Lecture VIII General Physics (PHY 2130) Solids and fluids  density and pressure  buoyant force  Archimedes’ principle  Fluids in motion

Lightning Review Last lecture: 1.Rotational dynamics torque and angular momentum two equillibrium conditions Review Problem: A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational inertia and her angular speed increases so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be 1. the same. 2. larger because she’s rotating faster. 3. smaller because her rotational inertia is smaller.

Example: Given: Moments of inertia: I 1 and I 2 Find: K 2 =? Rotational kinetic energy is We know that (a) angular momentum L is conserved and (b) angular velocity increases Thus, rotational kinetic energy must increase!

Solids and Fluids

Question What is a fluid? 1. A liquid 2. A gas 3. Anything that flows 4. Anything that can be made to change shape.

States of matter: Phase Transitions ICE WATER STEAM Add heat These are three states of matter (plasma is another one)

States of Matter ► Solid ► Liquid ► Gas ► Plasma

States of Matter ► Solid ► Liquid ► Gas ► Plasma  Has definite volume  Has definite shape  Molecules are held in specific location by electrical forces and vibrate about equilibrium positions  Can be modeled as springs connecting molecules

States of Matter ► Solid ► Liquid ► Gas ► Plasma CCCCrystalline solid Atoms have an ordered structure Example is salt (red spheres are Na+ ions, blue spheres represent Cl- ions) AAAAmorphous Solid Atoms are arranged randomly Examples include glass

States of Matter ► Solid ► Liquid ► Gas ► Plasma  Has a definite volume  No definite shape  Exist at a higher temperature than solids  The molecules “wander” through the liquid in a random fashion The intermolecular forces are not strong enough to keep the molecules in a fixed position The intermolecular forces are not strong enough to keep the molecules in a fixed position

States of Matter ► Solid ► Liquid ► Gas ► Plasma  Has no definite volume  Has no definite shape  Molecules are in constant random motion  The molecules exert only weak forces on each other  Average distance between molecules is large compared to the size of the molecules

States of Matter ► Solid ► Liquid ► Gas ► Plasma  Matter heated to a very high temperature  Many of the electrons are freed from the nucleus  Result is a collection of free, electrically charged ions  Plasmas exist inside stars or experimental reactors or fluorescent light bulbs! For more information:

Is there a concept that helps to distinguish between those states of matter?

Density ►T►T►T►The density of a substance of uniform composition is defined as its mass per unit volume: some examples: ►T►T►T►The densities of most liquids and solids vary slightly with changes in temperature and pressure ►D►D►D►Densities of gases vary greatly with changes in temperature and pressure (and generally 1000 smaller)UnitsSI kg/m 3 CGS g/cm 3 ( 1 g/cm 3 =1000 kg/m 3 )

Sometimes: Specific Gravity ► The specific gravity of a substance is the ratio of its density to the density of water at 4° C  The density of water at 4° C is 1000 kg/m 3 ► Specific gravity is a unitless ratio

Pressure ► Pressure of fluid is the ratio of the force exerted by a fluid on a submerged object to area UnitsSI Pascal (Pa=N/m 2 ) Example: 100 N over 1 m 2 is P=(100 N)/(1 m 2 )=100 N/m 2 =100 Pa.

Pressure and Depth ► If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium ► All points at the same depth must be at the same pressure (otherwise, the fluid would not be in equilibrium) ► Three external forces act on the region of a cross-sectional area A External forces: atmospheric, weight, normal

ConcepTest 1 You are measuring the pressure at the depth of 10 cm in three different containers. Rank the values of pressure from the greatest to the smallest: It’s the same in all three Please fill your answer as question 1 of General Purpose Answer Sheet 10 cm 1 23

Pressure and Depth equation ► P o is normal atmospheric pressure  x 10 5 Pa = 14.7 lb/in 2 ► The pressure does not depend upon the shape of the container  Other units of pressure: 76.0 cm of mercury One atmosphere 1 atm =1.013 x 10 5 Pa 14.7 lb/in 2

Example: Given: masses: h=100 m Find: P = ? Find pressure at 100 m below ocean surface.

Pascal’s Principle ► A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. ► The hydraulic press is an important application of Pascal’s Principle ► Also used in hydraulic brakes, forklifts, car lifts, etc. Since A 2 >A 1, then F 2 >F 1 !!!

Measuring Pressure ► The spring is calibrated by a known force ► The force the fluid exerts on the piston is then measured One end of the U-shaped tube is open to the atmosphere One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured The other end is connected to the pressure to be measured Pressure at B is P o +ρgh Pressure at B is P o +ρgh A long closed tube is filled with mercury and inverted in a dish of mercury A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρgh Measures atmospheric pressure as ρgh

How would you measure blood pressure? Has to be: (a) accurate (b) non-invasive (c) simple sphygmomanometer

Question Suppose that you placed an extended object in the water. How does the pressure at the top of this object relate to the pressure at the bottom? 1. It’s the same. 2. The pressure is greater at the top. 3. The pressure is greater at the bottom. 4. Whatever…

Buoyant Force This force is called the buoyant force. This force is called the buoyant force. What is the magnitude of that force? What is the magnitude of that force? P1AP1A P2AP2A mg

Buoyant Force ► The magnitude of the buoyant force always equals the weight of the displaced fluid ► The buoyant force is the same for a totally submerged object of any size, shape, or density ► The buoyant force is exerted by the fluid ► Whether an object sinks or floats depends on the relationship between the buoyant force and the weight

Archimedes' Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. buoyant force. This force is buoyant force. Physical cause: pressure difference between the top and the bottom of the object

Archimedes’ Principle: Totally Submerged Object ► The upward buoyant force is B=ρ fluid gV obj ► The downward gravitational force is w=mg=ρ obj gV obj ► The net force is B-w=(ρ fluid -ρ obj )gV obj Depending on the direction of the net force, the object will either float up or sink!

► The object is less dense than the fluid ρ fluid <ρ obj ► The object experiences a net upward force The net force is B-w=(ρ fluid -ρ obj )gV obj The object is more dense than the fluid ρ fluid >ρ obj The object is more dense than the fluid ρ fluid >ρ obj The net force is downward, so the object accelerates downward The net force is downward, so the object accelerates downward

Archimedes’ Principle: Floating Object ► The object is in static equilibrium ► The upward buoyant force is balanced by the downward force of gravity ► Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level

Question 1 Suppose that you have a steel bar. Will it float on water? Why?

Question 2 Suppose that you have a steel bar. Will it float on water? Why? How come that ships (which are made of steel) can float?

Question 3 Suppose that your friend gave you a necklace (crown, piece of yellow metal, …). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)?

Question 3 Suppose that your friend gave you a necklace (crown, piece of yellow metal, …). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)? Idea: determine density! Let’s weight the object in and outside the water container: If  is not the same as  gold, your friend is lying… Out:In: Out: In:

ConcepTest 2 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? 1. The glass without ice cubes. 2. The glass with ice cubes. 3. The two weigh the same. Please fill your answer as question 3 of General Purpose Answer Sheet

ConcepTest 3 Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.When the ice cubes melt, in which glass is the level of the water higher? 1. The glass without ice cubes. 2. The glass with ice cubes. 3. It is the same in both. Please fill your answer as question 5 of General Purpose Answer Sheet

Fluids in Motion: Streamline Flow ► Streamline flow  every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier  also called laminar flow ► Streamline is the path  different streamlines cannot cross each other  the streamline at any point coincides with the direction of fluid velocity at that point

Fluids in Motion: Turbulent Flow ► The flow becomes irregular  exceeds a certain velocity  any condition that causes abrupt changes in velocity ► Eddy currents are a characteristic of turbulent flow

Fluid Flow: Viscosity ► Viscosity is the degree of internal friction in the fluid ► The internal friction is associated with the resistance between two adjacent layers of the fluid moving relative to each other

Characteristics of an Ideal Fluid ► The fluid is nonviscous  There is no internal friction between adjacent layers ► The fluid is incompressible  Its density is constant ► The fluid is steady  Its velocity, density and pressure do not change in time ► The fluid moves without turbulence  No eddy currents are present

Equation of Continuity ► A 1 v 1 = A 2 v 2 ► The product of the cross- sectional area of a pipe and the fluid speed is a constant  Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter ► Av is called the flow rate

Bernoulli’s Equation ► Relates pressure to fluid speed and elevation ► Bernoulli’s equation is a consequence of Conservation of Energy applied to an ideal fluid ► Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner ► States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

How to measure the speed of the fluid flow: Venturi Meter ► Shows fluid flowing through a horizontal constricted pipe ► Speed changes as diameter changes ► Swiftly moving fluids exert less pressure than do slowly moving fluids