Rotational MotionRotational MotionNCEA AS 3.4NCEA AS 3.4

Text Chapter: 4Text Chapter: 4

The SunThe Sun► Use the information Use the information

you have absorbed you have absorbed this year to this year to estimate the size of estimate the size of the sun.the sun.

rGmv 1

►Need a hint?Need a hint?

kgx 301002.2

CD StudyCD Study► The CD reads from the inside to the outside.The CD reads from the inside to the outside.

They used to read 4.3 mega bytes per second.They used to read 4.3 mega bytes per second.► They require a constant linear speed of 1.4msThey require a constant linear speed of 1.4ms-1-1. . ► This disc needs to rotate at 500rpm at the start This disc needs to rotate at 500rpm at the start

and 200rpm at the finish.and 200rpm at the finish.► a) Convert 500rpm to radsa) Convert 500rpm to rads-1-1

► b) a CD can reach the correct b) a CD can reach the correct ωω in one in one revolution. What is revolution. What is

► c) What is the radius of the disc at the start?c) What is the radius of the disc at the start?► A particular CD (Bee Gees) has a playing time A particular CD (Bee Gees) has a playing time

of 72 minutes.of 72 minutes.► d) Convert 200rpm to radsd) Convert 200rpm to rads-1-1..

►e) Calculate the angular acceleration e) Calculate the angular acceleration as the disc plays from start to finish.as the disc plays from start to finish.

►f) Calculate the angle the disc moves f) Calculate the angle the disc moves through in this time.through in this time.

►g) Convert the angle to revolutions.g) Convert the angle to revolutions.►h) Calculate the radius of this disc (not h) Calculate the radius of this disc (not

Mo’s Rosa) at the finish.Mo’s Rosa) at the finish.

TypesTypes►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 forces Pure Rotation –2 equal & opposite forces

act at a perpendicular distance from the act at a perpendicular distance from the c.o.m (force couple) C.o.m remains c.o.m (force couple) C.o.m remains stationary, object spins around itstationary, object spins around it

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

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

AB

Angular DisplacementAngular Displacement► 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

rs

Angular DisplacementAngular Displacement► 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 ►Measured in radians per second (radsMeasured in radians per second (rads-1-1.).)►Average angular velocity calculated by:Average angular velocity calculated by:

takentimentdisplacemeangular 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(

tsv

rv

rs

trst

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:

takentimelocityangular vein changeonacceleratiangular

t

Angular AccelerationAngular Acceleration►Angular acceleration and linear Angular acceleration and linear

acceleration are linked by:acceleration are linked by: ra

SummarySummaryTranslational Rotational Equation

ss s=rs=r

vv v=rv=r

aa a=ra=r

GraphsGraphsAngular Displacement vs Time

0 10 20 30 40Time

Angu

lar D

ispl

acem

ent

Gradient =

angular velocity

GraphsGraphsGradient = angular accelerati

on Angular Velocity vs Time

0 1 2 3 4Time

Angu

lar V

eloc

ityArea 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

TorqueTorque► Torque is the Torque is the

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

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

metres (Nm)metres (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

ExampleExample►A mass of 0.1kg is A mass of 0.1kg is

used to accelerate used to accelerate a fly-wheel of a fly-wheel of radius 0.2m. The radius 0.2m. The mass accelerateds mass accelerateds downwards at downwards at 1ms1ms-2-2..

m

r

ExampleExample

NmmNrF

NFF

NmskgmgF

NmskgmaF

T

RW

w

R

18.02.09.0:is wheelon the torque theSo

9.01.01F: wheel)on the torque

theproviding ( force tension theSo1101.0

:bygiven is force weight The1.011.0

:is mass on the forceresultant thelaw 2nd sNewton' Using

T

2

1

m

FT

FWFR

TorqueTorque► Just as force causes linear acceleration, Just as force causes linear acceleration,

torque causes angular acceleration.torque causes angular acceleration.

►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

52

21

MRI

MRI

MRI

Solid CylinderSolid Cylinder

Hollow CylinderHollow Cylinder

Solid SphereSolid Sphere

1

12

1.1

100112

112

40.00.470

rads

ILskgmL

xxLmvrL

Assuming no linear motion of boat – not likely!

ProblemProblem►Cameron was pushing his friends on a Cameron 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-1-1..

►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 Lewis’ turn to push. Cameron Now it’s Lewis’ turn to push. Cameron and Chris decide to climb into the and Chris decide to climb into the centre of the roundabout instead of centre of the roundabout instead of sitting on the seats at the outside. This sitting on the seats at the outside. This reduces the inertia of the roundabout + reduces the inertia of the roundabout + friends to 7000kgmfriends to 7000kgm2.2.

► If Lewis pushes with the same force of If Lewis 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?speed of the roundabout be?

0.64rads-1

Problem Problem ► Jacob comes along Jacob comes along and decides to try and and decides to try and find out what angular find 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.their body weight.

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

1

1

2

2

6.25.19.3

9.35.1

701070

radsrvmsv

vrmvmgFC

Angular MomentumAngular Momentum►Any rotating object has angular Any rotating object has angular

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

Problem:Problem:►Lachie is listening to some records one Lachie is listening to some records one

Sunday afternoon. His turntable Sunday afternoon. His turntable (I=0.10kgm(I=0.10kgm22) is spinning freely (ie no ) is spinning freely (ie no motor) with an angular velocity of motor) with an angular velocity of 4rads4rads-1,-1, when he drops a Dire Straits when he drops a Dire Straits record (I=0.02kgmrecord (I=0.02kgm22) onto it from ) onto it from directly above. What is the angular directly above. What is the angular speed now?speed now?

3.33rads-1

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 make goes to the back wheel suddenly to make it spin one way, the bike tries to spin the it spin one way, the bike tries to spin the other way. The weight of the rider and other way. The weight of the rider and bike body supplies an external torque to bike body supplies an external torque to keep the front end of bike on the road.keep the front end of bike on the road.

Examples:Examples:

►Figure Skating – Ice-skaters go into Figure Skating – Ice-skaters go into a spin with arms outstretched and a a spin with arms outstretched and a fixed amount of L dependent on the fixed amount of L dependent on the torque used to get themselves 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.

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 the external torque is required to change the direction of the angular momentum. direction of the angular momentum.

Angular MomentumAngular Momentum►Linear Linear

momentum can momentum can be converted to be converted to angular angular momentummomentum

)( prLmvrL

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?

Solution:Solution:

12

12

12

12

121

11

05.0120060

IL

:be willsatellite theof speedangular The60 of L opposite

and equalgain willsatellite conserved, is L Since60 rockets 2by Multiply

305.120

:rocketeach from gas theof L momentumangular The201002.0

:rocketeach from gas theof p momentumlinear The

radskgmskgm

skgm

skgm

skgmmkgmsprL

kgmsmskgmvp

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:► Jethro was at a theme park and wanted Jethro was at a theme park and wanted

a go on the bumper boats. He runs at a go on the bumper boats. He runs at 4ms4ms-1-1 and jumps onto a floating boat of and jumps onto a floating boat of radius 1m, landing 40cm from the radius 1m, landing 40cm from the centre. If the boat + Jethro have an centre. If the boat + Jethro have an inertia of 100kgminertia of 100kgm22, what angular speed , what angular speed will they spin at? will they spin at?

►What assumption are we making here?What assumption are we making here?(Hint: what kind of motion will be produced?)(Hint: what kind of motion will be produced?)

1.12rads-1

Assuming no linear motion of boat – not likely!

1

12

1.1

100112

112

40.00.470

rads

ILskgmL

xxLmvrL

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 20kgmsHow much kinetic energy does a 20kgms-1-1

gear cog have if spinning at 4radsgear cog have if spinning at 4rads-1-1??

JIEk 16042021

21 22

Conversion of Energy - Conversion of Energy - ExampleExample

►A toy car operates using a A toy car operates using a flywheel.The car is pushed across the flywheel.The car is pushed across the floor a few times to start the wheel floor a few times to start the wheel spinning and when let go the spinning and when let go the rotational kinetic energy is converted rotational kinetic energy is converted to linear kinetic energy as the car to linear kinetic energy as the car moves forward.moves forward.

ProblemProblem►A 65kg trampolinist named Daniel is A 65kg trampolinist named Daniel 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 moes upward . As he moes upward he curls nto a ball and does a 360he curls nto a ball and does a 360°° 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 260096521

21 22

)(

Converted to gravitational and rotational kinetic energy

ProblemProblem► If he reaches a maximum height of 3m, how If he reaches a maximum height of 3m, how

much gravitational energy has he gained?much gravitational energy has he gained?

► At the top, he is spinning at 6radsAt the top, he is spinning at 6rads-1-1. What is . What is his inertia?his inertia?

JmghEp 195031065

1222

)(

)()(

36665022

65019502600

skgmE

I

JEEE

rotk

plinkrotk

Rolling Down SlopesRolling Down Slopes►Which will reach the bottom first?Which will reach the bottom first?

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.