Download The Vector Product Defined Ch 11: Question 3

The Vector Product and Torque
•
•
•
The torque vector lies in
a direction
perpendicular to the
plane formed by the
position vector and the
force vector
τ=rxF
The torque is the vector
(or cross) product of the
position vector and the
force vector
More About the Vector Product
•
•
The direction of C is
perpendicular to the plane
formed by A and B
Direction is given by the
right-hand rule
•
•
•
•
fingers in direction of A
curl into direction of B
thumb gives direction of crossproduct C
Be sure to use your right hand!
The Vector Product Defined
•
•
•
•
Given two vectors, A and B
The vector (cross) product of A and B is
defined as a third vector, C
C is read as “A cross B”
The magnitude of C is AB sin θ
•
θ is the angle between A and B
Ch 11: Question 3
•
Vector A is in the negative y direction,
and vector B is in the negative x
direction. What are the directions of
•
•
AxB
BxA
1
Ch 11: Question 2
•
•
Is the triple product, defined by A·(BxC)
a scalar or a vector quantity?
Explain why the operation (A·B)xC has
no meaning.
Torque and Angular Momentum
•
The torque is related to the angular
momentum
•
Similar to the way force is related to linear
momentum
Angular Momentum
•
The instantaneous angular
momentum L of a particle
relative to the origin O is
defined as the cross product
of the particle’s
instantaneous position vector
r and its instantaneous linear
momentum p
More About Angular Momentum
•
•
•
The SI units of angular momentum are
(kg. m2)/ s
Both the magnitude and direction of L depend
on the choice of origin
The magnitude of L = mvr sin φ
•
•
This is the rotational analog of Newton’s
Second Law
•
•
•
φ is the angle between p and r
The direction of L is perpendicular to the
plane formed by r and p
Στ and L must be measured about the same origin
This is valid for any origin fixed in an inertial frame
2
Angular Momentum of a
Rotating Rigid Object
Angular Momentum of a Particle,
Example
•
•
The vector L = r x p is
pointed out of the diagram
The magnitude is
L = mvr sin 90o = mvr
•
•
Each particle of the
object rotates in the xy
plane about the z axis
with an angular speed of
ω
•
sin 90o is used since v is
perpendicular to r
A particle in uniform circular
motion has a constant
angular momentum about an
axis through the center of its
path
The angular momentum
of an individual particle
is Li = mi ri2 ω
L and ω are directed
along the z axis
•
•
Angular Momentum of a
Rotating Rigid Object, cont
•
To find the angular momentum of the
entire object, add the angular momenta
of all the individual particles
Angular Momentum of a
Bowling Ball
•
•
•
This also gives the rotational form of
Newton’s Second Law
•
The momentum of
inertia of the ball is
2/5MR 2
The angular momentum
of the ball is Lz = Iω
The direction of the
angular momentum is in
the positive z direction
3
Conservation of Angular
Momentum
•
The total angular momentum of a system is
constant in both magnitude and direction if
the resultant external torque acting on the
system is zero
•
•
Conservation of Angular Momentum
If the mass of an isolated system undergoes
redistribution, the moment of inertia changes
•
•
Net torque = 0 -> means that the system is
isolated
Ii ωi = If ωf
Ltot = constant or
•
Li = Lf
•
Ch 11: Question 12
•
Often when a high diver wants to turn a
flip in midair, she draws her legs up
against her chest. Why does this make
her rotate faster? What should she do
when she wants to come out of her flip?
The conservation of angular momentum requires a
compensating change in the angular velocity
This holds for rotation about a fixed axis and for rotation
about an axis through the center of mass of a moving
system
The net torque must be zero in any case
Ch 11: Question 14
•
Stars originate as large bodies of slowly
rotating gas. Because of gravitation,
these clumps of gas slowly decrease in
size. What happens to the angular
speed of a star as it shrinks? Explain.
4
Conservation of Angular Momentum:
The Merry-Go-Round
Conservation Law Summary
For an isolated system (1) Conservation of Energy:
•
•
•
Ei = Ef
(2) Conservation of Linear Momentum:
•
pi = pf
(3) Conservation of Angular Momentum:
•
Li = Lf
The moment of inertia of the
system is the moment of
inertia of the platform plus
the moment of inertia of the
person
•
•
As the person moves toward
the center of the rotating
platform, the angular speed
will increase
•
Ch 11: Question 13
•
In some motorcycle races, the riders
drive over small hills, and the
motorcycle becomes airborne for a
short time. If the motorcycle racer
keeps the throttle open while leaving
the hill and going into the air, the
motocycle tends to nose upward. Why
does this happen?
Assume the person can be
treated as a particle
To keep L constant
Ch 11: Question 15
•
If global warming occurs over the next
century, it is likely that some polar ice
will melt and the water will be
distributed closer to the Equator. How
would this change the moment of inertia
of the Earth? Would the length of the
day increase or decrease?
5
Motion of a Top
•
•
•
•
The only external forces
acting on the top are the
normal force n and the
gravitational force M g
The direction of the angular
momentum L is along the
axis of symmetry
The right-hand rule indicates
that τ = r × F = r × M g is in
the xy plane
Causes precession, slow
rotation of top about z axis
Angular Momentum as a
Fundamental Quantity
•
•
•
The concept of angular momentum is also
valid on a submicroscopic scale
Angular momentum has been used in the
development of modern theories of atomic,
molecular and nuclear physics
In these systems, the angular momentum has
been found to be a fundamental quantity
•
•
Fundamental here means that it is an intrinsic
property of these objects
It is a part of their nature
Gyroscope in a Spacecraft
•
•
•
The angular momentum of
the spacecraft about its
center of mass is zero
A gyroscope is set into
rotation, giving it a nonzero
angular momentum
The spacecraft rotates in the
direction opposite to that of
the gyroscope so that the
total momentum of the
system remains zero.
Can also use gyroscopes to keep a spacecraft
pointed in a set direction (Hubble Space Telescope).
Fundamental Angular Momentum
•
Angular momentum has discrete values
•
•
•
quantized
These discrete values are multiples of a
fundamental unit of angular momentum
The fundamental unit of angular momentum
is h-bar
•
Where h is called Planck’s constant
6
Classical Ideas in Subatomic
Systems
•
Certain classical concepts and models
are useful in describing some features
of atomic and molecular systems
•
•
Proper modifications must be made
A wide variety of subatomic phenomena
can be explained by assuming discrete
values of the angular momentum
associated with a particular type of
motion
Niels Bohr
•
•
Niels Bohr was a Danish physicist
He adopted the (then radical) idea of
discrete angular momentum values in
developing his theory of the hydrogen
atom
•
Classical models were unsuccessful in
describing many aspects of the atom
Bohr’s Hydrogen Atom
•
The electron could occupy only those
circular orbits for which the orbital
angular momentum was equal to n h
•
•
where n is an integer
This means that orbital angular
momentum is quantized
7