Transcript Rotational Mechanics

Rotational Motion
NCEA AS 3.4
Text Chapter: 4
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
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
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.
Angular Momentum
► Linear
momentum
can be converted to
angular momentum
L  mvr
( L  pr )
r
v
m
Rotational Kinetic Energy
► The
energy of rotating objects Ek(rot)
Ek ( rot)
1 2
 Iw
2
Rolling Down Slopes
► Which
will reach the bottom first?