Transcript rotational motion quantities

MECHANICS 2
Rotational Motion
Teaching Ideas
• Outside – object rotating about a fixed
position (line of students link arms, one
end stays fixed and the rest sweep
around in a circle)
– Can you keep the line straight?
– Mix up where you are in the line to
experience the rotation from a different
position
– Is it easier to rotate about the person in
the centre of the line?
Difference between Rotational
motion and circular motion
New vocab!
• What are the 3 typical ways we can
describe the LINEAR motion of an
object?
Acceleration
Velocity
Displacement
Rotational motion words
θ
d = displacement(m)
v = velocity (ms-1)
θ= angular displacement (rad)
ω= angular velocity (rads-1)
 = ∆θ
∆t
v = ∆d
∆t
a = acceleration (ms-2) α = angular acceleration (rads-2)
a=∆v
α=∆ω
∆t
∆t
Relationship between distance and angular
d
displacement
d=r
θ
when θ in radians
v=r
a=rα
Teaching Idea
• Re-write equations of motion with the
new terminology for rotational motion
– Build up the four EoM using the jigsaw
pieces
– Replace the three (/four) variables with
our new variables
Constant Acceleration
vf = vi + at
ωf=ωi +αt
d = vi t + ½ at2
1
2
θ ωi t  αt
2
vf2 = vi 2 + 2ad ωf2=ωi2+2αθ
(vi  vf )t
d
2
(ωi ωf)t
θ
2
Combined motion
Cycloid motion
• http://www.youtube.com/watch?v=vkah
XgCaHho
• http://www.upscale.utoronto.ca/General
Interest/Harrison/Flash/ClassMechanic
s/RollingDisc/RollingDisc.html
In order to get something to accelerate
we need an unbalanced
FORCE
In order to get something to angular
accelerate we need an unbalanced
TORQUE
Force
τ=Fd
Perpendicular
distance to the
pivot
Pull at various angles on a roll of wire and predict
which way the roll of wire will roll.
Note the helpful dotted lines - they might
provide a clue.
Mass and Rotational inertia
MASS (m) is a measure of how hard
it is to accelerate of an object with
a force
ROTATIONAL INERTIA (I) is a
measure of how hard it is to angular
accelerate an object using a torque
Newton’s 2nd law: F=ma. Mass is
measured in kg
Newton’s 2nd law τ=Iα
Rotational inertia is measured in kg m2
Rotational inertia depends on the mass
of the object and on the distribution
of mass around the axis of rotation
MORE ROTATIONAL MOTION QUANTITIES
TRANSLATIONAL
F Force
m mass
F=ma
ROTATIONAL
MORE ROTATIONAL MOTION QUANTITIES
TRANSLATIONAL
ROTATIONAL
F Force (N)
τ Torque (Nm)
m mass (kg)
I rotational inertia
(kgm2)
F=ma
τ=Iα
Rotational inertia for
masses moving in a circle
I=mr2
radius
Rotational inertia for
masses moving in a circle
I=m1r12+ m2r22
Calculating rotational inertia
Either: Measure τ and α to find I
Or: use an equation based on the distribution of
mass
For a mass moving in a circle:
I  mr
2
For hollow cylinders or hoops
I  mr
2
For solid cylinders or disks
1 2
I  mr
2
For hollow spheres
2 2
I  mr
3
For solid spheres
2 2
I  mr
5
Conservation of angular momentum
The MOMENTUM (p=mv) of a system
objects doesn’t change unless there is
an an external FORCE. For a single
object m can’t change, and v can’t
change
Conservation of angular momentum
The MOMENTUM (p=mv) of a system
objects doesn’t change unless there is
an an external FORCE. For a single
object m can’t change, and v can’t
change
The ANGULAR MOMENTUM (L=Iω) of
a system of objects doesn’t change
unless there is an external TORQUE.
For a single object I CAN change, so ω
CAN change!
Helicopters have two rotors. One big one on the top,
and one small one on the tail.
If it there wasn’t a tail rotor what would happen
when the helicopter slowed down its main rotor in
mid air?
http://www.youtube.com/watch?v=-GTCvyPWzMk
http://www.youtube.com/watch?v=Ug6W7_tafnc
Angular momentum conservation
Li = Lf
Iiωi=Ifωf
Maximum rotational inertia
Minimum angular velocity
Minimum rotational inertia
Maximum angular velocity
Divers
• http://www.fandome.com/video/102554
/Best-of-Athens-2004-Olympic-Diving/
Cat lands on its feet
• http://nz.youtube.com/watch?v=lqsWz9
TAwBs
• http://www.upscale.utoronto.ca/General
Interest/Harrison/Flash/ClassMechanic
s/CatOnItsFeet/CatOnItsFeet.swf
Tornado
• http://www.youtube.com/watch?v=xCI1
u05KD_s
• Tornado tube
Explain how divers can start their dive spinning slowly and start
spinning quickly in mid-air.
R- Respond
The divers curl up to spin faster
R – Rule: Conservation of angular momentum
Angular momentum is a measure of how fast matter is
spinning.
It is defined as the rotational inertia multiplied by angular
velocity L=Iω. Rotational inertia depends on the mass and the
distribution of mass.
In the absence of external torques angular momentum is
conserved
R – Relate
Divers start their dive with their body stretched out. Their
mass is spread out so they have a high rotational inertia. They
curl their bodies into a ball which decreases their rotational
inertia as their mass is closer to their axis of rotation.
Since L=Iω is conserved, when they decrease their rotational
inertia their angular velocity has to increase to conserve
angular momentum.
R- Reread
Angular momentum with a bicycle
wheel
• http://www.youtube.com/watch?v=dVwK
E9yDqVo
Relationship between angular
momentum and translational momentum
L=pr=mvr
L = Angular momentum
v = velocity
r = perpendicular distance between
the direction of motion and the centre
of rotation.
r
Merry go round
• http://physics.weber.edu/amiri/directo
r/dcrfiles/momentum/merryGoRoundS.d
cr
TYPES OF ENERGY
Kinetic energy
Linear kinetic energy
EKlin=½ mv2
Rotational kinetic energy
EKrot=½ I ω2
Describe the energy transformations
involved in
1. A ball rolling down a hill
2. Our rotation contraption
3. A trampolinist doing a flip
Proving the equation for a ring
EK(LIN) = ½mv2
= ½m(r)2 (as v =r)
=½mr22
EK(ROT) = ½I2 (as I=mr2)
YO- YO PROBLEM
When a yo-yo is released it accelerates downward
at a constant rate until the string is all unwound.
1. Using forces explain why it accelerates down
2. Using torques explain why it rotates as it falls
3. What happens when the string is all unwound?
Explain using conservation of angular
momentum.
4. Before the yo-yo is released it has only
potential energy. Explain what happens to this
energy after the yo-yo is released.
2005 yoyo contest
• http://www.youtube.com/watch?v=XvG3
IK-hzRs
Rotational inertia and energy
A student started spinning slowly on the stool
with her arms out.
Her rotational inertia started off as 1.8 kg m2.
She pulled her arms in. Her rotational inertia
was then 1.2 kg m2.
Her initial angular velocity was 1.6 rad/s.
Calculate her final angular velocity.
Calculate her kinetic energy before and after
she moves her arms in. Are they the same? If
not explain where the change in energy comes
from or goes to.
Two cylinders with the same mass and radius.
Why does one roll faster than the other?
Both cylinders transform gravitational
potential energy into kinetic energy
mgh =½Iω2+½mv2
The cylinder with the mass
concentrated closer to the centre has a
smaller rotational inertia so more of its
energy goes into linear kinetic energy
and therefore it goes faster.
Why does a car wheel roll slower than a car?
Both objects transform gravitational
potential energy into kinetic energy
mgh =½Iω2+½mv2
For the car, most of the mass is simply
translating, and not rotating, so most of its
potential energy turns into the translational
kinetic energy giving it a larger speed. The
whole of the car wheel has to rotate so a
larger proportion of its energy goes into
linear kinetic energy and therefore it goes
faster.
Rotational Kinetic Energy
problems
• Examples page 140
• Q2-4 page 142
•
A student started spinning slowly on the stool with
her arms out.
Her rotational inertia started off as 1.8 kg m2. She
pulled her arms in. Her rotational inertia was then 1.2
kg m2.
Her initial angular velocity was 1.6 rad/s. Calculate
her final angular velocity.
Calculate her kinetic energy before and after she
moves her arms in. Are they the same? If not explain
where the change in energy comes from or goes to.
Rolling races
You have several things that can roll
What determines how fast a thing rolls down a slope?
Is it
– mass?
- radius?
- rotational inertia?
- shape?
- whether it is hollow or filled with stuff?
- colour?
- how aesthetically pleasing it is?
- something else?
Do some informal experimentation and write down your
conclusion on the whiteboard
Rotational inertia and energy
A student started spinning slowly on the stool
with her arms out.
Her rotational inertia started off as 1.8 kg m2.
She pulled her arms in. Her rotational inertia
was then 1.2 kg m2.
Her initial angular velocity was 1.6 rad/s, and
her final angular velocity was 2.4 rad/s.
Calculate her kinetic energy before and after
she moves her arms in. Are they the same? If
not explain where the change in energy comes
from or goes to.
Review quiz
1. What does rotational inertia depend
on?
2. A solid wheel of mass 83kg and radius
0.63m is rotated by a torque of 4.3
Nm. Calculate the rotational inertia
of the disk and calculate its angular
acceleration.
3. Draw and label the forces acting on
this plane moving in a circle. Make
the size of the arrows in proportion
to each other. In a different colour
draw in the total force.
1.
Review quiz
What does rotational inertia depend on?
Mass and the distribution of mass about
the axis of rotation
2. A solid wheel of mass 83kg and radius
0.63m is rotated by a torque of 4.3 Nm.
Calculate the rotational inertia of the
disk and calculate its angular
acceleration.
I=½mr2=½x83x0.632=16.5 kgm2.
α=τ/I=4.3/32.9=0.26rad s-2
3. Draw and label the forces acting on this
plane moving in a circle. Make the size of
the arrows in proportion to each other.
In a different colour draw in the total
Total
force.
Lift
Weight/
gravity