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Mastering Physics Mechanics 2 - assessed

Course

Physics for Scientists and Engineers (PHYS1001 )

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Question

Learning Goal:

To understand the definition and the meaning of moment of inertia; to be able to calculate the moments of inertia for a group of particles and for a continuous mass distribution with a high degree of symmetry.

By now, you may be familiar with a set of equations describing rotational kinematics. One thing that you may have noticed was the similarity between translational and rotational formulas. Such similarity also exists in dynamics and in the work­ energy domain.

For a particle of mass   moving at a constant speed  , the kinetic energy is given by the formula  . If we

consider instead a rigid object of mass   rotating at a constant angular speed  , the kinetic energy of such an object cannot

be found by using the formula   directly: different parts of the object have different linear speeds. However, they

all have the same angular speed. It would be desirable to obtain a formula for kinetic energy of rotational motion that is similar to the one for translational motion; such a formula would include the term   instead of .

Such a formula can, indeed, be written: for rotational motion of a system of small particles or for a rigid object with continuous mass distribution, the kinetic energy can be written as

.

Here,   is called the moment of inertia of the object (or of the system of particles). It is the quantity representing the inertia with respect to rotational motion.

It can be shown that for a discrete system, say of   particles, the moment of inertia (also known as rotational inertia) is given by

.

In this formula,   is the mass of the ith particle and   is the distance of that particle from the axis of rotation. For a rigid object, consisting of infinitely many particles, the analogue of such summation is integration over the entire object: 

.

In this problem, you will answer several questions that will help you better understand the moment of inertia, its properties, and its applicability. It is recommended that you read the corresponding sections in your textbook before attempting these questions.

Part A

On which of the following does the moment of inertia of an object depend?

Check all that apply.

ANSWER:

m v K= 12 mv 2

m ω

K= 12 mv 2

ω 2 v 2

K= 1 I

2 ω

2

I

n

I=∑

n

i=1mir

2 i

mi ri

I=∫r 2 dm

linear speed

linear acceleration

angular speed

angular acceleration

total mass

shape and density of the object

location of the axis of rotation

Correct

Unlike mass, the moment of inertia depends not only on the amount of matter in an object but also on the distribution of mass in space. The moment of inertia is also dependent on the axis of rotation. The same object, rotating with the same angular speed, may have different kinetic energy depending on the axis of rotation.

Consider the system of two particles, a and b, shown in the figure . Particle a has mass  , and particle b has mass .

Part B

What is the moment of inertia   of particle a? 

ANSWER:

Correct

Part C

Find the moment of inertia   of particle a with respect to the x axis (that is, if the x axis is the axis of rotation), the moment of inertia   of particle a with respect to the y axis, and the moment of inertia   of particle a with respect to the z axis (the axis that passes through the origin perpendicular to both the x and y axes).

Express your answers in terms of   and   separated by commas.

ANSWER:

Correct

m 2 m

I

mr 2

9 mr 2

10 mr 2

undefined: an axis of rotation has not been specified.

Ix

Iy Iz

m r

Ix,  ,   = Iy Iz mr 2 ,9mr 2 ,10mr 2

Correct

Part G

Now, using the results of Part F, find the total kinetic energy   of the system. Remember that both particles rotate about the y axis.

Express your answer in terms of  ,  , and .

ANSWER:

Correct

Not surprisingly, the formulas   and   give the same result. They should, of course, since the rotational kinetic energy of a system of particles is simply the sum of the kinetic energies of the individual particles making up the system.

Ka,   = Kb ,

9 m(ωr) 2

2

2 m(ωr) 2

2

K

mω r

K =  11 m(ωr)

2

2

K= 12 Iω 2 K= 12 mv 2

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Mastering Physics Mechanics 2 - assessed

Course: Physics for Scientists and Engineers (PHYS1001 )

138 Documents
Students shared 138 documents in this course

University: University of Western Australia

Was this document helpful?
09/04/2017 MasteringPhysics: Mechanics 2 - assessed
https://session.masteringphysics.com/myct/itemView?view=print&assignmentProblemID=77274440 1/4
Question7
LearningGoal:
Tounderstandthedefinitionandthemeaningofmomentofinertia;tobeabletocalculatethemomentsofinertiaforagroup
ofparticlesandforacontinuousmassdistributionwithahighdegreeofsymmetry.
Bynow,youmaybefamiliarwithasetofequationsdescribingrotationalkinematics.Onethingthatyoumayhavenoticed
wasthesimilaritybetweentranslationalandrotationalformulas.Suchsimilarityalsoexistsindynamicsandinthework
energydomain.
Foraparticleofmass movingataconstantspeed ,thekineticenergyisgivenbytheformula .Ifwe
considerinsteadarigidobjectofmass rotatingataconstantangularspeed ,thekineticenergyofsuchanobjectcannot
befoundbyusingtheformula directly:differentpartsoftheobjecthavedifferentlinearspeeds.However,they
allhavethesameangularspeed.Itwouldbedesirabletoobtainaformulaforkineticenergyofrotationalmotionthatis
similartotheonefortranslationalmotion;suchaformulawouldincludetheterm insteadof .
Suchaformulacan,indeed,bewritten:forrotationalmotionofasystemofsmallparticlesorforarigidobjectwithcontinuous
massdistribution,thekineticenergycanbewrittenas
.
Here, iscalledthemomentofinertiaoftheobject(orofthesystemofparticles).Itisthequantityrepresentingtheinertia
withrespecttorotationalmotion.
Itcanbeshownthatforadiscretesystem,sayof particles,themomentofinertia(alsoknownasrotationalinertia)isgiven
by
.
Inthisformula, isthemassoftheithparticleand isthedistanceofthatparticlefromtheaxisofrotation.
Forarigidobject,consistingofinfinitelymanyparticles,theanalogueofsuchsummationisintegrationovertheentireobject:
.
Inthisproblem,youwillanswerseveralquestionsthatwillhelpyoubetterunderstandthemomentofinertia,itsproperties,
anditsapplicability.Itisrecommendedthatyoureadthecorrespondingsectionsinyourtextbookbeforeattemptingthese
questions.
PartA
Onwhichofthefollowingdoesthemomentofinertiaofanobjectdepend?
Checkallthatapply.
ANSWER:
m
v
K
=
m
1
2
v
2
m
ω
1
2
v
ω
2
v
2
K
=
I
1
2
ω
2
I
n
I
=
∑
n
i
=1
m
i
r
2
i
m
i
r
i
I
=
∫
dm
r
2
linearspeed
linearacceleration
angularspeed
angularacceleration
totalmass
shapeanddensityoftheobject
locationoftheaxisofrotation

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