9-1 CHAPTER 9 Mass and Mass-Related Parameters © 2011 Cengage Learning Engineering. All Rights Reserved

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CHAPTER 9Mass and Mass-Related Parameters

© 2011 Cengage Learning Engineering. All Rights Reserved.

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Outline

In this chapter we will

• introduce the concept of mass in terms of a quantitative measure of the amount of atoms and molecules possessed by a substance

• define mass-related engineering quantities density, specific gravity, mass moment of

inertia, momentum, mass flow rate

• discuss conservation of mass


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Why Is the Concept of Mass Important?

• It provides a quantitative measure of the amount of atoms and molecules possessed by a substance

• It provides a measure of resistance to translation motion

• It is important in determining the momentum of moving objects

• It is important in describing material properties such as density and specific gravity


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What Is Mass?

• Provides a quantitative measure of how many molecules or atoms are in a given object

• A physical variable that provides a measure of how light or heavy things are


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Mass – A Physical Variable

• All objects and living things are made of matters

• Matter is made of atoms or chemical elements

• Atoms are made of smaller particles called electrons, protons, and neutrons

• Atoms are the basic building block of all matter

• Matter can exist in 4 states, depending on its own and surrounding conditions – solid, liquid, gaseous, or plasma


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• heavy

• light

• massive

• tiny

In the earlier days, we did not fully understand the concept of gravity. Hence, there was no clear distinction between mass and weight

How Do We Describe Mass?

descriptive but not exact


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How Do We Describe Mass?


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How do we describe mass?

• The gravitational pull on the moon is about 1/6th of that on the earth.

• If your mass on the earth is 100 kg, what is your mass on the moon?


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Mass

Base SI Unit – kilogram (kg)

kilogram (kg) – a unit of mass in SI; it is equal to the mass of the international prototype of the kilogram

Mass provides a quantitative measure of how many molecules or atoms are in a given object


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Other Mass Units

• pound mass (lbm)

• slug (which system of units is this?)


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Measurement of Mass

• In practice, mass of an object is measured indirectly through its weight

• Weight of an object on earth is the force that is exerted on the mass due to gravitational pull of the earth


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• Density – ratio of the mass of an object to the volume that it occupies

• Specific volume – inverse of density

Mass-Related Parameters

volume

massdensity

mass

volumevolumespecific


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• Specific gravity – compares the density of a material to the density of water

• Specific weight – another way to measure how truly heavy or light a material is for a given volume

Mass-Related Parameters

4@waterofdensity

materialaofdensitygravityspecific

gravity todueon acceleratidensity volume

weightweightspecific


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Density, Specific Gravity, and Specific Weight of Some Solid Materials

Values shown are at room temperature or at the specified temperature


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Density, Specific Gravity, and Specific Weight of Some Fluids

Values shown are at room temperature or at the specified temperature


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Mass Flow Rate

• Mass flow rate tells engineers how much material is being used or moved over a period of time so that they can replenish the supply of material

• Engineers use flowmeters to measure volume or mass flow rate


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Mass Flow Rate

• Mass flow rate – amount of mass that flows through something per unit of time

• Units: kg/s, kg/min, kg/h, slugs/s, lbm/s

time

mass rate flow mass


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Mass Flow Rate and Volume Flow Rate

RATEFLOW VOLUMEdensity RATEFLOW MASS

time

volumedensity

time

volumedensity

time

mass rate flow mass


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Mass Moment of Inertia

• Mass moment of inertia provides a measure of how hard it is to rotate something with respect to a center of rotation

The further away the mass is located from the center of rotation, the harder it is to rotate the mass about the given center of rotation


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For a single particle m, located at a distance r from the axis of rotation z-z, the mass moment of inertia is defined by

mrI zz2


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dmrI 2zz

For a system of 3 mass particles shown, the mass moment of inertia about the z-z axis is

32

322

212

1 mrmrmrI zz

In general the mass moment of inertia about the z-z axis of a continuous body is determined from


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Mass Moment of Inertia of Some Typical Objects

disk

2

2

1mRI zz

2

2

1mRI zz

circular cylinder sphere

2

5

2mRI zz

2

12

1mWI zz

thin rectangular plate


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Example 9.1 – Mass Moment of Inertia

kg5.123m 01571.0m

kg7860volumedensity mass

volume

mass density

m 01571.0m 2m 1.04

length4

volume

33

322

d

Given: a 2-m long steel shaft with a diameter (d) of 10 cm; density of steel = 7860 kg/m3

Find: mass moment of inertia

Solution:

Moment of inertia of the shaft about the longitudinal axis becomes,

222 mkg 154.0m 05.0kg 5.1232

1

2

1 mRI zz


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Momentum


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• Momentum – a physical variable that is defined as the product of mass and velocity

• Momentum’s direction is the same as the direction of the velocity vector or the direction of the moving object

Momentum

VmL


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Momentum

How can a small pebble break my wind shield?


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Example 9.2 – Linear Momentum

Given: a person with a mass of 80 kg and running at a speed of 3 m/s; a car with a mass of 2000 kg and moving at a speed of 30 m/s in the same direction as the person

Find: the linear momentum of the person and the car

Solution:

m/skg 000,60m/s 30kg 2000 :car

m/skg 240m/s 3kg 80 :person

VmL

VmL


Momentum

Engineering Fundamentals, By Saeed Moaveni, Fourth Edition, Copyrighted 2011 9-28

Momentum


Momentum


Momentum


The rate at which a fluid enters a control volume minus the rate at which the fluid leaves the control volume should be equal to the rate of accumulation or depletion of the mass of fluid within the given control volume

= the tub

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Conservation of Mass – Keeping Track of Mass


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• The concept of conservation of mass can also be applied to a concept called queuing

a broad area in mathematics, operation research, engineering management, and traffic management

a study of • people waiting in service lines• products waiting in assembly lines• digital information waiting to move through

computer networks


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outlet. Tank B has an outlet that discharges the water at 1 kg/s. Assume density of water is 1000 kg/m3.

Find: the amount of water stored in each tank after 5 minutes; how long will it take to fill the tanks completely if the volume of each tank is 12 m3?

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Example 9.3 – Conservation of Mass

Given: two identical size tanks A and B are filled with water at a rate of 2 kg/s. Tank A has no


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(rate at which fluid enters a control volume)

-(rate at which fluid leaves the control volume)

=

(rate of accumulation or depletion of the mass of fluid within the given control volume)

Solution:

Using the conservation of mass,

Tank A: no water leaves the tank

kg 600min 1

s 60min 5kg/s 2 volumecontrol theinside mass of change

thenminutes 5after

kg/s 2in time change

volumecontrol theinside mass of change0kg/s 2


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Solution (continued):

Time to fill tank A:

min 100 s 6000kg/s 2

kg 000,12

rate flow mass

mass tank thefill totime

kg 000,12m 12kg/m 1000

volumedensity tank theinside water theof mass33


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kg 300min 1

s 60min 5kg/s 1 volumecontrol theinside mass of change

thenminutes 5after

kg/s 1in time change

volumecontrol theinside mass of changekg/s 1kg/s 2


Tank B: water leaves the tank at the rate of 1 kg/s:

Time to fill tank B:

min 200 s 12000kg/s 1

kg 000,12

rate flow mass

mass tank thefill totime


Given: The gasoline consumption of a car is 15 km/liter when the car is moving at a speed of 90 km/h. The specific gravity of gasoline is 0.72.

Find: The amount of gasoline, in kg, burned every hour if there were one million of these cars on the road?

Solution:

First, we will use Equation (9.2) to compute the density of gasoline.

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Example 9.4 – Mass Flow Rate

33gasoline kg/m 720 gasoline ofdensity

kg/m 1000

gasoline ofdensity 72.0SG


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Example 9.4 – Mass Flow Rate

liter/h 6km/liter 15

km/h 90 rate flow-volume


The volume-flow rate of fluid for a single car is determined from:

Next, we will use Equation (9.6) to calculate the mass-flow rate of the fuel per car

kg/h 32.4liters 1000

m 1

h

liter 6

m

kg 720 rate flow-mass

rate flow volumedensity rate flow-mass3

3

For one million cars 4,320,000 kg of gasoline is burned each hour!


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Summary

• You should have a good understanding of mass and its important role in engineering applications and analysis.

• You should know that in engineering to show how light or heavy materials are, we use properties such as density, specific volume, specific weight, and specific gravity.

• You should also know the definition of these properties.


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Summary

• You should have a good understanding of mass flow rate and its relationship to volume flow rate.

• You should realize that mass provides a measure of resistance to translational motion.

• You should also know the role the distribution of mass plays in rotational motion in terms of mass moment of inertia


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Summary

• You should know how we define momentum for a moving object.

• You should know how to use the conservation of mass to keep track of mass entering, leaving, accumulating, or depleting for a control volume


Documents

9-1 CHAPTER 9 Mass and Mass-Related Parameters © 2011 Cengage Learning Engineering. All Rights Reserved