Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Rotational Dynamics
• Kinetic Energy
• Angular Momentum

2
Reminder

• Exam 3 in class Thursday, November 17
• HW 9 due Wednesday, November 16
• Blue homework book HW - 65, 66, 72
• MPHW 6 due Friday, November 18
• - assignment will be online on Friday, November
  11

elastic energy, linear momentum, collisions,
center of mass, equilibrium of rigid bodies,
torque, rotational dynamics, angular momentum
3
Recap

• Torque ? tendency of force to cause rotation
• Angular velocity, acceleration for rigid body
  rotating about axis

4
A disk is rotating at a constant rate about a
vertical axis through its center. Point Q is
twice as far from the center as point P. The
angular velocity of Q is

• twice as big as P
• the same as P
• half as big as P
• none of the above

5
A disk is rotating at a constant rate about a
vertical axis through its center. Point Q is
twice as far from the center as point P. The
linear velocity of Q is

• twice as big as P
• the same as P
• half as big as P
• none of the above

6
Rotating disk demo
7
Computing torque
F
t? Fd Frsinq ?? (F sinq)r?

q
r
d
component force at 90o to position vector times
distance
O
8
Balancing stick demo
9
Rotational Motion
w
Particle i
ri
vi ri w at 90o to ri
Fi
pivot
mi
Newtons 2nd law
miDvi/DtFiT ? component at 900 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 Ma
F
11
Moment of Inertia
I must be defined with respect to a particular
axis
12
Demo

• Spinning a weighted bar moments of inertia

13
Moment of Inertia of Continuous Body
Dm a 0
14
Tabulated Results for Moments of Inertia of some
rigid, uniform objects
(from p.342 of University Physics, Young
Freedman)
15
Parallel-Axis Theorem
CM
Smallest I will always be along axis passing
through CM
16
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 9.2
• Use Table 9.2 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

17
Example
a
axis
Find moment of inertia of the object, consisting
of the square plate with mass ms2kg and width
a0.5m and of the disk with mass md1.5kg and
diameter equal to a, about the axis of rotation
perpendicular to the plate and going through the
center of the side of the square plate as
indicated.
a
a
18
Conditions for equilibrium of an extended object

• For an extended object that remains at rest and
  does not rotate

• The net force on the object has to be zero.
• The net torque on the object has to be zero.

19
Beam problem
CM of beam
N
r
r
rm
x
m
M?
Mb 2m
Vertical equilibrium?
Rotational equilibrium?
20
Suppose M replaced by M/2 ?
vertical equilibrium?
rotational dynamics? net torque? which way
rotates? initial angular acceleration?
21
Moment of Inertia ?
I Smiri2 replace plank by point mass situated
at CM depends on pivot position! I
Hence a??t???
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Constant angular acceleration

• Assume a is constant
• Dw/Dt a i.e (wF - wI)/t a
• wF wI at
• Now (wF wI)/2 wav if constant a

Then with qF - qI wavt
qF qI wIt 1/2 a t2
23
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 ?
10 cm
24
Rotational Kinetic Energy
K Si(1/2?mivi2 (1/2)w2Simiri2 Hence K
(1/2)Iw2 Energy rigid body possesses by virtue
of rotation
25
Simple problem
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 ?
26
cylindercable problemenergy method

• Use work-KE theorem
• work W ?
• Moment of inertia of cylinder ?
• from table
• ?

27
cylindercable problemconst 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??
FW
radius R
28
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
  .

29
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 Iw
30
Demos

• Rotating stand plus dumbbells spin faster when
  arms drawn in

31
Rotational Motion
w
Particle i
ri
vi ri w at 90o to ri
Fi
pivot
mi
Newtons 2nd law
miDvi/DtFiT ? component at 900 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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Points to note

• If text 0, L Iw is constant in time
• conservation of angular momentum
• Internal forces/torques do not contribute to
  external torque.
• L mvr if v is at 900 to r for single particle
• L r x p -- general result (x vector cross
  product)

33
The angular momentum L of a particle

• is independent of the specific choice of origin
• is zero when its position and momentum vectors
  are parallel
• is zero when its position and momentum vectors
  are perpendicular
• is zero if the speed is constant

34
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