Physics 211: Lecture 18 Todays Agenda

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Physics 211 Lecture 18Todays Agenda

• More about rolling
• Direction and the right hand rule
• Rotational dynamics and torque
• Work and energy with example

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Rotational v.s. Linear Kinematics

• Angular Linear

And for a point at a distance R from the rotation
axis
x ?R???????????v ?R??????????a ?R
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Rolling Motion
Roll objects down ramp

• Objects of different I rolling down an inclined
  plane

?K - ?U Mgh
v 0 ?? 0 K 0
R
M
h
v ?R
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Rolling...

• If there is no slipping

v
?
v
2v
v
Where v ?R
In the lab reference frame
In the CM reference frame
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Rolling...
hoop c 1 disk c 1/2 sphere c
2/5 etc...
Use v ?R and I cMR2 .
c
c
So
c
c
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Direction of Rotation

• In general, the rotation variables are vectors
  (have direction)
• If the plane of rotation is in the x-y plane,
  then the convention is
• CCW rotation is in the z direction
• CW rotation is in the - z direction

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Direction of RotationThe Right Hand Rule

• To figure out in which direction the rotation
  vector points, curl the fingers of your right
  hand the same way the object turns, and your
  thumb will point in the direction of the rotation
  vector!
• We normally pick the z-axis to be the rotation
  axis as shown.
• ??? ?z
• ?? ?z
• ?? ?z
• For simplicity we omit the subscripts unless
  explicitly needed.

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Example

• A flywheel spins with an initial angular velocity
  ?0 500 rad/s. At t 0 it starts to slow down
  at a rate of 0.5 rad/s2. How long does it take
  to stop?

?

• Use to find when ? 0
  

• So in this case

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Lecture 18, Act 1Rotations

• A ball rolls across the floor, and then starts up
  a ramp as shown below. In what direction does
  the angular acceleration vector point when the
  ball is on the ramp?

(a) down the ramp (b) into the page (c) out
of the page
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Lecture 18, Act 1Solution

• Using your right hand rule, a is out of the page!

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Rotational DynamicsWhat makes it spin?

• Suppose a force acts on a mass constrained to
  move in a circle. Consider its acceleration in
  the direction at some instant
• a? ?r
• Now use Newtons 2nd Law in the ?direction
• F? ma? m?r
• rF? mr2?

F

F?
a?
m
r

• Multiply by r

?
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Rotational DynamicsWhat makes it spin?

• rF? mr2? use
• Define torque ? rF?.
• ? is the tangential force F?times the lever arm
  r.
• Torque has a direction
• z if it tries to make the systemspin CCW.
• - z if it tries to make the systemspin CW.

F
F?
a?
m
r
?
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Rotational DynamicsWhat makes it spin?

• So for a collection of many particles arranged
  in a rigid configuration

?i
I
m4
F1
m1
F4
r1
?
r4
m3
r2
r3
m2
F2
F3
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Rotational DynamicsWhat makes it spin?

• ????????????????????? ?NET
  I???????????

• This is the rotational analogue of FNET ma
  
• Torque is the rotational analogue of force
• The amount of twist provided by a force.
• Moment of inertia I is the rotational analogue of
  mass.
• If I is big, more torque is required to achieve
  a given angular acceleration.
• Torque has units of kg m2/s2 (kg m/s2) m Nm.

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Torque

• ? rF?

• Recall the definition of torque

• r F sin ?
• r sin ? F
• ? rpF
• Equivalent definitions!

rp distance of closest approach
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Torque

• ? r Fsin ?
• So if ? 0o, then ? 0
• And if ? 90o, then ? maximum

F
r
F
r
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Lecture 18, Act 2Torque

• In which of the cases shown below is the torque
  provided by the applied force about the rotation
  axis biggest? In both cases the magnitude and
  direction of the applied force is the same.

(a) case 1 (b) case 2 (c) same
L
F
F
L
axis
case 1
case 2
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Lecture 18, Act 2Solution

• Torque F x (distance of closest approach)

• The applied force is the same.
• The distance of closest approach is the same.

F
F
L
L
case 1
case 2
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Torque and the Right Hand Rule

• The right hand rule can tell you the direction of
  torque
• Point your hand along the direction from the axis
  to the point where the force is applied.
• Curl your fingers in the direction of the force.
• Your thumb will point in the directionof the
  torque.

F
y
r
x
?
z
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The Cross Product

• We can describe the vectorial nature of torque in
  a compact form by introducing the cross
  product.
• The cross product of two vectors is a third
  vector
• A X B C
• The length of C is given by
• C AB sin ?
• The direction of C is perpendicular to the plane
  defined by A and B, and inthe direction defined
  by the right handrule.

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The Cross Product

• Cartesian components of the cross product
• C A X B
• CX AY BZ - BY AZ
• CY AZ BX - BZ AX
• CZ AX BY - BX AY

B
A
C
Note B X A - A X B
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Torque the Cross Product

• So we can define torque as
• ? r X F
• rF sin ?
• ?X rY FZ - FY rZ y FZ - FY z
• ?Y rZ FX - FZ rX z FX - FZ x
• ?Z rX FY - FX rY x FY - FX y

F
?
r
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Comment on ? I?

• When we write ? I? we are really talking about
  the z component of a more general vector
  equation. (Recall that we normally choose the
  z-axis to be the the rotation axis.)
• ?z Iz?z
• We usually omit the z subscript for simplicity.

?z
Iz
z
?z
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Example

• To loosen a stuck nut, a (stupid) man pulls at an
  angle of 45o on the end of a 50 cm wrench with a
  force of 200 N.
• What is the magnitude of the torque on the nut?
• If the nut suddenly turns freely, what is the
  angular acceleration of the wrench? (The
  wrenchhas a mass of 3 kg, and its shape is that
  of a thin rod).

45o
F 200 N
L 0.5 m
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Example
Wrench w/ bolts

• Torque ? LFsin ? (0.5 m)(200 N)(sin 45)
  70.7 Nm

• If the nut turns freely, ? I?
• We know ? and we want ?, so we need to figure
  out I.

45o
F 200 N
L 0.5m
?
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Work

• Consider the work done by a force F acting on an
  object constrained to move around a fixed axis.
  For an infinitesimal angular displacement d?
• dW F.dr FR d? cos(?)
• FR d? cos(90-?)
• FR d? sin(?)
• FR sin(?) d?
• ? ?dW ? d?
• We can integrate this to find W ??
• Analogue of W F ?r
• W will be negative if ? and ? have opposite
  signs!

?
F
?
R
dr R d?
d?
axis
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Work Kinetic Energy

• Recall the Work/Kinetic Energy Theorem ?K
  WNET
• This is true in general, and hence applies to
  rotational motion as well as linear motion.
• So for an object that rotates about a fixed axis

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Example Disk String

• A massless string is wrapped 10 times around a
  disk of mass M 40 g and radius R 10 cm. The
  disk is constrained to rotate without friction
  about a fixed axis though its center. The string
  is pulled with a force F 10 N until it has
  unwound. (Assume the string does not slip, and
  that the disk is initially not spinning).
• How fast is the disk spinning after the string
  has unwound?

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Disk String...

• The work done is W ???
• The torque is ?? RF (since ? 90o)
• The angular displacement ? is2? rad/rev x 10 rev.

• So W (.1 m)(10 N)(20??rad) 62.8 J

??
?
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Disk String...
Flywheel, pulley, mass

• WNET W 62.8 J ?K

M
R
?
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Lecture 18, Act 3Work Energy

• Strings are wrapped around the circumference of
  two solid disks and pulled with identical forces
  for the same distance. Disk 1 has a bigger
  radius, but both have the same moment of inertia.
  Both disks rotate freely around axes though
  their centers, and start at rest.
• Which disk has the biggest angular velocity after
  the pull ?

w2
w1
(a) disk 1 (b) disk 2 (c) same
F
F
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Lecture 18, Act 3Solution

• The work done on both disks is the same!
• W Fd

• The change in kinetic energy of each will
  therefore also be the same since W DK.

w2
w1
F
F
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Spinning Disk Demo
I

• We can test this with our big flywheel.

negligiblein this case
m
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Recap of todays lecture

• More about rolling
• Direction and the right hand rule
• Rotational dynamics and torque
• Work and energy with example