Physics 1501: Lecture 30, Pg 1 Physics 1501: Lecture 30 Today’s Agenda l Homework #10 (due Friday Nov. 18) l Midterm 2: Nov. 16 l Honor’s student … l

Physics 1501: Lecture 30, Pg 1

Physics 1501: Lecture 30Physics 1501: Lecture 30TodayToday’’s Agendas Agenda

Homework #10

(due Friday Nov. 18)

Midterm 2: Nov. 16 Honor’s student … Topics

Doppler effectFluids:static conditionsPressurePascal’s Principle (hydraulic lifts etc.)Archimedes’ Principle


Recap & Useful Formulas:Recap & Useful Formulas:

y

x

A

Waves on a string General harmonic waves

tension

mass / length


Speed of SoundSpeed of Sound For a pulse propagating along a string

In general, the speed of sound depends onElastic propertyInertial property

v

tension F

mass per unit length

For ideal gas

CP / CV


Speed of SoundSpeed of Sound For a compressible fluid

Compressibility B plays the role of tension FThe density plays the role of

Bulk modulus

where Y : Young’s modulus

where

For a solid rod

Some numbersAir (0oC) : 331 m/sAir (100oC) : 386 m/sWater (25oC) : 1 490 m/sAluminum : 5 100 m/s


Waves, Wavefronts, and RaysWaves, Wavefronts, and Rays If the power output of a source is constant, the total

power of any wave front is constant. The Intensity at any point depends on the type of wave.


Intensity of sounds Intensity of sounds The amplitude of pressure wave depends on

Frequency of harmonic sound waveSpeed of sound v and density of medium of medium Displacement amplitude s0 of element of medium

Intensity of a sound wave is

Proportional to (amplitude)2

True in general (not only for sound)

Threshold of human hearing: I0 = 10-12 W/m2


Intensity of sounds Intensity of sounds Sound intensity level is given in decibels (dB)

On a log-scale with 0 for threshold of human hearingThreshold of human hearing: I0 = 10-12 W/m2

Some examples:

Sound pressure intensityintensity

amplitude (Pa) (W/m2) level (dB)

Threshold of hearing 3 10-5 10-12 0Classroom 0.01 10-7 50City street 0.3 10-4 80Car without muffler 3 10-2 100Indoor concert 30 1 120Jet engine at 30 m.100 10 130


Doppler effect Doppler effect If the source of sound is moving

Toward the observer seems smallerAway from observer seems larger

If the observer is movingToward the source seems smallerAway from source seems larger

If both are moving

Ex: police car, train, etc.


Fluids : Chapter 15Fluids : Chapter 15

At ordinary temperature, matter exists in one of three states

Solid - has a shape and forms a surfaceLiquid - has no shape but forms a surfaceGas - has no shape and forms no surface

What do we mean by “fluids”?Fluids are “substances that flow”…. “substances that

take the shape of the container”Atoms and molecules are free to move.No long range correlation between positions.


FluidsFluids What parameters do we use to describe fluids?

Density

units :kg/m3 = 10-3 g/cm3

(water) = 1.000 x103 kg/m3 = 1.000 g/cm3

(ice) = 0.917 x103 kg/m3 = 0.917 g/cm3

(air) = 1.29 kg/m3 = 1.29 x10-3 g/cm3

(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3


FluidsFluids

A

n

Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface.

Force (a vector) in a fluid can be expressed in terms of pressure (a scalar) as:

units : 1 N/m2 = 1 Pa (Pascal)1 bar = 105 Pa1 mbar = 102 Pa1 torr = 133.3 Pa

1atm = 1.013 x105 Pa = 1013 mbar = 760 Torr = 14.7 lb/ in2 (=PSI)

What parameters do we use to describe fluids?Pressure


Young’s modulus: measures the resistance of a solid to a change in its length.

Shear modulus: measures the resistance to motion of the planes of a solid sliding past each other.

Bulk modulus: measures the resistance of solids or liquids to changes in their volume.

Some definitionsSome definitions

L L

F

F1

F2V

V - V

F

Elastic properties of solids :

elasticity in length

elasticity of shape (ex. pushing a book)

volume elasticity


FluidsFluids

Bulk Modulus

LIQUID: incompressible (density almost constant) GAS: compressible (density depends a lot on pressure)

Bulk modulus (Pa=N/m2)

H2O SteelGas (STP)Pb


When the pressure is much less than the bulk modulus of the fluid, we treat the density as constant independent of pressure:

incompressible fluid For an incompressible fluid, the

density is the same everywhere, but the pressure is NOT!

Pressure vs. DepthIncompressible Fluids (liquids)

Consider an imaginary fluid volume (a cube, face area A)The sum of all the forces on this volume must be ZERO as

it is in equilibrium: F2 - F1 - mg = 0


If the pressures were different, fluid would flow in the tube!

However, if fluid did flow, then the system was NOT in equilibrium since no equilibrium system will spontaneously leave equilibrium.

Pressure vs. Depth For a fluid in an open container

pressure same at a given depth independent of the container

Fluid level is the same everywhere in a connected container, assuming no surface forces

Why is this so? Why does the pressure below the surface depend only on depth if it is in equilibrium?

Imagine a tube that would connect two regions at the same depth.


Lecture 30, Lecture 30, ACT 1ACT 1PressurePressure

What happens with two fluids?? Consider a U tube containing liquids of density 1 and 2 as shown:

Compare the densities of the liquids:

A) 1 < 2 B) 1 = 2 C) 1 > 2

1

2

dI


PascalPascal’’s Principles Principle

So far we have discovered (using Newton’s Laws):Pressure depends on depth: p = gy

Pascal’s Principle addresses how a change in pressure is transmitted through a fluid.

Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel.

Pascal’s Principle explains the working of hydraulic lifts i.e. the application of a small force at one place can result

in the creation of a large force in another.Does this “hydraulic lever” violate conservation of energy?

» Certainly hope not.. Let’s calculate.



Consider the system shown:A downward force F1 is applied to

the piston of area A1.This force is transmitted through the

liquid to create an upward force F2.Pascal’s Principle says that

increased pressure from F1 (F1/A1) is transmitted throughout the liquid.

F2 > F1 : Have we violated conservation of energy??



Consider F1 moving through a distance d1.

How large is the volume of the liquid displaced?

Therefore the work done by F1 equals the work done by F2

We have NOT obtained “something for nothing”.

This volume determines the displacement of the large piston.


A1 A10

A2 A10

A1 A20

M

M

MdC

dB

dA

A) dA = (1/2)dB B) dA = dB C) dA = 2dB

» If A10 = 2A20, compare dA and dC.

A) dA = (1/2)dC B) dA = dC C) dA = 2dC

Lecture 30, Lecture 30, ACT 2ACT 2HydraulicsHydraulics

Consider the systems shown to the right.In each case, a block of mass M is placed

on the piston of the large cylinder, resulting in a difference dI in the liquid levels.

» If A2 = 2A1, compare dA and dB.


At what depth is the water pressure two atmospheres? It is one atmosphere at the surface. What is the pressure at the bottom of the deepest oceanic trench (about 104 meters)?

P2 = P1 + gd

2.02105 Pa = 1.01105 Pa

+ 103 kg/m3*9.8m/s2*d

d = 10.3 m

P2 = 1.01105 Pa + 103 kg/m3*9.8m/s2*104 m

= 9.81107 Pa = 971 Atm

Solution:

d is the depth.

The pressure increases

one atmosphere for

every 10m.

This assumes that water

is incompressible.

For d = 104 m:

Example Problems (1)Example Problems (1)


Two masses rest on two pistons in a U-shaped tube of water, as shown. If M1 = 3 kg and M2 = 4.5 kg, what is

the height difference, h? The area of piston 1 is A1 = 200

cm2, and the area of piston 2 is A2 = 100 cm2.

Example Problems (2)Example Problems (2)

Solution:

M1

M2h

The pressure difference due to the two masses must be balanced by the water pressure due to h.

Note that g does not appear in the answer.

M1g/A1 + gh = M2g/A2

h = (M2/A2-M1/A1)/

= 0.3 m


Using Fluids to Measure PressureUsing Fluids to Measure Pressure

Barometer

Use Barometer to measure Absolute Pressure

Top of tube evacuated (p=0)

Bottom of tube submerged into pool of mercury open to atmosphere (p=p0)

Pressure dependence on depth:


Using Fluids to Measure PressureUsing Fluids to Measure Pressure

1 atm = 760 mm (29.9 in) Hg = 10.3 m (33.8 ft) H20

p0

h

Manometerp1

Use Manometer to measure Gauge Pressure

Measure pressure of volume (p1) relative to atmospheric

pressure (gauge pressure)

The height difference (h) measures the gauge pressure


ArchimedesArchimedes’’ Principle Principle

Suppose we weigh an object in air (1) and in water (2).How do these weights compare?

W2?W1

W1 < W2 W1 = W2 W1 > W2

Why?

» Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object.


The buoyant force is equal to the difference in the pressures times the area.

W2?W1

Archimedes:

The buoyant force is equal to the weight of the liquid displaced.

The buoyant force determines whether an object will sink or float. How does this work?

ArchimedesArchimedes’’ Principle Principle


Sink or Float?Sink or Float?

The buoyant force is equal to the weight of the liquid that is displaced.

If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink.

F mgB

y

We can calculate how much of a floating object will be submerged in the liquid:

Object is in equilibrium


The Tip of the IcebergThe Tip of the Iceberg

What fraction of an iceberg is submerged?

F mgB

y


Lecture 30, Lecture 30, ACT 3ACT 3BuoyancyBuoyancy

A lead weight is fastened to a large styrofoam block and the combination floats on water with the water level with the top of the styrofoam block as shown.

If you turn the styrofoam+Pb upside down, what happens?

styrofoam

Pb

A) It sinks C)B) D)styrofoam

Pb


ACT 3-AACT 3-AMore Fun With BuoyancyMore Fun With Buoyancy

Two cups are filled to the same level with water. One of the two cups has plastic balls floating in it.

Which cup weighs more?

Cup I Cup II

(A) Cup I (B) Cup II (C) the same (D) can’t tell


ACT 3-BACT 3-BEven More Fun With BuoyancyEven More Fun With Buoyancy

A plastic ball floats in a cup of water with half of its volume submerged. Next some oil (oil < ball < water) is slowly added to the container until it just covers the ball.

Relative to the water level, the ball will: water

oil

(A) move up (B) move down (C) stay in same place

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Physics 1501: Lecture 30, Pg 1 Physics 1501: Lecture 30 Today’s Agenda l Homework #10 (due Friday Nov. 18) l Midterm 2: Nov. 16 l Honor’s student … l