accounting for angular momentum chapter 21. objectives understand the basic fundamentals behind...
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Objectives Understand the basic fundamentals behind
angular momentum Be able to define measures of rotary
motion and measures of rotation
Accounting for Angular Momentum Linear momentum deals with objects
moving in a straight line. Angular momentum deals with objects that
rotate or orbit.
Angular MotionConsider a person on a
spinning carnival ride.
Position, velocity, and acceleration are a function of angular, rather than linear measurements.
r
q1
q2
Timepasses
Angular Position and Speed Position can be described by q
Angular displacement, = Dq q2 - q1
Average angular speed is
Instantaneous angular speed is
r
s
radius
length arc
ttt
12
12
dt
d
tt
lim0
Angular Acceleration Average angular acceleration is
Instantaneous angular acceleration is
ttt
12
12
dt
d
tt
lim0
Analogy to Linear Motion Linear
x = xo + vot + ½aot2
dx/dt = v = vo + aot
dv/dt = a = ao
Angular
q = qo + wot + ½aot2
dq/dt = w = wo + aot
dw/dt = a = ao
Pair Exercise #1 A carnival ride is rotating at 2.0 rad/s. An
external torque is applied that slows the ride down at a rate of –0.05 rad/s2. How long does it take the ride to come to
rest? How many revolutions does it make while
coming to rest?
Measures of RotationFrequency, f is the number of cycles (revolutions) per second
Unit: Herzt or Hz = 1 cycle/second
n revolutions sweep out q radians. There are 2p radians per cycle (revolution). Therefore
Frequency is given by
πn
n
π 2radians 2
revolution 1
dtd
dtd
dtdn
f2
121
2
f 2
Angular frequency, ω, is the amount of radians per second.
Measures of RotationPeriod, T, is the time it takes to complete one
cycle (seconds per revolution)
21
f
T
Pairs Exercise #2
A carnival ride completes 2 revolutions per second. What is its frequency, period, and angular velocity?
More Basic Equations Speed (magnitude of velocity) is
Using some basic calculus and algebra you can find (Section 20.1.3):
rdt
dsv VSpeed
22
rr
v
dt
dva
Centripetal Forces
Applying Newton’s second law:
This is the inward force required to keep a mass in a circular orbit. If the force stops being applied, the mass will fly off tangentially.
22
mrr
vmmaF
Centrifugal Force
From the Inertial Frame The ball is seen to rotate. Centripetal force keep the ball rotating
From a co-rotating frame: a rotating object appears to have an outward force acting on it. This fictitious centrifugal force has the same magnitude as the centripetal force, but acts in the opposite direction.
The centripetal (not the centrifugal) force can be used to design a centrifuge, which separates materials based upon density differences.
Angular momentum (particles) Angular momentum (L) is a vector quantity The direction is perpendicular to the plane of the
orbit and follows the right hand rule
2mrrrmrmvL
Moment of Inertia
The mass moment of inertia is define as:
Thus angular momentum can be expressed as:
Which is analogous to linear momentum:
2mrI
IL
mvp
Angular Momentum for BodiesFinding the angular momentum of a rotating body requires you to integrate over the volume of the body. (see Figure 21.7)
L
dhddrrdL
dhdrrdrrdL
dmrvdL
0
H
0
2
0
R
0
3
Angular Momentum for Bodies This gets messy for most shapes (even simple ones),
so tables with moments of inertia have be developed for standard shapes (see Table 21.2).
From AutoCAD you can use the moment of inertia from Mass Properties times the density of the material.
Once you have the moment of inertia, the angular momentum is found from the formula:
IL
Parallel Axis Theorem Because we want properties about an axis
other than the center of mass, we must use the Parallel Axis Theorem, which states that
Iaxis=Icg+MD2
where Iaxis and ICG are parallel and D is the distance between them. M is the mass of the part.
Using the Theorem With the mass and CG known, and the desired
axis known, we can find the correct moment of inertia.
The axis we want is through the point 55, 25 mm. Because the CG’s X and Y are 35.54, 25 mm, the distance between them is 19.46 mm.
So the the correct moment is 275.46 kg•mm2 + 0.401 kg • (19.46 mm)2
= 427.4 kg•mm2
Torque Torque is a twisting force. It is created by
applying a force to an object in an attempt to make the object rotate. (see Figure 21.8)
Like momentum, it is a vector quantity.
rFT
Conservation of Angular Momentum Angular momentum is a conserved
quantity. Changing angular momentum of a system
Changing mass
(see Figure 21.9) Applying an external torque
Torque and Newton When Torque is applied to an object, it changes
the angular momentum of the object in a manner similar to a force applied to a body changes its linear momentum.
ma
dt
dvm
dt
mvd
dt
dpF
Idt
dI
dt
Id
dt
dLT
Pair Exercise #3 The previous AutoCAD part is to be spun
by applying a torque of 0.001N·m to the hole for 10 seconds.
What is its angular velocity (in rpm’s) at the end of the 10-s period?
Systems without Net Momentum Input
Many times you do not have unbalanced torque or mass transfer. In this case the UAE simplifies to:
Final Amount=Initial Amount This is how ice skaters spin faster.
Pair Exercise #4 A space satellite has an electric motor in it
with a flywheel attached (see Figure 21.11). The motor causes the flywheel to rotate at 10 rpm. If the moment of inertia of the flywheel is 10 kg·m2 and the satellite is 10000 kg·m2, how long must the motor run to twist the satellite by 10 degrees?