In many instances, what is pressure at the surface of a liquid?
1 atm
1 atm = ? Pa
1.013x10⁵
meters equivalent to 1 cm³
1/1,000,000 m³
meters equivalent to 1 cm²
1/10,000 m²
meters equivalent to 1 cm
1/100 m
What is Bernoulli's equation involving absolute pressure, density and velocity of a fluid, gravitational acceleration, and height of a fluid relative to some reference height?
P₁+1/2pv²₁+pgh₁=P₂+1/2pv²₂+pgh₂
What is the formula for the shear modulus (S) that includes force, area, movement in the direction of the force, and height?
S=(F/A)/(x/h)