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3. Fluid Mechanics

- In this chapter, we'll discuss some of the concepts relating to fluids.
- There are differences between liquids and gases, but this chapter focuses on the similarities of all fluids.

- The density of the substance is more useful in fluid mechanics than the concept of mass.

- When r is a constant, m is equal to rV.

- The fluid exerts contact force on the object if we place it in it.

- The SI unit for pressure is the newton per square meter.

- A nickel on a table exerts 140 Pa of pressure, just due to its weight alone, because one pascal is a very tiny amount of pressure.
- You'll often see pressures expressed in kPa, where 1 kPa is 10 3 Pa, or in MPa, where 1 MPa is 10 6 Pa.
- The atmosphere is a common unit for pressure.
- The atmospheric pressure at sea level is 101,300 Pa.

- A column made of cement has a base area of less than 2 m.

- The force the column exerts on the ground is equal to its weight, so we'll find the pressure it exerts by dividing it by the base area.

- You can use the unit m/s 2 for g, but it is simpler to use the unit N/kg for g.

- Imagine a tank with a lid on top filled with liquid.
- There is a thin sheet of metal hanging from this lid.

- The force that pushes down on the metal sheet comes from the weight of the liquid.

- The liquid is at rest, so this is known as hydrostatic pressure.

- The density of the liquid and the depth below the surface are what determines the hydrostatic pressure due to the liquid.
- The shape of the container doesn't matter.
- If all the containers in the figure below are filled with the same liquid, then the pressure is the same at every point along the horizontal dashed line.

- If the liquid in the tank were open to the atmosphere, the total pressure in the tank would be equal to the pressure on the surface.

- The magnitude of the force per area is called pressure.
- It doesn't have a direction.
- The direction of the force is determined by the pressure on the small surface.
- The pressure at Point A is the same as the pressure at Point B because they are at the same depth.

- The mass of an object does not affect pressure.
- An elephant and a feather feel the same amount of pressure.
- The way they react to this pressure is different.

- The force is pushing on whichever side of the surface it is on.

- We need to put a block in our tank of fluid.
- There is more pressure on the bottom of the block than there is on the top.
- The block is rectangular and the top and bottom have the same area so there is more force pushing up on it.
- The net force on the block is upward because of the pressure on the other four sides.

- The net upward force is referred to as the F buoy.

- The strength of the force on the object is the same as the force on the fluid.

- The V sub is the volume of the fluid displaced when an object is partially or completely submerged.
- The weight of the fluid displaced is determined by the mass of the fluid displaced and the density of the fluid.

- When an object floats, its submerged volume makes the force it feels balance its weight.
- If an object's density is r object and its total volume is V, its weight will be r object Vg.
- The force it feels is fluid.

- The fraction of the object's volume submerged is the same as the ratio of the object's density to the fluid's density.
- If the density of the fluid is 1/3 the density of the object, then the object will float and 1/3 will be submerged.

- The object will sink if it is denser than the fluid.
- Even if the entire object is submerged, its weight is still greater than the force of the water.
- If an object has the same density as the fluid, it will be happy hovering over it.

- Let's go through the steps of an object floating and then an object as it sinks and hits the bottom of a tank.
- An object is floating in a liquid in the diagram on the next page.

- As the object sinks to the bottom of the tank, it has a density greater than the fluid density.

- The forces are always directed in opposite directions because the force of gravity is defined to point downwards.
- The result of the opposing gravity is that any object in a fluid appears to weigh less than when it is not in the fluid, and as the fluid density increases, the apparent weight decreases.
- The normal force of an object is referred to as the apparent weight.

- An object with a mass of 150 kilogram and a volume of 0.75 m 3 floating in ethyl alcohol has a density of 800 kilogram per m 3.

- This means that 1/3 of the object's volume is below the surface of the fluid.
- Make sure you understand what the question is asking.
- Typically you use this equation to find the volume submerged, but this question asks for the volume not to be submerged.

- A brick is dropped into a swimming pool full of water.

- Net force is zero when the brick is on the bottom of the pool.

- The apparent weight of the brick is 15 N because we have F buoy + F N + F g.

- A balloon has a volume of 0.03 m.

- The balloon will feel a force upward and a force downward.

- If you let go of the balloon's string, it will float away.

- Consider a pipe that is moving.
- The volume flow rate is the volume of fluid that passes a point.
- The flow rate is expressed in m 3 /s in SI units.
- To find the volume flow rate, we just need to divide the pipe size by the average speed of the flow at that point.
- The size of the pipe is related to the direction of the flow.

- Don't confuse volume flow rate with flow speed because volume flow rate tells us how much fluid flows per unit time.
- The volume flow rate is calculated in meters per second, while the flow speed is calculated in meters per second.

- The volume flow rate must be the same everywhere along the pipe if the pipe is carrying an incompressible fluid.
- A liquid and a gas are both incompressible.

- The flow speed increases where the pipe narrows and decreases where the pipe widens because Av is a constant.
- The flow speed is proportional to the cross-sectional area or the square of the circle of the pipe.

- A circular pipe carries water.
- The flow speed is 6 m/s at a point in the pipe.

- The second approach is to use proportional reasoning.
- The flow speed is proportional to the cross-sectional area of the pipe.
- If the pipe's radius goes from 2 cm to 1 cm, then A goes from 4 to 2.

- v will increase if A decreases by a factor of 4.
- The flow speed at a point where the pipe's radius is less than 1 cm will be 6 m/s.

- The Bernoulli's Equation is the most important equation in fluid mechanics.
- The conditions that make fluid flow ideal are described first.

- The fluid is stable.

- If the pressure changes are small, this works well for liquids and gases.

- The fluid's density is low.

- Think of Viscosity as internal friction for fluids.
- There's more resistance to a flow of maple syrup than there is to a flow of water.
- When applied to a flow of water, Bernoulli's Equation would give good results, but not if it were applied to a flow of maple syrup.

- The flow is orderly.

- A streamline is a line in the stream in a tube carrying a fluid.
- If we were to inject a drop of dye into a clear glass pipe carrying water, we would see a streak of dye in the pipe.

- The fluid moves smoothly through the tube when the flow is streamlined.

- Bernoulli's Equation can be applied to any pair of points along a streamline if the flow rate is steady.
- Let us know the density of the fluid.
- Point 1 and Point 2 are what we want to compare.
- Let y 1 and y 2 be the heights of the points above the horizontal reference level.

- This equation is very similar to a previous one.

- Bernoulli's Equation is a statement of energy use.
- There are similarities between r gy and mgh as well as between rv 2 and mv 2.

- Imagine punching a small hole in the side of the open tank of liquid.
- Bernoulli's Equation can be used to figure out how fast the liquid will leave the hole.

- Point 1 is at the surface of the liquid and Point 2 is at the hole where the water shoots out.
- The emerging stream at Point 2 is open to the air, so it's at atmospheric pressure as well, since the tank is open.
- The terms P 1 and P 2 are not in Bernoulli's Equation.
- Since the area at Point 1 is larger than the area at Point 2, we can assume that the speed at which the water shoots out of the hole is lower.
- The water level at the top of the tank is constant, and the speed of the water at the top is zero.

- Torricelli's Theorem is what this is called.

- In the figure below, a pump forces water at a constant flow rate through a pipe whose cross-sectional area gradually decreases: at the exit point, A has decreased to 1/3 its value.

- Bernoulli's Equation will be applied to Point 1 and the exit point.
- The level of Point 1 will be the horizontal reference level.
- If the cross-sectional area of the pipe decreases by a factor of 3 between Points 1 and 2, the flow speed must increase by a factor of 3.

- The total pressure at point 1 is P 1.
- The gauge pressure is P tot - P atm.

- Water gushes out through a hole on the side of an above-ground pool.

- The distance from the surface of the pool to the hole is the Torricelli's Theorem.
- If the pool is 2.5 m deep, then the puncture is 2.5 m below the surface of the water.

- The equation above tells us that the speed increases as the cross-sectional area of the pipe decreases.

- The pressure is lower when the flow speed is greater.

- The Bernoulli Effect is illustrated in the figure below.

- Due to the fact that the flow speed at Point 2 is greater than at Point 1, the height of the liquid column above Point 2 is less than the height of the liquid column above Point 1.

- Many everyday phenomena are attributed to the Bernoulli Effect.
- It allows airplanes to fly, curve balls to curve, and tennis balls to hit with top spin to drop quickly.
- Sky divers and motorcycle riders wear jackets that seem to deflate as they move through the air.
- The jacket expands because the air trapped inside is at a higher pressure than the air outside.
- The drop in air pressure that accompanies high winds in a tornado is an example.
- If high winds streak across the roof of a house with windows closed, the outside air pressure can be reduced so much that the air pressure inside the house can be great enough to blow the roof off.

- The upper side of the wing has a longer air flow so it takes more time to reach the edge.
- The top air is moving faster than the bottom if it takes the same time.
- The air on the bottom has more force than the air on the wing.

- Chapter 12 contains solutions.

- A cube is suspended from a scale.
- A beaker of water is lowered into a cube.

- A block of Styrofoam with a density of r S and volume V is pushed under the surface of a liquid with a density of r L.

- A ball is tied to a string and placed in a container of water.
- The water is slowly drained from the container until the water level is below the ball shown but the draining stops when there is still water in the container.
- The string is being measured.

- When an object is submerged in water and weighed, it weighs 100 N less than when it's weighed in air.

- The flow speed at Point X is 6 m/s in the pipe shown above.

- A portion of a conduit for water is shown in the figure.

- The diameter of the hose is 10 times that of the nozzle through which the water leaves.

- The tank is open to the atmosphere and filled with a liquid of density r L. A block of density r B, which is greater than r L, is suspended from a string.
- The top of the block is below the liquid.

- In each of the following, write your answer in the simplest way possible.

- The figure below shows a large, cylindrical tank of water, open to the atmosphere, filled with water to depth D. The tank's radius is R. The point where the emerging stream strikes the level ground is marked by a small circular hole in the side of the tank.

- The water level in the tank can be assumed to be negligible.

- If the stream of water coming from the second hole also lands at Point X, you'll find h in terms of D.

- Your answer should be written in terms of r, R, h, and g.

- The pipe is fitted with a U-tube.
- The fluid of density r F flows at a constant flow rate and with negligible viscosity through the pipe, which causes it to narrow from a cross-sectional area A 1 at Point 1 to a smaller cross-sectional area A 2 at Point 2.
- The upper portion of the U-tube contains the same fluid that's flowing through the pipe, while the lower portion contains a higher density fluid.
- At Point 1 in the pipe, the pressure is P 1 and the flow speed is v 1; at Point 2 in the pipe, the pressure is P 2 and the flow speed is v 2.
- The fluid within the U-tube is not moving.

- Write your answer in terms of P 1, r F, h 1 and g.

- Write your answer in terms of P 2, r F, r V, h 2, d and g.

- The Density is given by r. The pressure is given by P. If r gh is the pressure below the surface of the fluid and P0 is the pressure above it, hydrostatic pressure can be found.
- The upward force of the fluid is known as the buoyant force.
- The density of the fluid and the volume of the fluid displaced are used to calculate the buoyant force.
- The flow rate through a pipe is constant according to the Continuity Equation.
- The idea is that a larger cross-sectional area of pipe will experience fluids traveling at a lower speed.
- Bernoulli's Equation is a statement of how to conserve energy.

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