8. rigid body motion

7/24/2019 8. Rigid Body Motion

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Question 8: Rigid Body Motion

Please remember to photocopy 4 pages onto one sheet by going A3A4 and using back to back on the photocopier

Page Commencement date

Questions covered

Introduction

Formulae

The Parallel and Perpendicular Axes Theorems

Introduction to Derivations

Moment of inertia for a rod

Moment of inertia for a square lamina

Moment of inertia for a disc

Moment of inertia for a triangle

Derivations exam questions

Part b! Introduction toPeriodic Time

Periodic Time"xam #uestions

Part b! Introduction toAngular Velocity

Angular Velocity "xam #uestions

Angular $elocit%plusPeriodic Time "xam#uestions

Annulus

Translational "nerg%

&ddballs'uide to ans(ering individual higher level

exam questions )**+ ,++- ...

&ther miscellaneous points

*********** Higher Level Marking Schemes to e !rovided se!arately *************

Questions to make you thinkA /spinning0 cric1et ball often moves faster after pitching on a smooth (ic1et2 3ave %ou noticed that. 4hat do %outhin1 is the reason.

,


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The follo(ing information is on page -5 of the log tables and (e also need to prove some of theserelationships2Relevant "ormulae:

Moments o" #nertia

Point mass md)

6niform rod7 length )l About axis through centre perpendicular to rod ,85ml)

About axis at one end perpendicular to rod 985ml)

6niform lamina7 length )l About axis through centre in the plane of the lamina ,85ml)

About axis along one end in the plane of the lamina 985ml)

6niform disc7 radius r About axis through centre perpendicular to disc,

8)mr)

About a diameter ,89mr)

6niform hoop7 radius r About axis through centre perpendicular to hoop mr)

About a diameter ,8)mr)

6niform solid sphere7 radius r About a diameter )8-mr)

Parallel Axis Theorem I : Ic ; md)

Perpendicular Axis Theorem I< : Ix ; I%

The centre of gravit% of a triangular lamina is on a line on a bisector and is one third up from the base2

Com!ound $%ects

It =ust happens to be the case /fortunatel%0 that the moment of inertia of a s%stem made up of severalsimple parts about a particular axis is simpl% the sum of the individual moments of inertia /about that

axis02

>o for example to calculate the moment of inertia of the s%stem on the right about pointA7 simpl%add together the moment of inertia of the rod about A andthe moment of inertia of the disc aboutA2

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dm

axis

r

# & m r'

Background

If an ob=ect is moving from one place to another the associated 1inetic energ% is 1no(n as translationalenerg% /"?0 and is represented b% the term @ mv)

If instead the ob=ect is rotating about a fixed axis then the associated 1inetic energ% is 1no(n as rotational1inetic energ% /"0 and is represented b% the term @ I )

(o derive the e)!ression R& + #'

>tart (ith the usual expression for translational 1ineticenerg%! "1 : @ mv)

But the 1inetic energ% of the ob=ect is simpl% the sum of the1inetic energ% of all the small parts2

4e represent this as follo(s!

"1 : @ dm v)

5


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Cdm /short for delta mE0 means a ver% small mass mE and /short for sigmaE0 means the sum ofE

But v : r "1 : @ dm r))

It turns out that the term dm r2represents ho( difficult the ob=ect is to rotate about an axis andbecause it is of such significance it is given its o(n name the moment of inertia2

I : moment of inertia and represents ho( difficult it is to turn the ob=ect2

k & + # '

The mass mof an ob=ect is a measure of ho( difficult it is to acceleratethat ob=ect2>imilarl% the moment of InertiaIof an ob=ect is a measure of ho( difficult it is to rotatethat ob=ect2

Got surprisingl%7 the t(o factors that the moment of inertia of an ob=ect depend on are,2 the mass of the ob=ect and)2 the distance bet(een the ob=ect and the point or axis of rotation2

In this chapter (e focus on ho( difficult it is to rotate a variet% of shapes2 Most of them are t(oHdimensional7 %et still have a mass2 It ma% stri1e %ou as unusual that a )HD shape can have a mass7 but there%ou go2 In Applied Maths (e li1e to first simplif% things as much as (e can and onl% after(ards do (e(orr% about the real (orld2

eminds me of the stor% of the ph%sicist (ho believed he could solve all the (orlds economic problems24hen as1ed ho( he (ould go about this he replied! "as%2 First7 imagine all the (orlds economic problemsto be a perfect circle 2 2 2E2

There are actuall% a lot of realH(orld applications to figuring out ho( difficult it is to rotate an ob=ect2 Forexample if %ou are designing a door for an aircraft %ou need to ma1e sure that it can be opened b% a t%pical

person2 >o %ou could either ta1e the "ngineering approach7 (hich is to first build the door and then test it7 or%ou could tr% to mathematicall% (or1 out in advance ho( difficult a given design (ould be to open and thencompare it (ith 1no(n values of (hat the average human can deal (ith2

Ho, to go "rom -.. & + mv'to R.. & + #'

,++ /b0 ,+J- /b0A lamina is rotating (ith angular velocit% K about an axis perpendicular to its plane2If the moment of inertia of the lamina about the axis is I7 prove that the 1inetic energ% is @IK)2

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L

N

d

Ic

(he Parallel and Per!endicular /)es (heorems

Per!endicular /)is (heorem:

IfX7 YandZare mutuall% perpendicular axes through a point in a lamina7 (ithXand Yl%ing in the plane of thelamina7 then I< : Ix ; I%7 (here these represents the moments of inertia about the respective axes2

Diagram!

Parallel /)is (heorem:

If Ic is the moment of inertia of a rigid bod% of mass mabout an axis through its centre of gravit%7 I is the moment ofinertia of the bod% about a parallel axis and d is the distance bet(een these t(o axes7 thenI : Ic ; md)2

Diagram!

0B:Gote the Ic part of this you must use the moment o" inertia through an a)ispassing through the centre of gravityo" the o%ect1no matter ho( tempting it is to use another axis /%oull still ma1e this mista1e but at least no( I can sa%I told %ou so20

-

(he Parallel /)is (heorem mustinclude an a)is

!assing through the centre o" gravity o" the o%ect


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2ind the moment o" #nertia o" each o" the "ollo,ing o%ects1 aout the !oint s!eci"ied

(i!s

Oabel each rotation as one of the follo(ing t(o options2,2 A spin/the axis is perpendicular to the plane7 li1e a nail into the page02

3ere the ob=ect spins in the plane of the page2

)2 Aflip/because the axis is along the planeE li1e a ruler l%ing on the page023ere the ob=ect flips out of the page2

Rod

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A uniform rod of mass 5m and length ,2) metres can turn freel% in a vertical planeabout a hori


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isc

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Prove that the moments of inertia of a uniform circular disc of radius *R9 m and mass - 1g about an axis oqthrough its centre oand perpendicular to the disc is *R9 1g m)2

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Find the moment of inertia of a uniform circular disc of radius rand mass m(hich can move freel% about asmooth pivot at a point aon its circumference2

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Find the moment of inertia of a circular sheet of cardboard of radius rand mass m(hichrotates freel% in its o(n plane7 (hich is vertical7 about a hori


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Com!ound $%ects

The moment of inertia of a s%stem made up of several simple parts about a particular axis is simpl%the sum of the individual moments of inertia /about that axis02

>o for example to calculate the moment of inertia of the s%stem on the right about pointA7 simpl%add together the moment of inertia of the rod about A andthe moment of inertia of the disc aboutA2

Rod and Point Mass

78 5c6

Find the moment of inertia of a uniform rod Sa"7 of mass mand length )l7 (hich is freeto rotate in a vertical plane about a fixed hori


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Rod and isc

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A uniform rod of length Qlis attached to the rim of a uniform disc of diameter )l2The rod is collinear (ith a diameter of the disc /see diagram02The disc and the rod are both of mass m2Ualculate the moment of inertia of the compound bod% about a perpendicular axis through the end A2

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A uniform rod of mass mand length cm has a uniform disc of mass mand radius ,) cmattached to one end2The rod and disc are in the same plane and the rod is collinear (ith a diameter of the disc/see diagram02Find the moment of inertia of a the compound bod% (hich is set in motion about an axisthrough q(hich is perpendicular to the plane of the rod and disc7

7=

A pendulum consists of a rod pq of mass m and length 5r attached to the rim of a discof mass )m and radius r7 as sho(n2Find the moment of inertia of the compound bod% (hich is set in motion about an axisthrough p7 (hich is perpendicular to the plane of the rod and the disc2

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A pendulum of a cloc1 consists of a thin uniform rod a"of mass M and length Qlto (hich is rigidl% attacheda uniform circular disc of mass 9M and radius l(ith the centre of the disc being at the point con a"(here"c: l26sing the parallel axis theorem for the disc7 sho( that the moment of inertia of the pendulum about an axisat aperpendicular to the plane of the disc is ,,9 Ml)2

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/nnulus

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An annulus is created (hen a central hole of radius " is removed from a uniform circular disc of radius a2The mass of the annulus /shaded area0 is$2>ho( that the moment of inertia of the annulus about an axis through its centre and

perpendicular to its plane isM(a2+b2)

2

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The diagram sho(s a t(oHdimensional (heel /shaded area0 and four spo1es arrangedinside the (heel as sho(n2The inner and outer radii of the (heel are Qa and a7 respectivel%2"ach spo1e is of mass m and length Qa2The total mass of the (heel and four spo1es is ,m2

/i0 >ho( that the mass per unit area of the (heel /shaded area0 is ))

m

a 2/ii0 >ho( that the total moment of inertia of the (heel and four spo1es about an axis through the centre and

perpendicular to the plane of the (heel is J9ma)2

78;5a6

Prove that the moment of inertia of a uniform annulus of internal diameterp7 external diameter 5pand mass9m7 about an axis through its centre perpendicular to its plane is -mp)2 /see tables 0

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Part >?s

The part bs can be subHdivided into t(o t%pes of problem,2 Find periodic time T)2 Find angular velocit%

4hen loo1ing over the questions begin b% identif%ing the questions as either T%pe , or T%pe )2

Periodic (ime

The periodic time is the time a pendulum (ould ta1e for one complete oscillation /over and bac1 again02

A simple pendulum consists of a point mass hanging from a ver% light string2A compound pendulum consists of a t(oHdimensional ob=ect (hich ma% or ma% not be composed of morethan one part2

2ormula "or !eriodic time

Sim!le Pendulum Com!ound Pendulum

# total moment of inertiaM Mass of the system /a ver% common mista1e is for students to leave this as m e2g2 if the mass of the

s%stem is 5m then substitute 5m for M in the formula above20

h (he distance "rom the centre of gravity of the system to the axis.

The best (a% to find this distance is to put the s%stem on its side /(ith the pivot on the left handside0 and use the fact that the sum of the individual moments equals the moment of the entireob=ect2

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Periodic (ime: )am Questions

5i6 @ust "ind !eriodic time

/ good clucking rilliant idea is to %ust "ind h "or all the o%ects in the ne)t "e, !ages1 then come ack

here and do each Auestion out "rom scratch.

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A uniform rod of length Qlis attached to the rim of a uniform disc of diameter )l2The rod is collinear (ith a diameter of the disc /see diagram02The disc and the rod are both of mass m2Ualculate the moment of inertia of the compound bod% about a perpendicular axis through the end A2

g

l

)*

,)5)

If the compound bod% ma1es small oscillations in a vertical plane about a hori


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5ii6 2ind the !eriodic time o" the eAuivalent sim!le !endulum

Uompare the equation to the equation for a simple pendulum and solve2

Cbegin b% cancelling the )W and the square root

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A uniform rod of mass mand length cm has a uniform disc of mass mand radius ,) cmattached to one end2The rod and disc are in the same plane and the rod is collinear (ith a diameter of the disc/see diagram02If the compound bod% is set in motion about an axis through q(hich is perpendicular to the

plane of the rod and disc7

/i0 find the period of small oscillations correct to t(o decimal places2/ii0 find the length of the equivalent simple pendulum2

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A uniform rod Sab of length )pand of mass 5mhas a mass mattached to it at adistanceyfrom a2/i0 Prove that the moment of inertia of this s%stem about a smooth hori


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A uniform circular lamina7 of mass m and radius r7 can turn freel% about a hori


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5iii6 2ind the length o" the com!ound !endulum that corres!onds to the minimum !eriodic time

,2 >quare T)2 Ualculate dT)8dx /%ou (ill need to use the quotient rule0252 Oet this expression equal * and solve to findx2

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A circular sheet of cardboard of radius rrotates freel% in its o(n plane7 (hich is vertical7about a hori


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A uniform square lamina of side )ais freel% pivoted at a point in one diagonal andoscillates in its o(n plane2Prove that (hen the period of small oscillations is a minimum the distance of the pivotfrom the centre isx(here 5x): )a)2

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A uniform rod of mass mis free to rotate in a vertical plane about an axis (hich is perpendicular to the rodand *25) m from its centre of gravit%2For small oscillations the rod has the same period as a simple pendulum of length *2- m2/i0 Find the length of the rod2/ii0 For (hat other distance bet(een the axis and the centre of gravit% (ill the period be the same./iii0 4here must the axis be located to give a minimum period.

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A thin uniform rod AB of mass m7 and length )acan turn freel% in a vertical plane7

about a fixed horiho( that the moment of inertia of the s%stem about the axis is )m/a); ,)x)0/ii0 The s%stem ma1es small oscillations2

Find the period and sho( that the period is a minimum (henx: a892

78ho( that its moment of inertia about "cis Qm'/ii0 Prove that the moment of inertia of a"cabout an axis through aperpendicular to the

l f " i )9

8. rigid body motion

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