Rotational Motion

Rotational MotionRotational MotionNCEA AS 3.4NCEA AS 3.4

Text Chapter: 9Text Chapter: 9

Types of MotionTypes of Motion

►Pure Translation –force acts through the Pure Translation –force acts through the centre of mass, C.o.m moves.centre of mass, C.o.m moves.

►Pure Rotation –2 equal & opposite Pure Rotation –2 equal & opposite forces act at a perpendicular distance forces act at a perpendicular distance from the c.o.m (force couple) C.o.m from the c.o.m (force couple) C.o.m remains stationary, object spins around remains stationary, object spins around itit

►Mixture – single force acts, NOT Mixture – single force acts, NOT through c.o.m, object moves and through c.o.m, object moves and rotates around c.o.mrotates around c.o.m

What do you understand about What do you understand about radians?radians?

Angular DisplacementAngular Displacement

► Although both Although both points A & B have points A & B have turned through the turned through the same angle, A has same angle, A has travelled a greater travelled a greater distance than B distance than B

► A must have had A must have had the greater linear the greater linear speed speed

B

A

A

B


► Symbol Symbol ► Measured in Measured in radians radians

(rad)(rad)► Angular displacement Angular displacement

is related to linear is related to linear distance by:distance by:

r

r

s

r

s


► Remember from Remember from Maths:Maths: How to put your How to put your

calculator into radian calculator into radian mode?mode?

How many radians How many radians are in a full circle?are in a full circle?

r

rs 2

Angular VelocityAngular Velocity

►Symbol Symbol omega) w (double u)omega) w (double u)►Measured in radians per second (radsMeasured in radians per second (rads-1-1.).)►Average angular velocity calculated by:Average angular velocity calculated by:

takentime

ntdisplacemeangular locityangular ve average

t

Angular VelocityAngular Velocity

► To put it another To put it another way:way:

► So angular velocity So angular velocity is related to linear is related to linear velocity by:velocity by:

r

r

s

) (since

) since(

t

sv

r

vr

s

tr

st

rv

Angular AccelerationAngular Acceleration►Changing angular velocityChanging angular velocity►Symbol: Symbol: ►Measured in radians per second Measured in radians per second

squared (radssquared (rads-2-2.).)►Calculated by:Calculated by:

takentime

locityangular vein changeonacceleratiangular

t

Angular AccelerationAngular Acceleration

►Angular acceleration and linear Angular acceleration and linear acceleration are linked by:acceleration are linked by: ra

Try TheseTry These

► An old record player rotates at 33 rpm. An old record player rotates at 33 rpm. ► What is the displacement per min?What is the displacement per min?► What distance would a fly sitting 10cm fromWhat distance would a fly sitting 10cm from

the centre travel in a minute?the centre travel in a minute?► How fast is the earth travelling through How fast is the earth travelling through

spacespaceif it is 0.15Tm from the sunif it is 0.15Tm from the sun

►33x2x33x2x=207rads=207rads-1-1

►x 0.1 = 21mx 0.1 = 21m►How fast is the earth travelling How fast is the earth travelling

through spacethrough spaceif it is 0.15Tm from the sunif it is 0.15Tm from the sun

r

s

SummarySummary

Translational Rotational Equation

dd d=rd=r

vv v=rv=r

aa a=ra=r

GraphsGraphs

Angular Displacement vs Time

0 10 20 30 40

Time

Ang

ular

Dis

plac

emen

tGradien

t = angular velocity

GraphsGraphsGradient = angular accelerati

on Angular Velocity vs Time

0 1 2 3 4

Time

Ang

ular

Vel

ocit

yArea under

graph = angular

displacement

Kinematic EquationsKinematic Equations

►Recognise Recognise these??:these??:

►Use them the Use them the same way you did same way you did last year.last year.

t

tt

t

if

i

if

if

2

22

21

22

The CDThe CD

► A CD reads from the inside to outsideA CD reads from the inside to outside► They used to read 4.3 megabits per second They used to read 4.3 megabits per second

per channel (probably much higher now)per channel (probably much higher now)► They require a constant linear velocity of They require a constant linear velocity of

1.40ms1.40ms-1-1

► The disc needs to rotate at 500rpm at the The disc needs to rotate at 500rpm at the start and 200rpm at the finish.start and 200rpm at the finish.

1) Convert 500rpm into rads1) Convert 500rpm into rads-1-1 14.52

60

2500 radsx

2) A CD can reach correct 2) A CD can reach correct in 1 in 1 revolution revolution what is what is ??

2

22

1

218

2

214.520?

radsgettosolveandindatatheput

radrevrads

of

fo

3) What is the radius of the disc at the start?3) What is the radius of the disc at the start?

2.7cm2.7cm

4) Convert 200rpm to rads4) Convert 200rpm to rads-1-1

A particular CD has a 72 minute playing timeA particular CD has a 72 minute playing time

5) Calculate 5) Calculate as the disc plays from start to finish. as the disc plays from start to finish.

6) Calculate the angle the disc moves through during 6) Calculate the angle the disc moves through during this time this time

7) Convert this to revolutions7) Convert this to revolutions

8) Calculate the radius of the disc at the finish. 8) Calculate the radius of the disc at the finish.

5) -7.3x105) -7.3x10-3-3radsrads-2-2

6) 158000 approx radians6) 158000 approx radians

7) 25000 rev7) 25000 rev

8) 6.7cm8) 6.7cm

21rads21rads-1-1

Answer this by yourselfAnswer this by yourself

Bill and Ben are riding on a merry go round.Bill and Ben are riding on a merry go round.Bill rides on a horse at the outer rim, twice as Bill rides on a horse at the outer rim, twice as

far from the center as Ben, who rides on far from the center as Ben, who rides on an inner horse. When the merry go round an inner horse. When the merry go round is rotating at a constant angular speed, isis rotating at a constant angular speed, is

Bill’s tangential speed is Bill’s tangential speed is a)a) Twice Ben’sTwice Ben’sb)b) The sameThe samec)c) Half of Ben’sHalf of Ben’sd)d) Impossible to determine.Impossible to determine.

Co-operative Co-operative ChallengeChallenge

Make an order of Make an order of magnitude estimatemagnitude estimate of the of the number of revolutions through which a number of revolutions through which a typical car tyre turns in a year. State the typical car tyre turns in a year. State the quantities you measure or estimate and quantities you measure or estimate and their values.their values.

Answer this by yourselfAnswer this by yourself

Bill and Ben are riding on a merry go round.Bill and Ben are riding on a merry go round.Bill rides on a horse at the outer rim, twice as Bill rides on a horse at the outer rim, twice as

far from the center as Ben, who rides on far from the center as Ben, who rides on an inner horse. When the merry go round an inner horse. When the merry go round is rotating at a constant angular speed, is rotating at a constant angular speed, Bill’s angular speed isBill’s angular speed is

a)a) Twice Ben’sTwice Ben’sb)b) The sameThe samec)c) Half of Ben’sHalf of Ben’sd)d) Impossible to determine.Impossible to determine.

Solve ThisSolve This

►What is the angular speed and What is the angular speed and acceleration of the second hand?acceleration of the second hand?

The tub of a washer goes into a spin cycle, The tub of a washer goes into a spin cycle, starting from rest and gaining angular speed starting from rest and gaining angular speed steadily for 8.00s, at which time it is turning steadily for 8.00s, at which time it is turning at 5.00rev/s. At this point Eiva opens the lid, at 5.00rev/s. At this point Eiva opens the lid, and a safety switch turns off the washer. and a safety switch turns off the washer. The tub smoothly slows to rest in The tub smoothly slows to rest in 12.0s.Through how many revolutions does 12.0s.Through how many revolutions does the tub turn while it is in motion? the tub turn while it is in motion?

Solve ThisSolve This

50 revs50 revs

5.00rev/s 5.00rev/s = 31.4rads= 31.4rads-1-1

tif

2

Try ThisTry This

Estimate or calc. the mass of the sun

2.02x102.02x103030kgkg

Remember this?Remember this? Satellites Satellites

By equating Newton’s Law with centripetal By equating Newton’s Law with centripetal force and cancelling:force and cancelling:

r

Gmv

r

vm

r

mGm

1

22

221

ie. All satellites must have the same orbital ie. All satellites must have the same orbital speed for a given radius regardless of their speed for a given radius regardless of their mass.mass.

TorqueTorque► Torque is the Torque is the

turning effect of a turning effect of a force couple.force couple.

► Symbol: Symbol: ► Measured in Newton Measured in Newton

meters (Nm)meters (Nm)► Acts clockwise or Acts clockwise or

anticlockwiseanticlockwise► Force and distance Force and distance

from pivot must be from pivot must be perpendicularperpendicular

r

F

Fr►Does this cause pure rotation?Does this cause pure rotation?

NONO

What is Torque?What is Torque?

►Objective to find the relationship

between and

Find the relationship between and

► So what are we So what are we going to do?going to do?

► This is great This is great training for 3.1training for 3.1

m

r

►Assumption: In this demo it is OKAssumption: In this demo it is OKto let Fto let FTT=F=Fww for small accelerations. for small accelerations.

►Normally FNormally FRR=F=Fww-F-FTT

► FFTT causes the torque causes the torque

22

1 tti

= 0.0045 + 0.012

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 1 2 3 4 5 6 7 8 9

To

rqu

e(N

m)

Angular Acceleration(rad/s/s)

Torque vs. Angular Acceleration

TorqueTorque

► Just as force causes linear acceleration, Just as force causes linear acceleration, torque causes angular acceleration.torque causes angular acceleration.

►Compare with F=maCompare with F=ma►So what is this “I” thing anyway….So what is this “I” thing anyway….

I

Rotational InertiaRotational Inertia►Symbol: ISymbol: I►Measured in kgmMeasured in kgm22

►Rotational inertia is a measure of how Rotational inertia is a measure of how hard it is to get an object spinning.hard it is to get an object spinning.

► It depends on:It depends on: MassMass How the mass is distributed about the axis How the mass is distributed about the axis

of rotationof rotation 2mrI

Examples of Inertia’sExamples of Inertia’s

2

2

2

5

2

2

1

MRI

MRI

MRI

Solid CylinderSolid Cylinder

Hollow CylinderHollow Cylinder

Solid SphereSolid Sphere

Your InertiaYour Inertia

►Calculate the approximate I value for Calculate the approximate I value for your body. State any assumptions you your body. State any assumptions you make.make.

ExampleExample

►A mass of 0.10kg is A mass of 0.10kg is used to accelerate a used to accelerate a fly-wheel of radius fly-wheel of radius 0.20m. The mass 0.20m. The mass accelerated accelerated downwards at downwards at 1.0ms1.0ms-2-2..

►What is the torque?What is the torque?► g=10msg=10ms-2-2 today today

m

r

ExampleExample

m

FT

FW

FR

:is mass on the force

resultant thelaw 2nd sNewton' Using

NmskgmaFR 1.011.0 2

:bygiven is force weight TheNmskgmgFw 1101.0 2

: wheel)on the torque

theproviding ( force tension theSo

NFF RW 9.01.01FT

:is wheelon the torque theSo

NmmNrFT 18.02.09.0

ProblemProblem

►Bruce was pushing his friends on a Bruce was pushing his friends on a roundabout (radius 1.5m) at the local roundabout (radius 1.5m) at the local park with a steady force of 120N. After park with a steady force of 120N. After 25s it has reached a speed of 0.60rads25s it has reached a speed of 0.60rads--

11..►What is the torque he is applying?What is the torque he is applying?►What is the angular acceleration of the What is the angular acceleration of the

roundabout?roundabout?►What is the rotational inertia of the What is the rotational inertia of the

roundabout & friends?roundabout & friends?

180Nm

0.024rads-2.

7500kgm2

ProblemProblem

►Now it’s Ivys’ turn to push. Bruce and Now it’s Ivys’ turn to push. Bruce and Ben decide to climb into the centre of Ben decide to climb into the centre of the roundabout instead of sitting on the the roundabout instead of sitting on the seats at the outside. This reduces the seats at the outside. This reduces the inertia of the roundabout + friends to inertia of the roundabout + friends to 6000kgm6000kgm2.2.

► If Ivy pushes with the same force of If Ivy pushes with the same force of 120N for 25s, what will the final angular 120N for 25s, what will the final angular speed of the roundabout be now?speed of the roundabout be now?

0.75rads-1

Answer Answer thesethese

1.1. Which quantities Which quantities affect rotational affect rotational inertia? inertia?

2.2. What are the units of What are the units of I?I?

3.3. Where does a force Where does a force act for an object to be act for an object to be travelling with travelling with translational motion translational motion only?only?

4.4. Explain why you are Explain why you are able to travel at a able to travel at a higher speed around a higher speed around a corner if it is banked.corner if it is banked.

Mass and it’s distributionMass and it’s distribution

KgmKgm22

Through COMThrough COMThe horizontal componentThe horizontal componentof the reaction force of the reaction force provides an additional provides an additional centripetal force whichcentripetal force whichallows a vehicle to travelallows a vehicle to travelfaster without slipping atfaster without slipping ata tangent.a tangent.

Angular MomentumAngular Momentum

►Any rotating object has angular Any rotating object has angular momentum, much the same as any momentum, much the same as any object moving in a straight line has object moving in a straight line has linear momentum.linear momentum.

►Angular momentum depends on:Angular momentum depends on: The angular velocity The angular velocity The rotational inertia IThe rotational inertia I

►Symbol: LSymbol: L►Measured in kgmMeasured in kgm22ss-1-1

IL

Angular momentum Angular momentum

►Angular momentum is conserved as Angular momentum is conserved as long as…..long as…..

►There are no external torques acting.There are no external torques acting.

Examples:Examples:

►Balancing on a bicycle – If a stationary Balancing on a bicycle – If a stationary bike wheel is supported on one side of bike wheel is supported on one side of the axle, it tips over. If the bike wheel is the axle, it tips over. If the bike wheel is spinning, it will balance easily when spinning, it will balance easily when supported on only one side. A large supported on only one side. A large external torque is required to change external torque is required to change the direction of the angular momentum. the direction of the angular momentum.

Examples:Examples:

►Helicopters: The blades spin one way Helicopters: The blades spin one way so the helicopter body tries to spin so the helicopter body tries to spin the other way – not much use! So we the other way – not much use! So we have to supply an external torque have to supply an external torque (from tail rotor) to keep the body still.(from tail rotor) to keep the body still.

Examples:Examples:

►Motorbikes (doing wheelies!) – As power Motorbikes (doing wheelies!) – As power goes to the back wheel suddenly to goes to the back wheel suddenly to make it spin one way, the bike tries to make it spin one way, the bike tries to spin the other way. The weight of the spin the other way. The weight of the rider and bike body supplies an external rider and bike body supplies an external torque to keep the front end of bike on torque to keep the front end of bike on the road.the road.

Problem:Problem:► Tom is listening to some records one Sunday Tom is listening to some records one Sunday

afternoon. His new turntable (I=0.10kgmafternoon. His new turntable (I=0.10kgm22) is ) is spinning freely (ie no motor) with an angular spinning freely (ie no motor) with an angular velocity of 4.0radsvelocity of 4.0rads-1-1, when he drops a record , when he drops a record (I=0.02kgm(I=0.02kgm22) onto it from directly above. ) onto it from directly above. What is the angular speed now?What is the angular speed now?

3.3 rads-1

124.0

0.41.0

skgm

x

IL

x

IL

12.04.0

Examples:Examples:

►Figure Skating – Ice-skaters go into Figure Skating – Ice-skaters go into a spin with arms outstretched and a spin with arms outstretched and a fixed amount of L dependent on a fixed amount of L dependent on the torque used to get themselves the torque used to get themselves spinning. (Once spinning, no spinning. (Once spinning, no external torque) If they then draw external torque) If they then draw in their arms, their inertia in their arms, their inertia decreases, so their angular speed decreases, so their angular speed increases in order to keep the total increases in order to keep the total momentum conserved.momentum conserved.

RoundaboutRoundabout

► 7 Physics students get on a 7 Physics students get on a roundabout and get it spinning. They roundabout and get it spinning. They then climb towards the middle. then climb towards the middle.

►Discuss what happens.Discuss what happens.Discuss means Discuss means lots of detail lots of detail

Problem Problem ► Sean comes along and Sean comes along and

decides to try and find decides to try and find out what angular out what angular speed he would need speed he would need to spin the roundabout to spin the roundabout at to make everyone at to make everyone fall off. Assume a 70kg fall off. Assume a 70kg person can hold on person can hold on with a force equal to with a force equal to their body weight. g is their body weight. g is still 10msstill 10ms-1-1

1

1

2

2

6.2

5.1

9.3

9.3

5.1

701070

rads

r

v

msv

v

r

mvmgFC

►Hint: want one?Hint: want one?►What speed would What speed would give you a centripetal give you a centripetal force = weight force??force = weight force??

Angular MomentumAngular Momentum

►Linear Linear momentum can momentum can be converted to be converted to angular angular momentummomentum

)( prL

mvrL

m

v

r

ExampleExample

►A satellite in orbit needs to be turned A satellite in orbit needs to be turned around. This is done by firing two around. This is done by firing two small “retro-rockets” attached to the small “retro-rockets” attached to the side of the satellite. These rockets side of the satellite. These rockets fire 0.2kg of gas each at 100msfire 0.2kg of gas each at 100ms-1. -1.

►The satellite has an inertia of The satellite has an inertia of 1200kgm1200kgm22 and the rockets are and the rockets are positioned at a radius of 1.5mpositioned at a radius of 1.5m

►What speed will the satellite turn at?What speed will the satellite turn at?

SolutionSolution:::rocketeach from gas theof p momentumlinear The

11 201002.0 kgmsmskgmvp

:rocketeach from gas theof L momentumangular The121 305.120 skgmmkgmsprL

:be willsatellite theof speedangular The

60 of L opposite

and equalgain willsatellite conserved, is L Since

60 rockets 2by Multiply

12

12

skgm

skgm

12

12

05.01200

60

I

L

radskgm

skgm

ExtraExtra

►How would you stop the satellite from How would you stop the satellite from rotating once it was in the correct position??rotating once it was in the correct position??

Fire an equal burst of gas from the rockets in the opposite direction to the original.

Problem:Problem:►Nik 70kg was at a theme park and wanted a Nik 70kg was at a theme park and wanted a

go on the bumper boats. He runs at 4.0msgo on the bumper boats. He runs at 4.0ms-1-1 and jumps onto a floating boat of radius and jumps onto a floating boat of radius 1.0m, landing 40cm from the centre. If the 1.0m, landing 40cm from the centre. If the boat + Nik have an inertia of 100kgmboat + Nik have an inertia of 100kgm22, , what angular speed will they spin at? what angular speed will they spin at?

►What assumption are we making here?What assumption are we making here?►Nik is 70kg, Nik can run 4.0msNik is 70kg, Nik can run 4.0ms-1-1, , all linear all linear

momentum is changed into rotationalmomentum is changed into rotational..

1

12

1.1

100112

112

40.00.470

rads

IL

skgmL

xxL

mvrL

Assuming no linear motion of boat – not likely!

Rotational Kinetic EnergyRotational Kinetic Energy

►The energy of rotating objects EThe energy of rotating objects Ek(rot)k(rot)

2)( 2

1 IE rotk

ExampleExample

► How much kinetic energy does a 20kgmHow much kinetic energy does a 20kgm22 gear cog have if spinning at 4.0radsgear cog have if spinning at 4.0rads-1-1??

JIEk 1604202

1

2

1 22

Rolling Down SlopesRolling Down SlopesType 1

►Which will reach the bottom first? Why?Which will reach the bottom first? Why?

Rolling DownhillRolling Downhill

►The ball.The ball.►Why?Why?►All have the same EAll have the same Epp to begin with. to begin with.

►The hollow cylinder has the largest I so The hollow cylinder has the largest I so gains the most Egains the most Ek(rot) k(rot) and the least Eand the least Ek(lin)k(lin)..

► It will have the smallest acceleration of It will have the smallest acceleration of rolling – ie will be rolling downhill slower rolling – ie will be rolling downhill slower than the others at any given time.than the others at any given time.

ProblemProblem► Johnny the 65kg trampolinist is Johnny the 65kg trampolinist is

bouncing on his trampoline so that at bouncing on his trampoline so that at the instant he leaves the mat he is the instant he leaves the mat he is travelling at 9mstravelling at 9ms-1-1. As he moves . As he moves upward he curls into a ball and does a upward he curls into a ball and does a 360360°° front flip. front flip.

►How much linear kinetic energy does How much linear kinetic energy does he have as he leaves the mat?he have as he leaves the mat?

►What happens to this energy?What happens to this energy?

JmvE link 26009652

1

2

1 22)(

Converted to gravitational and rotational kinetic energy

ProblemProblem

► If he reaches a maximum height of 3.0m, If he reaches a maximum height of 3.0m, how much gravitational energy has he how much gravitational energy has he gained?gained?

► At the top, he is spinning at 6.0radsAt the top, he is spinning at 6.0rads-1-1. What . What is his inertia?is his inertia?

JmghEp 191138.965

1222

)(

)()(

406

68922

68919112600

skgmE

I

JEEE

rotk

plinkrotk

Co-operative ChallengeCo-operative ChallengeTeachers ChoiceTeachers Choice

A box of objects that can roll down a slope are supplied.A box of objects that can roll down a slope are supplied.

Investigate and experiment to develop a rule. Investigate and experiment to develop a rule.

Aim:Aim: To provide a written rule with sound Physics logic to To provide a written rule with sound Physics logic to back it up that will allow you to correctly identify which of back it up that will allow you to correctly identify which of any 2 chosen objects will roll fastest down the slope.any 2 chosen objects will roll fastest down the slope.

No texts, just you and your group.No texts, just you and your group.

Hubble TelescopeHubble Telescope

The Hubble telescope is a satellite which uses3 flywheels within its structure as a means oforienting itself. The axes of these wheels are mutually perpendicular as shown:

How could you move this telescope?

Flywheel Diagram showing the positions of the 3 mutually

perpendicular flywheels


Electric motors and brakes, powered by electricitygenerated by solar panels, are used to start andstop them. When a flywheel starts rotating, the satellite rotates in the opposite direction.When the desired position is reached, the brakesstop the flywheel and the satellite stops rotating. In operation, a flywheel turns at 181rpm and has a rotational inertia of 0.589kgm2.


Explain, referring to the appropriate principle ofrotational motion, how a flywheel turning canrotate the satellite.

As there are no externally applied torques, theangular momentum of the system is conserved.The initial angular momentum of the system is zero.Hence the final angular momentum is zero. The angular momentum of the flywheel in onedirection must be matched by an angular momentumof the Hubble Telescope in the opposite direction.


Show that the flywheel’s angular Show that the flywheel’s angular velocity is 19.0rads-1. velocity is 19.0rads-1.

260

181


Calculate the angular momentum Calculate the angular momentum possessed by the turning flywheel. possessed by the turning flywheel.

122.11

0.19589.0

skgm

IL


Deduce the angular momentum of the Hubble telescope as the flywheel is spinning.

122.11 skgmLHubble in the opposite direction to flywheel


The Hubble telescope is rotated by 1/6 of a The Hubble telescope is rotated by 1/6 of a revolution. What angular displacement revolution. What angular displacement

does it turn through?does it turn through?

rad3

26

1

or 1.05 rad


If it takes 10.0 minutes to complete this turn, what is the Hubble telescope’s constant angular velocity?

131075.1

6010

05.1

rads

t


Calculate Hubble telescope’s rotational inertia.

2

3

6400

1075.1

2.11

kgm

I

IL


The flywheel’s motor has a braking system which operates on its spindle bringing it to a halt with an angular acceleration of -8.56rads-2. The spindle has a radius of 20.0mm.Calculate the retarding torque required to be applied by the motor’s brake system.

Nm

I

04.5

56.8589.0


Calculate the retarding force needed to be applied by the braking system.

N

F

Fr

252020.0

04.5

Hubble TelescopeHubble TelescopeWhile the Hubble telescope is in its new position it observes a star known to have an inner core of radius 2.00x107m that rotates once every 45 days. While the Hubble telescope is oriented to observethis star, it undergoes a supernova explosion- much of its outer mass explodes to form a Nebula,but the inner core collapses to form a neutron star6.00 km in radius.

Hubble TelescopeHubble TelescopeThe formula for rotational inertia for a uniformsphere is I = 2/5mr2.The newly formed neutron star rotates at a differentfrequency to the original star.

Calculate this new frequency.

Hz

f

f

frfr

fmrfmr

II

86.2

36002445)1000.6(

)1000.2(

36002445

1)1000.2()1000.6(

25

22

5

2

23

27

2

272

3

12

122

2

12

122

2

1122

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Rotational Motion