Goals:

1
Lecture 16

• Goals

• Chapter 12
• Extend the particle model to rigid-bodies
• Understand the equilibrium of an extended
  object.
• Analyze rolling motion
• Understand rotation about a fixed axis.
• Employ conservation of angular momentum concept

• Assignment
• HW7 due March 25th
• After Spring Break Tuesday
• Catch up

2
Rotational Dynamics A childs toy, a physics
playground or a students nightmare

• A merry-go-round is spinning and we run and jump
  on it. What does it do?
• What principles would apply?
• We are standing on the rim and our friends spin
  it faster. What happens to us?
• We are standing on the rim a walk towards the
  center. Does anything change?

3
Rotational Variables

• Rotation about a fixed axis
• Consider a disk rotating aboutan axis through
  its center
• How do we describe the motion
• (Analogous to the linear case )

4
Rotational Variables...

• Recall At a point a distance R away from the
  axis of rotation, the tangential motion
• x ? R
• v ? R
• a ? R

5
Comparison to 1-D kinematics

• Angular Linear

And for a point at a distance R from the rotation
axis
x R ????????????v ? R ??????????aT ? R
Here aT refers to tangential acceleration
6
Exercise Rotational Definitions

• A friend at a party (perhaps a little tipsy) sees
  a disk spinning and says Ooh, look! Theres a
  wheel with a negative w and positive a!
• Which of the following is a true statement about
  the wheel?

• The wheel is spinning counter-clockwise and
  slowing down.
• The wheel is spinning counter-clockwise and
  speeding up.
• The wheel is spinning clockwise and slowing down.
• The wheel is spinning clockwise and speeding up

7
Example Wheel And Rope

• A wheel with radius r 0.4 m rotates freely
  about a fixed axle. There is a rope wound around
  the wheel.
• Starting from rest at t 0, the rope is
  pulled such that it has a constant acceleration
  aT 4 m/s2.
• How many revolutions has the wheel made after
  10 seconds?
• (One revolution 2? radians)

aT
r
8
Example Wheel And Rope

• A wheel with radius r 0.4 m rotates freely
  about a fixed axle. There is a rope wound around
  the wheel. Starting from rest at t 0, the rope
  is pulled such that it has a constant
  acceleration aT 4 m/s2. How many revolutions
  has the wheel made after 10 seconds?
  (One revolution 2? radians)
• Revolutions R (q - q0) / 2p and aT a r
• q q0 w0 Dt ½ a Dt2 ?
• R (q - q0) / 2p 0 ½ (aT/r) Dt2 / 2p
• R (0.5 x 10 x 100) / 6.28

aT
r
9
System of Particles (Distributed Mass)

• Until now, we have considered the behavior of
  very simple systems (one or two masses).
• But real objects have distributed mass !
• For example, consider a simple rotating disk and
  2 equal mass m plugs at distances r and 2r.
• Compare the velocities and kinetic energies at
  these two points.

w
1
2
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System of Particles (Distributed Mass)

• Twice the radius, four times the kinetic energy
• The rotation axis matters too!

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A special point for rotationSystem of Particles
Center of Mass (CM)

• If an object is not held then it will rotate
  about the center of mass.
• Center of mass Where the system is balanced !
• Building a mobile is an exercise in finding
• centers of mass.

mobile
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System of Particles Center of Mass

• How do we describe the position of a system
  made up of many parts ?
• Define the Center of Mass (average position)
• For a collection of N individual point like
  particles whose masses and positions we know

RCM
m2
m1
r2
r1
y
x
(In this case, N 2)
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Sample calculation

• Consider the following mass distribution

XCM (m x 0 2m x 12 m x 24 )/4m meters YCM
(m x 0 2m x 12 m x 0 )/4m meters XCM 12
meters YCM 6 meters
14
System of Particles Center of Mass

• For a continuous solid, convert sums to an
  integral.

dm
r
y
where dm is an infinitesimal mass element.
x
15
Connection with motion...

• So for a rigid object which rotates about its
  center of mass and whose CM is moving
• For a point p rotating

16
Rotation Kinetic Energy

• Consider the simple rotating system shown below.
  (Assume the masses are attached to the rotation
  axis by massless rigid rods).
• The kinetic energy of this system will be the sum
  of the kinetic energy of each piece
• K ½ m1v12 ½ m2v22 ½ m3v32 ½ m4v42

m4
m1
r1
?
r4
r2
m3
r3
m2
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Rotation Kinetic Energy

• Notice that v1 w r1 , v2 w r2 , v3 w r3 ,
  v4 w r4
• So we can rewrite the summation
• We recognize the quantity, moment of inertia or
  I, and write

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Calculating Moment of Inertia

• where r is the distance from the mass
  to the axis of rotation.

Example Calculate the moment of inertia of four
point masses (m) on the corners of a square whose
sides have length L, about a perpendicular axis
through the center of the square
m
m
L
m
m
19
Calculating Moment of Inertia...

• For a single object, I depends on the rotation
  axis!
• Example I1 4 m R2 4 m (21/2 L / 2)2

I1 2mL2
I 2mL2
I2 mL2
m
m
L
m
m
20
Home Exercise Moment of Inertia

• A triangular shape is made from identical balls
  and identical rigid, massless rods as shown. The
  moment of inertia about the a, b, and c axes is
  Ia, Ib, and Ic respectively.
• Which of the following is correct

(A) Ia gt Ib gt Ic (B) Ia gt Ic gt Ib (C)
Ib gt Ia gt Ic
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Home Exercise Moment of Inertia

• Ia 2 m (2L)2 Ib 3 m L2 Ic m (2L)2
• Which of the following is correct

a
(A) Ia gt Ib gt Ic (B) Ia gt Ic gt Ib (C)
Ib gt Ia gt Ic
L
b
L
c
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Moments of Inertia

• For a continuous solid object we have to add up
  the mr2 contribution for every infinitesimal mass
  element dm.
• An integral is required to find I

• Some examples of I for solid objects

• Solid disk or cylinder of mass M and radius
  R, about perpendicular axis through its center.
• I ½ M R2

Use the table
23
Moments of Inertia

• Some examples of I for solid objects

Thin hoop (or cylinder) of mass M and radius R,
about an axis through it center, perpendicular
to the plane of the hoop is just MR2
R
R
Thin hoop of mass M and radius R, about an axis
through a diameter.
24
Exercise Rotational Kinetic Energy

• We have two balls of the same mass. Ball 1 is
  attached to a 0.1 m long rope. It spins around at
  2 revolutions per second. Ball 2 is on a 0.2 m
  long rope. It spins around at 2 revolutions per
  second.
• What is the ratio of the kinetic energy
• of Ball 2 to that of Ball 1 ?

• ¼
• ½
• 1
• 2
• 4

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Exercise Rotational Kinetic Energy

• K2/K1 ½ m wr22 / ½ m wr12 0.22 / 0.12 4
• What is the ratio of the kinetic energy of Ball 2
  to that of Ball 1 ?
• (A) 1/4 (B) 1/2 (C) 1 (D) 2 (E)
  4

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Rotation Kinetic Energy...

• The kinetic energy of a rotating system looks
  similar to that of a point particle
• Point Particle Rotating System

v is linear velocity m is the mass.
? is angular velocity I is the moment of
inertia about the rotation axis.
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Moment of Inertia and Rotational Energy

• So where

• Notice that the moment of inertia I depends on
  the distribution of mass in the system.
• The further the mass is from the rotation axis,
  the bigger the moment of inertia.
• For a given object, the moment of inertia depends
  on where we choose the rotation axis (unlike the
  center of mass).
• In rotational dynamics, the moment of inertia I
  appears in the same way that mass m does in
  linear dynamics !

28
Exercise Work Energy

• Strings are wrapped around the circumference of
  two solid disks and pulled with identical forces,
  F, for the same linear distance, d. Disk 1 has a
  bigger radius, but both are identical material
  (i.e. their density r M / V is the same). Both
  disks rotate freely around axes though their
  centers, and start at rest.
• Which disk has the biggest angular velocity
  after the drop?

W F d ½ I w2 (A) Disk 1 (B) Disk 2 (C)
Same
w2
w1
F
F
start
d
finish
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Exercise Work Energy

• Strings are wrapped around the circumference of
  two solid disks and pulled with identical forces
  for the same linear distance. Disk 1 has a
  bigger radius, but both are identical material
  (i.e. their density r M/V is the same). Both
  disks rotate freely around axes though their
  centers, and start at rest.
• Which disk has the biggest angular velocity
  after the drop?

W F d ½ I1 w12 ½ I2 w22 w1 (I2 / I1)½ w2
and I2 lt I1 (A) Disk 1 (B) Disk 2 (C) Same

w2
w1
F
F
start
d
finish
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Lecture 16

• Assignment
• HW7 due March 25th
• For the next Tuesday
• Catch up

31
Lecture 16

• Assignment
• HW7 due March 25th
• After Spring Break Tuesday Catch up