Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Ch. 12
• Torque
• Rotational energy
• Rolling
• Angular momentum
• FHW15 11, 13, 29, 44, 45, 63, 64, 69, 70, 71
• Final Exam Monday 12/8/2008 1015 1215 Stolkin

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Rotations about fixed axis

• Linear speed v (2pr)/T ?r. Quantity ? is
  called angular velocity
• ? is a vector! Use right hand rule to find
  direction of ?.
• Angular acceleration a ?d?/dt is also a vector!
• ? and ? parallel ? angular speed increasing
• ? and ? antiparallel ? angular speed decreasing

3
Relating linear and angular kinematics

• Linear speed v (2pr)/T ?r
• Tangential acceleration atan r?
• Radial acceleration arad v2/r ?2r

4
Rotational Motion
w
Particle i
Fi
ri
vi ri w at 90º to ri
pivot
mi
Newtons 2nd law
miDvi/Dt FiT ? component at 90º to ri
Substitute for vi and multiply by ri
miri2Dw/Dt FiT ri ti
Finally, sum over all masses
?Dw/Dt) S miri2 Sti tnet
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Discussion
?Dw/Dt) S miri2 tnet
a - angular acceleration
Moment of inertia, I
I a tnet compare this with Newtons 2nd law M
a Fnet
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Moment of Inertia
I must be defined with respect to a particular
axis
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Moment of Inertia of Continuous Body
Dm a 0
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Parallel-Axis Theorem
CM
Smallest I will always be along axis passing
through CM
10
Practical Comments on Calculation of Moment of
Inertia for Complex Object

• To find I for a complex object, split it into
  simple geometrical shapes that can be found in
  Table 12.2
• Use Table 12.2 to get ICM for each part about
  the axis parallel to the axis of rotation and
  going through the center-of-mass
• If needed use parallel-axis theorem to get I for
  each part about the axis of rotation
• Add up moments of inertia of all parts

11
Beam resting on pivot
CM of beam
N
r
r
rm
x
m
M ?
Mb 2m
SF
Vertical equilibrium?
S?
Rotational equilibrium?
M N
12
Suppose M replaced by M/2 ?
SF

• vertical equilibrium?

• rotational dynamics?
• net torque?
• which way rotates?
• initial angular acceleration?

S?
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Moment of Inertia?
I Smiri2 depends on pivot position! I
Hence a??t???
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Constant angular acceleration

• Assume a is constant
• ??/?t a i.e., (?f - ?i)/t a
• ?f ?i at
• Now (?f ?i)/2 ?av if constant a

Then with qf - qi wavt
qf qi wit (1/2) a t2
15
Problem slowing a DVD
wI 27.5 rad/s, a? -10.0 rad/s2

• how many revolutions per second?
• linear speed of point on rim?
• angular velocity at t 0.3s ?
• when will it stop?

27.5 rad/s (1 rev/2? rad)
10 cm
vt ?r
?f ?i ?t
t (?f - ?i)/?
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Rotational Kinetic Energy

• K Si(1/2?mivi2 (1/2)w2Simiri2
• Hence
• K (1/2)I w2
• This is the energy that a rigid body possesses
  by virtue of rotation

17
Spinning a cylinder
Cable wrapped around cylinder. Pull off
with constant force F. Suppose unwind a distance
d of cable
2R
F

• What is final angular speed of cylinder?

• Use work-KE theorem
• W Fd Kf (1/2)I ?2
• Mom. of inertia of cyl.? -- from table (1/2)mR2
• from table (1/2)mR2
• ? 2Fd/(mR2/2)1/2 4Fd/(mR2)1/2

18
cylindercable problem -- constant acceleration
method
extended free body diagram
N
F
no torque due to N or FW why direction of N
? torque due to t? FR hence a?
FR/(1/2)MR2 2F/(MR) ? w????t
4Fd/(MR2) 1/2
FW
radius R
?? (1/2)?t2 d/R t (MR/F)(d/R)1/2
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Angular Momentum

• can define rotational analog of linear
• momentum called angular momentum
• in absence of external torque it will be
  conserved in time
• True even in situations where Newtons laws fail
  .

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Definition of Angular Momentum
Back to slide on rotational dynamics miri2Dw/D
t ti Rewrite, using li miri2w?? Dli/Dt
ti Summing over all particles in body DL/Dt
text L angular momentum I w
w
Fi
ri
pivot
mi
21
An ice skater spins about a vertical axis through
her body with her arms held out. As she draws her
arms in, her angular velocity

• 1. increases
• 2. decreases
• 3. remains the same
• 4. need more information

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Angular Momentum 1.
q
r
p
O
Point particle L rpsin(q) mrv
sin(q) vector form ? L r x p direction
of L given by right hand rule (into paper here)
L mvr if v is at 900 to r for single particle
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Angular Momentum 2.
w
o
rigid body L I w?(fixed axis of
rotation) direction along axis into paper
here
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Rotational Dynamics

• t Ia
• ?L/ ?t t
• These are equivalent statements
• If no net external torque t ?0
• L is constant in time
• Conservation of Angular Momentum
• Internal forces/torques do not contribute
• to external torque.

25
Bicycle wheel demo

• Spin wheel, then step onto platform
• Apply force to tilt axle of wheel

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Linear and rotational motion

• Torque
• Angular acceleration
• Angular momentum
• Kinetic energy

• Force
• Acceleration
• Momentum
• Kinetic energy

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General motion of extended objects

• Net force ? acceleration of CM
• Net torque about CM ? angular acceleration
  (rotation) about CM
• Resultant motion is superposition of these two
  motions
• Total kinetic energy K KCM Krot

28
A hammer is held horizontally and then released.
Which way will it fall?
29
Three identical rectangular blocks are at rest on
a flat, frictionless table. The same force is
exerted on each of the three blocks for a very
short time interval. The force is exerted at a
different point on each block, as shown. After
the force has stopped acting on each block, which
block will spin the fastest?

• 1. A.
• 2. B.
• 3. C.
• 4. A and C.

Top-view diagram
30
Three identical rectangular blocks are at rest on
a flat, frictionless table. The same force is
exerted on each of the three blocks for a very
short time interval. The force is exerted at a
different point on each block, as shown. After
each force has stopped acting, which blocks
center of mass will have the greatest speed?

• 1. A.
• 2. B.
• 3. C.
• 4. A, B, and C have the same C.O.M. speed.

Top-view diagram
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Rolling without slipping
translation
rotation
vcm
acm
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Rolling without slipping
N
?F maCM ?? Ia Now aCM Ra if no
slipping So, m aCM and F
F
W
q
33
A ribbon is wound up on a spool. A person pulls
the ribbon as shown. Will the spool move to the
left, to the right, or will it not move at all?

• 1. The spool will move to the left.
• 2. The spool will move to the right.
• 3. The spool will not move at all.

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A ribbon is wound up on a spool. A person pulls
the ribbon as shown. Will the spool move to the
left, to the right, or will it not move at all?

• 1. The spool will move to the left.
• 2. The spool will move to the right.
• 3. The spool will not move at all.

35
A ribbon is wound up on a spool. A person pulls
the ribbon as shown. Will the spool move to the
left, to the right, or will it not move at all?

• 1. The spool will move to the left.
• 2. The spool will move to the right.
• 3. The spool will not move at all.

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Pulling ribbon at a special angle
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