Download Angular Kinetics II

Angular Kinetics II
KIN 335 Spring 2005
What you should know
• Definition of Moment of Inertia
• Moment of inertia and movement
performance.
• Definition of Angular Momentum
• Angular impulse-momentum relationship
• Newton’s law of angular motions.
• Comparison between linear and angular
kinetics
Moment of inertia
• Inertia
– The property of an object that resists ______ in motion
– Linear inertia Î quantified as an object’s _______
– More massive object Î Difficult to speed up, slow
down, or change the direction
– Angular inertia Î quantified as an object’s
________________
– More angular inertia Î Difficult to speed up or slow
down the rotation or change the axis of rotation of an
object
Moment of inertia
• Moment of inertia
– The quantity that describes angular inertia
Ia = _________
¾ Ia = moment of inertia about axis a
¾ mi = mass of particle i
¾ ri = radius (distance) from particle i to axis of rotation a
– Unit : kg·m2
Ia = __________
¾ mi = mass of particle i
¾ ka = radius of gyration about axis a
¾ radius of gyration : a length measurement that represents how
far from the axis of rotation all of the object’s mass must be
concentrated to create the same resistance to change in angular
motion as the object had in its original shape.
Characteristics of Moment of inertia
• It’s function of _____ and __________________
• More sensitive to the radius. Why?
• Objects may have many different moments of
inertia because an object may rotate about many
different axes of rotation.
• A human’s moment of inertia about any axis is
________ Îmore than one value depending on
the body postures.
• Figure skater, divers, gymnasts, skaters, long
jumper, …..
Corking a wooden bat?
• The illegal practice of cutting out a cylinder of wood from
the barrel of the bat and replacing it with lighter material.
The effect is to decrease the bat's mass and moment of
inertia about an axis through its handle - bat can be
accelerated more effectively.
• tuck vs. layout position of
a diver or gymnast
– Athlete rotates about an
axis through his/her center
of gravity. Tuck concentrates body mass
closer to axis; Layout body mass concentrated
away from axis; changes in
moment of inertia
ultimately affect body's rate
of rotation (see
conservation of angular
momentum discussions)
• changes in position of a
runner's leg during swing
phase
– Hip represents an important
axis of rotation for the entire
leg as it is swung forward.
Runner naturally tucks leg
(flexes knee) early in swing the effect is to concentrate the
mass of the leg more closely to
the hip axis, thereby reducing
the moment of inertia of the
leg about an axis through the
hip. Advantage - less
resistance to angular
acceleration of the leg during
swing.
Angular Momentum
• Momentum
– Quantity of motion
¾ “…Chicago Bulls Gaining Momentum…”
– Linear motion Î L= ____ (unit : kg·m/s)
– Angular motion Î H= I·ω (unit : _______)
¾ Quantity of __________________
¾ Function of moment of inertia and angular velocity
¾ Vector quantity
• Angular momentum of human body
– The sum of the angular momentum of all the body
segments gives an approximation of the angular
momentum of the entire body.
Newton’s 1st law
• For linear motion
– Every body continues in its state of rest, or of uniform
motion in a straight line, unless it is compelled to
change that state by force impressed upon it. Î Law of
inertia
• For angular motion
– The angular momentum of an object remains ________
unless a net external torque is exerted on it.
(ΣT)⋅(∆t) =∆(I⋅ω)Î 0 =∆(I⋅ω) Î _______________
– Controlling _________________ Î speed up or slow
down angular velocity.
– Conservation of angular momentum – figure skater,
diver, gymnasts,….
• Backward one and a
half diving
• Layout –
Tuck – Layout
Newton’s 2nd law
• For linear motion
– If a net external force is exerted on an object, the object
will accelerate in the direction of the net external force,
and its acceleration will be proportional to the net
external force and inversely proportional to its mass.
Σ F = m⋅a
• For angular motion
– If a net external torque is exerted on an object, the
object will accelerate angularly in the direction of the
net external torque, and its angular acceleration will be
directly proportional to the net external torque and
inversely proportional to its moment of inertia.
_________________
Impulse-Momentum Relationship
• Rewriting of Newton’s 2nd Law
ΣF = m⋅a Î ΣF = m⋅∆v/∆t Î(ΣF)⋅(∆t) = m⋅∆v
– Impulse = (ΣF)⋅(∆t)
– Change of momentum = m⋅∆v
• Angular motion
– Angular Impulse = ____________
– Change of angular momentum = ________ = I⋅ ∆ω
(ΣT)⋅(∆t) =∆(I⋅ω)=I⋅∆ω ÎΣT =I⋅(∆ω/∆t)Î ΣT = I⋅α
• Force = the rate of change of _______ momentum
• Torque = the rate of change of _______ momentum
Newton’s 3rd law
• For linear motion
– For every force exerted by one body on another, the
other body exerts an equal force back on the first body
but in the opposite direction
– Key note : the forces are ______ but _______ direction
• For angular motion
– For every torque exerted by one body on another, the
other body exerts an equal torque back on the first body
but in the opposite direction
– Key note : the torques acting on the two objects have
____________________ but opposite direction
Newton’s 3rd law
• Internal forces
• Internal muscle torques
• Tightrope walker
Linear vs. Angular Kinetics
Liner
Angualr
Quantity
Symbol
SI Unit
Quantity
Inertia
(mass)
m
Kg
Moment of
Inertia
Force
F
N
Torque
T = F ·r
Linear
Momentum
L = m·v
Angular
Momentum
H = I·ω
Impulse
(ΣF)(∆t)
N·s
Angular
Impulse
Symbol
SI Unit
I=
Nm·s
Q1. At the instant of takeoff, a 60-kg diver’s angular
momentum about his transverse axis is 20 kg·m2/s. His radius
of gyration about the transverse axis is 1.0 m at this instant.
During the dive, the diver tucks and reduces his radius of
gyration about the transverse axis is to 0.5 m.
a. At takeoff, what is the diver’s angular velocity about the
transverse axis?
b. After the diver tucks, what is his angular velocity about the
transverse axis?
Q. What average amount of force must be applied by the
elbow flexors inserting at an average perpendicular distance
of 1.5 cm from the axis of rotation at the elbow over a period
of 0.3 seconds to stop the motion of the 3.5 kg arm swing
with an angular velocity of 5 rad/s when k = 20 cm ? k =
radius of gyration.