PHY131H1F - Class 18

PHY131H1F - Class 18
PHY131H1F - Class 18
Today:
• Today, Chapter 11:
• Angular velocity and Angular
acceleration vectors
• Torque and the Vector Cross
Product
• Angular Momentum
• Conservation of Angular
Momentum
• Gyroscopes and Precession
Funny animation of skater from http://i.imgur.com/VgBOkoc.gif
Torque of a quick push
A student gives a quick push to
a puck that can rotate in a
horizontal circle on a frictionless
table. After the push has ended,
the puck’s angular speed
A. Steadily increases.
B. Increases for awhile, then holds steady.
C. Holds steady.
D. Decreases for awhile, then holds steady.
E. Steadily decreases.
1
Class 18 Optional Preclass Quiz on
MasteringPhysics
 This was due this morning at 10:00am
 94% of students got: A uniform disk, a uniform hoop, and a
uniform solid sphere are released at the same time at the top
of an inclined ramp. They all roll without slipping. They
reach the bottom of the ramp in the order: sphere (2/5
MR2), disk (1/2 MR2), hoop (MR2).
 Basically the larger the rotational inertia, the slower the
object rolls down the hill (because it requires more static
friction to get it rolling).
Class 18 Optional Preclass Quiz Student
Comments
 Would you rather be rick rolled everyday for the rest of your
life or curve up our midterm marks? Choose only one.
 Harlow answer: I choose the curve for sure.
 My grades are telling me to be a trophy husband, but my
looks are telling me to study harder.
 I found the right hand rule confusing and would like that to
be explained. The section about gyroscopes was interesting.
 If two people are rotating but in opposite directions, is one of
their angular momentum the negative of the other?
 Harlow answer: Correct.
2
Class 18 Optional Preclass Quiz Student
Comments
 [part 1:] Since static friction is always associated with not
moving is it impossible for static friction to do work?
 Harlow answer: Correct. Static friction from a non-moving
surface cannot do work.
 [part 2:] If the net force causing a car to move forwards is
static friction what force is doing the work on the car?
 Harlow answer: The car is doing work on itself! Chemical
energy is being converted to mechanical energy internally.
The static friction is important only to provide a rolling
without slipping constraint. Another way to look at this is that
energy is not being transferred from the road to the car.
The Vector Description of Rotational Motion
 One-dimensional motion uses a scalar velocity v
and force F.
 A more general understanding of motion
requires vectors and .
 Similarly, a more general description of rotational
motion requires us to replace the scalars  and 
with the vector quantities and .
 Doing so will lead us to the concept of angular
momentum.
3
The Angular Velocity Vector
 The magnitude of the
angular velocity
vector is .
 The angular velocity
vector points along
the axis of rotation in
the direction given by
the right-hand rule as
illustrated.
The Cross Product of Two Vectors
The scalar product (dot) is one way to multiply two
vectors, giving a scalar. A different way to multiple
two vectors, giving a vector, is called the cross
product.
If vectors and have angle  between them,
their cross product is the vector:
4
The Right-Hand Rule
The cross product is perpendicular to the plane of
and . The right-hand rule for the direction comes in
several forms. Try them all to see which works best
for you.
Note that
Instead,
.
.
The Torque Vector
We earlier defined torque τ = rFsinϕ. r and F are
the magnitudes of vectors, so this is a really a
cross product:
A tire wrench
exerts a torque
on the lug nuts.
5
Angular Momentum of a Particle
A particle of mass m is moving. The particle’s
momentum vector makes an angle  with the
position vector.
Angular Momentum of a Particle
Why this definition?
If you take the time derivative of and use the
definition of the torque vector, you find:
Torque causes a particle’s angular momentum
to change. This is the rotational equivalent of
and is a general statement of Newton’s
second law for rotation.
6
Angular Momentum of a Rigid Body
For a rigid body, we can add
the angular momenta of all
the particles forming the
object. If the object rotates
 on a fixed axle, or
 about an axis of
symmetry
then it can be shown that
And it’s still the case that
.
Angular Momentum
• Angular momentum
 rotational inertia  rotational velocity
𝐿 = 𝐼𝜔
– This is analogous to
Linear momentum  mass  velocity
𝑝 = 𝑚𝑣
7
• A bicycle is traveling toward the right.
• What is the direction of the angular
momentum of the wheels?
A. left
B. right
C. into page
D. out of page
E. up
Conservation of Angular Momentum
An isolated system that experiences no net torque
has
and thus the angular momentum vector
constant.
is a
8
The Law of Conservation of Momentum
• If there is no net external force on a
system, then its momentum is a constant.
The Law of Conservation of Energy
• If there is no work or heat being
exchanged with a system and its
surroundings, then its energy is constant.
The Law of Conservation of Angular Momentum
• If there is no net external torque on a
system, then its angular momentum is a
constant.
Angular Momentum
CHECK YOUR NEIGHBOR
Suppose you are swirling a can around and suddenly
decide to pull the rope in halfway; by what factor would
the speed of the can change?
A. Double
B. Four times
C. Half
D. One-quarter
9
Angular Momentum
CHECK YOUR NEIGHBOR
Suppose you are swirling a can around and suddenly
decide to pull the rope in halfway; by what factor would
the speed of the can change?
A. Double
B. Four times
C. Half
D. One-quarter
Conservation of Angular Momentum
Example:
• When the professor pulls the weights
inward, his rotational speed increases!
10
Linear / Rotational Analogy
Linear
  
• s, v, a

• Force: F
• Mass: m
• Newton’s
2nd law:

 F
a  net
m
Rotational Analogy
• θ, ω, α
• Torque: τ
• Rotational Inertia: I
• Kinetic
energy:
1
K cm  mv 2
2


• Momentum: p  mv


 net
I
1
K rot  I 2
2


L  I

Two buckets spin around in
a horizontal circle on
frictionless bearings.
Suddenly, it starts to rain.
As a result,
A.The buckets speed up because the potential energy
of the rain is transformed into kinetic energy.
B.The buckets continue to rotate at constant angular
velocity because the rain is falling vertically while the
buckets move in a horizontal plane.
C.The buckets slow down because the angular
momentum of the bucket + rain system is conserved.
D.The buckets continue to rotate at constant angular
velocity because the total mechanical energy of the
bucket + rain system is conserved.
E.None of the above.
11
EXAMPLE 12.19 Two Interacting Disks
Precession of a Gyroscope
 Consider a horizontal
gyroscope, with the disk
spinning in a vertical plane,
that is supported at only one
end of its axle, as shown.
 You would expect it to simply
fall over—but it doesn’t.
 Instead, the axle remains horizontal, parallel to the ground,
while the entire gyroscope slowly rotates in a horizontal plane.
 This steady change in the orientation of the rotation axis is
called precession, and we say that the gyroscope precesses
about its point of support.
 The precession frequency Ω is much less that the disk’s
rotation frequency ω.
12
Gravity on a Nonspinning Gyroscope
 Shown is a nonspinning
gyroscope.
 When it is released, the net
torque is entirely
gravitational torque.
 Initially, the angular
momentum is zero.
 Gravity acts to increase the
angular momentum gradually
in the direction of the torque,
which is the -direction.
 This causes the gyroscope
to rotate around x and fall.
Gravity on a Spinning Gyroscope
 Shown is a gyroscope initially
spinning around the z-axis.
 Initially, gravity acts to
increase the angular
momentum slightly in the
direction of the torque, which
is the -direction.
 This causes the gyroscopes
angular momentum to shift
slightly in the horizontal plane.
 The gravitational torque vector
is always perpendicular to the
axle, so dL is always
perpendicular to L.
13
Precession of a Gyroscope
 The precession frequency of a gyroscope, in rad/s, is
Ω=
𝑀𝑔𝑑
𝐼𝜔
 Here M is the mass of the gyroscope, I is its
rotational inertia, and d is the horizontal distance of
the center of mass from the support point.
 The angular velocity of the spinning gyroscope is
assumed to be much larger than the precession
frequency; ω >> Ω.
Nuclear Magnetic Resonance
 A proton in the nucleus of an atom is like a little spinning top.
 When placed in a strong static magnetic field, the magnetic
force produces a torque on the proton, which causes it to
precess.
 The precession frequency is in the radio-frequency range,
which allows the proton to absorb and re-emit radio-waves.
 This allows doctors to image inside the human body using
completely harmless radio waves.
14
Before Class 19 on Monday
 Please read chapter 12 on Static Equilibrium. The
preclass quiz is due Monday morning.
 Problem Set 8 on Chapters 10 and 11 is due
Tuesday Nov. 24 at 11:59pm. Problem sets have
Tuesday deadlines now!
 Something to think about over the weekend: The
supports to the diving board provide a vertical force
on the board so the girl will not fall. What are the
directions of the force on the board at point 1 and
point 2: up or down? Why?
1
2
15
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