Transcript q - MACscience

Rotational Motion
NCEA AS 3.4
Text Chapter: 4
The Sun
► Use
the information
you have absorbed this
year to estimate the
size of the sun.
►Need
Gm1
v
r
30
2.02 x10 kg
a hint?
► The
CD Study
CD reads from the inside to the outside.
They used to read 4.3 mega bytes per second.
► They require a constant linear speed of 1.4ms-1.
► This disc needs to rotate at 500rpm at the start and
200rpm at the finish.
► a) Convert 500rpm to rads-1
► b) a CD can reach the correct ω in one
revolution. What is a?
► c) What is the radius of the disc at the start?
► A particular CD (Bee Gees) has a playing time of
72 minutes.
► d) Convert 200rpm to rads-1.
► e)
Calculate the angular acceleration as the
disc plays from start to finish.
► f) Calculate the angle the disc moves
through in this time.
► g) Convert the angle to revolutions.
► h) Calculate the radius of this disc (not Mo’s
Rosa) at the finish.
Types
► Pure
Translation –force acts through the
centre of mass, C.o.m moves.
► Pure Rotation –2 equal & opposite forces act
at a perpendicular distance from the c.o.m
(force couple) C.o.m remains stationary,
object spins around it
► Mixture – single force acts, NOT through
c.o.m, object moves and rotates around
c.o.m
Angular Displacement
► Although
both points A
& B have turned
through the same
angle, A has travelled
a greater distance than
B
► A must have had the
greater linear speed
A
B
q
B
A
Angular Displacement
q
► Measured in radians
(rad)
► Angular displacement
is related to linear
distance by:
► Symbol
s
q
r
s
r
q
r
Angular Displacement
► Remember
Maths:
r
s  2r
from
 How to put your
calculator into radian
mode?
 How many radians are
in a full circle?
Angular Velocity
w
► Measured in radians per second (rads-1.)
► Average angular velocity calculated by:
► Symbol
angular displaceme nt
average angular ve locity 
time taken
w
q
t
Angular Velocity
► To
s
q
put it another way:
q
w
t
s
s
w
(since q  )
r t
r
v
s
w  (since v  )
r
t
r
r
► So
angular velocity is
related to linear
velocity by:
v  rw
Angular Acceleration
► Changing
angular velocity
► Symbol: a
► Measured in radians per second squared
(rads-2.)
► Calculated by:
change in angular ve locity
angular accelerati on 
time taken
w
a
t
Angular Acceleration
► Angular
a  ra
acceleration and linear acceleration
are linked by:
Summary
Translational
Rotational
Equation
s
q
s=rq
v
w
v=rw
a
a
a=ra
Graphs
Angular Displacement
Angular Displacement vs Time
0
10
20
Time
30
Gradient
=
angular
velocity
w
40
Graphs
Gradient =
angular
acceleration
a
Angular Velocity
Angular Velocity vs Time
Area under
graph =
angular
displacemen
tq
0
1
2
Time
3
4
Kinematic Equations
► Recognise
these??:
► Use them the same
way you did last
year.
w f  wi  a t
w  w  2aq
2
f
2
i
q  wi t  2 a t
1
2
 w f  wi 
t
q  
2


Torque
is the turning
effect of a force.
► Symbol: t
► Measured in Newton
metres (Nm)
► Acts clockwise or
anticlockwise
► Force and distance
from pivot must be
perpendicular
F
► Torque
r
t  Fr
Example
►A
mass of 0.1kg is
used to accelerate a
fly-wheel of radius
0.2m. The mass
accelerateds
downwards at 1ms-2.
r
m
Example
Using Newton' s 2nd law the resultant
force on the mass is :
FT
FR  ma  0.1kg  1ms 1  0.1N
The weight force is given by :
Fw  mg  0.1kg  10ms  2  1N
m
So the tension force ( providing the
torque on the wheel) :
FT  FW  FR  1  0.1  0.9 N
So the torque on the wheel is :
t  FT  r  0.9 N  0.2m  0.18 Nm
FW
FR
Torque
► Just
as force causes linear acceleration,
torque causes angular acceleration.
t  Ia
► So
what is this “I” thing anyway….
Rotational Inertia
► Symbol:
I
► Measured in kgm2
► Rotational inertia is a measure of how hard
it is to get an object spinning.
► It depends on:
 Mass
 How the mass is distributed about the axis of
rotation
I   mr
2
Examples of Inertia’s
Solid Cylinder
1
2
I  MR
2
Hollow Cylinder I  MR
Solid Sphere
2
2
2
I  MR
5
L  mvr
L  70 x 4.0 x0.40
2 1
L  112kgm s
L  Iw
112  100w
w  1.1rads
1
Assuming no linear motion of boat – not likely!
Problem
► Cameron
was pushing his friends on a
roundabout (radius 1.5m) at the local park
with a steady force of 120N. After 25s it has
reached a speed of 0.60rads-1.
► What is the torque he is applying? 180Nm
► What is the angular acceleration of the
roundabout?
0.024rads-2.
► What is the rotational inertia of the
roundabout+friends?
7500kgm2
Problem
► Now
it’s Lewis’ turn to push. Cameron and
Chris decide to climb into the centre of the
roundabout instead of sitting on the seats at
the outside. This reduces the inertia of the
roundabout + friends to 7000kgm2.
► If Lewis pushes with the same force of
120N for 25s, what will the final angular
speed of the roundabout be?
0.64rads-1
► Jacob
comes along and
decides to try and find
out what angular speed
he would need to spin
the roundabout at to
make everyone fall off.
Assume a 70kg person
can hold on with a force
equal to their body
weight.
► Hint: What speed would
give you a centripetal
force = weight force??
Problem
mv
FC  mg 
r
2
70v
70 10 
1. 5
1
v  3.9ms
v 3 .9
w 
r 1 .5
1
 2.6rads
2
Angular Momentum
► Any
rotating object has angular momentum,
much the same as any object moving in a
straight line has linear momentum.
► Angular momentum depends on:
 The angular velocity w
 The rotational inertia I
► Symbol:
L
► Measured in kgm2s-1
L  Iw
Angular momentum
► Angular
as…..
► There
momentum is conserved as long
are no external torques acting.
Problem:
► Lachie
is listening to some records one
Sunday afternoon. His turntable
(I=0.10kgm2) is spinning freely (ie no
motor) with an angular velocity of 4rads-1,
when he drops a Dire Straits record
(I=0.02kgm2) onto it from directly above.
What is the angular speed now?
3.33rads-1
Examples:
► Helicopters:
The blades spin one way so
the helicopter body tries to spin the other
way – not much use! So we have to supply
an external torque (from tail rotor) to keep
the body still.
Examples:
► Motorbikes
(doing wheelies!) – As power
goes to the back wheel suddenly to make it
spin one way, the bike tries to spin the
other way. The weight of the rider and bike
body supplies an external torque to keep
the front end of bike on the road.
Examples:
► Figure
Skating – Ice-skaters go into a
spin with arms outstretched and a
fixed amount of L dependent on the
torque used to get themselves
spinning. (Once spinning, no external
torque) If they then draw in their
arms, their inertia decreases, so their
angular speed increases in order to
keep the total momentum conserved.
Examples:
► Balancing
on a bicycle – If a stationary bike
wheel is supported on one side of the axle,
it tips over. If the bike wheel is spinning, it
will balance easily when supported on only
one side. A large external torque is required
to change the direction of the angular
momentum.
Angular Momentum
► Linear
momentum
can be converted to
angular momentum
L  mvr
( L  pr )
r
v
m
Example
►A
satellite in orbit needs to be turned
around. This is done by firing two small
“retro-rockets” attached to the side of the
satellite. These rockets fire 0.2kg of gas
each at 100ms-1.
► The satellite has an inertia of 1200kgm2
and the rockets are positioned at a radius
of 1.5m
► What speed will the satellite turn at?
Solution:
The linear momentum p of the gas from each rocket :
p  mv  0.2kg 100ms 1  20kgms1
The angular momentum L of the gas from each rocket :
L  pr  20kgms1 1.5m  30kgm2 s 1
Multiply by 2 rockets  60kgm2 s 1
Since L is conserved, satellite will gain equal and
opposite L of 60kgm2 s 1
The angular speed of the satellite will be :
L 60kgm2 s 1
1
w 
 0.05rads
2
I 1200kgm
Extra
► How
would you stop the satellite from
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:
► Jethro
was at a theme park and wanted a
go on the bumper boats. He runs at 4ms-1
and jumps onto a floating boat of radius
1m, landing 40cm from the centre. If the
boat + Jethro have an inertia of 100kgm2,
what angular speed will they spin at?
1.12rads-1
► What
assumption are we making here?
(Hint: what kind of motion will be produced?)
Assuming no linear motion of boat – not likely!
L  mvr
L  70 x 4.0 x0.40
2 1
L  112kgm s
L  Iw
112  100w
w  1.1rads
1
Assuming no linear motion of boat – not likely!
Rotational Kinetic Energy
► The
energy of rotating objects Ek(rot)
Ek ( rot)
1 2
 Iw
2
Example
► How
much kinetic energy does a 20kgms-1 gear
cog have if spinning at 4rads-1?
1 2 1
2
Ek  Iw   20  4  160 J
2
2
Conversion of Energy - Example
►A
toy car operates using a flywheel.The car
is pushed across the floor a few times to
start the wheel spinning and when let go
the rotational kinetic energy is converted to
linear kinetic energy as the car moves
forward.
Problem
►A
65kg trampolinist named Daniel is
bouncing on his trampoline so that at the
instant he leaves the mat he is travelling at
9ms-1. As he moes upward he curls nto a
ball and does a 360° front flip.
► How much linear kinetic energy does he
have as he leaves the mat?
Ek (lin)
► What
1 2 1
2
 mv   65  9  2600 J
2
2
happens to this energy?
Converted to gravitational and rotational kinetic energy
Problem
► If
he reaches a maximum height of 3m, how much
gravitational energy has he gained?
E p  mgh  65 10  3  1950 J
► At
the top, he is spinning at 6rads-1. What is his
inertia?
Ek ( rot)  Ek (lin)  E p  2600  1950  650 J
I
2 Ek ( rot)
w
2
2  650
2 1

 36kgm s
2
6
Rolling Down Slopes
► Which
will reach the bottom first?
Rolling Downhill
► The
ball.
► Why?
► All have the same Ep to begin with.
► The hollow cylinder has the largest I so
gains the most Ek(rot) and the least Ek(lin).
► It will have the smallest acceleration of
rolling – ie will be rolling downhill slower
than the others at any given time.