More regarding Moment of Inertia More regarding moment of...
Transcript of More regarding Moment of Inertia More regarding moment of...
1
Angular Kinetics of Human Movement
Angular analogues to of Newtonrsquos Laws of Motion
1 An rotating object will continue to rotate unless acted upon by an external torque
2 An external torque will cause angular acceleration (ΣT = Iα)
3 For every torque an equal and opposite reaction torque will exist
Resistance to Acceleration
Resistance to linear acceleration inertia or mass
Resistance to angular acceleration angular inertia or moment of inertia (I)
What directly influences I I = mk2
mass and the position of the mass relative to the axis of rotation both affect I
More regarding Moment of InertiaA relatively practical approach
IAXIS = (mBODY)(k2)
k indicates radius of gyration which is an experimentally determined length that applies to the whole object
k will change as the position of the mass relative to the axis of rotation changes (or if you simply move the axis location)
More regarding moment of inertiabull Body segments have different axes of rotation and a
corresponding I for each plane of motion
bull The whole human body rotating free of external force also has a I that is relative to one of the three cardinal axes
ApplicationsWhy would someone choke up on a bat use an aluminum bat or cork a wooden bat Is there a legal alternative
Why might the Big Bertha
driver result in better drives
ApplicationsTuck vs Layout for a diver or gymnast
A runners leg during swing phase
tucklayout
hip
2
Angular Momentum (H)Momentum
bull For linear motion L = mv
bull For angular motion H = I OrH = (mk2)
bull Factors that affect angular momentum (H)ndash mass of the object (m)
ndash distribution of mass relative to axis or rotation (k)
ndash angular velocity of the object ()
Units for angular momentum kgm2
s
Newtonrsquos Laws of Motion Angular Analogues
Newtonrsquos First Law
A rotating body will maintain a state of constant rotational motion unless acted on by an external torque
This law forms the basis for the principle of the conservation of angular momentum
Conservation of Angular Momentum
Similar to L total H of a given system remains constant in the absence of external torques
However I and can change
(recall that H = I )
Conservation of Angular MomentumA 60 kg diver is in a layout position with radius of gyration of 05 m as he leaves the board with an angular velocity of 4 rads What is the diverrsquos angular velocity when he assumes a tuck position reducing hisher radius of gyration to 025 m
k = 05 m
k = 025 m
Conservation of Angular Momentum
First find H when diver leaves the board
H = mk2
H = (60 kg)(05 m)2(4 rads) = 60 kgm2s
H is constant so now find when k is reduced to 025 m
60 kgm2s = (60 kg)(025 m)2
= 16 rads
Total body angular momentum is constant while the body is freely suspended in the air (no external torques)helliphellipbut you can
bull Transfer angular velocity from one part of the body to another
bull Change the total body axis of rotation
Transfer of Angular Momentum
3
Transfer of Angular MomentumEarly in the dive H is concentrated in upper body while later in the dive H is concentrated in the lower body
HTOTAL (constant)
HLOWER (changes)
HUPPER (changes)
How does the transfer of H between body segments affect the COG trajectory
Center of gravity still follows a parabolic path while the jumper is airborne
Newtonrsquos Second Law A net torque produces angular acceleration (also a change for H) of a body that is directly proportional to the magnitude of torque in the same direction as the torque and inversely proportional to the bodyrsquos moment of inertia
ΣT = I (compare with ΣF = m a)
Newtonrsquos Laws of Motion Angular Analogues
Change in Angular Momentum
bull How does someone change H To help answer this question recall how someone changes L
bull Linear and angular impulse
ndash Linear Impulse = force time = F t
ndash Angular Impulse = torque time = T t
bull Impulse-momentum relationships
ndash Linear Ft = M F t = (mv)2 ndash (mv)1
ndash Angular Tt = H T t = (I )2 ndash (I )1
Practice ProblemsAs she initiates a twisting jump Mikirsquos angular momentum (about her long axis) increases from 0 to 50 kgmiddotm2s in 025 s During this time her moment of inertia (about her long axis) is 22 kgmiddotm2
1How large must the angular impulse have been Answer 50 Nmmiddots
2How large is the corresponding torque Answer 200 Nm
3How fast is Mikirsquos angular velocity (about her long axis) at the end of the 025 s Answer 227 rads
Miki Ando is one of a few female skaters to have landed a quadruple
jump (a salchow) in competition She first completed the jump at the 2002
ISU Junior Grand Prix Final in the Netherlands at age 15
Practice ProblemsIn preparation to rotate four times in the air during a single jump Miki must increase ω about her long axis
1 Can Miki manipulate I (about her long axis) If so how
2 Is it beneficial for Miki to manipulate I (about her long axis) If so why
3 What might the overall effect of an increased or decreased I (about her long axis) be on her final score
Japanese figure skater Miki Ando is the only female skater to have landed a
quadruple jump (a salchow) in competition She first completed the
jump at the 2002 ISU Junior Grand Prix Final in the Netherlands at age 15
4
Another Practice Problem Related to Law 2
The Answer
Newtonrsquos Third Law
For every torque exerted by one body on another there is an equal and opposite torque exerted by the second body on the first
Newtonrsquos Laws of Motion Angular Analogues
Centripetal ForceCentripetal force (Fc)is what keeps objects moving along a curved path (ie centripetal force produces the centripetal component of acceleration)
Fc = mac = m(v2 r)
and since v = r
then Fc also equals m r 2
Centripetal Force (FCP in this figure) Summarybull Moment of inertia (I) or angular inertia depends on
mass and mass location relative to the axis of rotation
bull Angular momentum (H = I ω) is conserved in the absence of external torques
bull Angular impulse is required to alter H
bull For every torque an equal and opposite torque will exist
bull Centripetal force is required for an object to rotate
5
Practicehellip
Answer each configuration has ~70 units of angular momentum
frontal A 727 kg shot makes seven complete revolutions during its 25 second flight If its radius of gyration is 254 cm what is its angular momentum
a 00825 kgmiddotm2sb 738 kgmiddotm2sc 4616 kgmiddotm2sd None of the above
A 727 kg shot makes seven complete revolutions during its 25 second flight Its radius of gyration is 254 cm
What would happen to ω if the ball had more m while conserving H
It would decrease
What would happen to H if ω and k were increased
It would increase
A 727 kg hammer on a 1 m wire is released with a linear velocity of 28 ms What reaction force is exerted on the thrower by the hammer at the instant before release
a 05 Nb 1000 Nc 1604 Nd None of the above
2
Angular Momentum (H)Momentum
bull For linear motion L = mv
bull For angular motion H = I OrH = (mk2)
bull Factors that affect angular momentum (H)ndash mass of the object (m)
ndash distribution of mass relative to axis or rotation (k)
ndash angular velocity of the object ()
Units for angular momentum kgm2
s
Newtonrsquos Laws of Motion Angular Analogues
Newtonrsquos First Law
A rotating body will maintain a state of constant rotational motion unless acted on by an external torque
This law forms the basis for the principle of the conservation of angular momentum
Conservation of Angular Momentum
Similar to L total H of a given system remains constant in the absence of external torques
However I and can change
(recall that H = I )
Conservation of Angular MomentumA 60 kg diver is in a layout position with radius of gyration of 05 m as he leaves the board with an angular velocity of 4 rads What is the diverrsquos angular velocity when he assumes a tuck position reducing hisher radius of gyration to 025 m
k = 05 m
k = 025 m
Conservation of Angular Momentum
First find H when diver leaves the board
H = mk2
H = (60 kg)(05 m)2(4 rads) = 60 kgm2s
H is constant so now find when k is reduced to 025 m
60 kgm2s = (60 kg)(025 m)2
= 16 rads
Total body angular momentum is constant while the body is freely suspended in the air (no external torques)helliphellipbut you can
bull Transfer angular velocity from one part of the body to another
bull Change the total body axis of rotation
Transfer of Angular Momentum
3
Transfer of Angular MomentumEarly in the dive H is concentrated in upper body while later in the dive H is concentrated in the lower body
HTOTAL (constant)
HLOWER (changes)
HUPPER (changes)
How does the transfer of H between body segments affect the COG trajectory
Center of gravity still follows a parabolic path while the jumper is airborne
Newtonrsquos Second Law A net torque produces angular acceleration (also a change for H) of a body that is directly proportional to the magnitude of torque in the same direction as the torque and inversely proportional to the bodyrsquos moment of inertia
ΣT = I (compare with ΣF = m a)
Newtonrsquos Laws of Motion Angular Analogues
Change in Angular Momentum
bull How does someone change H To help answer this question recall how someone changes L
bull Linear and angular impulse
ndash Linear Impulse = force time = F t
ndash Angular Impulse = torque time = T t
bull Impulse-momentum relationships
ndash Linear Ft = M F t = (mv)2 ndash (mv)1
ndash Angular Tt = H T t = (I )2 ndash (I )1
Practice ProblemsAs she initiates a twisting jump Mikirsquos angular momentum (about her long axis) increases from 0 to 50 kgmiddotm2s in 025 s During this time her moment of inertia (about her long axis) is 22 kgmiddotm2
1How large must the angular impulse have been Answer 50 Nmmiddots
2How large is the corresponding torque Answer 200 Nm
3How fast is Mikirsquos angular velocity (about her long axis) at the end of the 025 s Answer 227 rads
Miki Ando is one of a few female skaters to have landed a quadruple
jump (a salchow) in competition She first completed the jump at the 2002
ISU Junior Grand Prix Final in the Netherlands at age 15
Practice ProblemsIn preparation to rotate four times in the air during a single jump Miki must increase ω about her long axis
1 Can Miki manipulate I (about her long axis) If so how
2 Is it beneficial for Miki to manipulate I (about her long axis) If so why
3 What might the overall effect of an increased or decreased I (about her long axis) be on her final score
Japanese figure skater Miki Ando is the only female skater to have landed a
quadruple jump (a salchow) in competition She first completed the
jump at the 2002 ISU Junior Grand Prix Final in the Netherlands at age 15
4
Another Practice Problem Related to Law 2
The Answer
Newtonrsquos Third Law
For every torque exerted by one body on another there is an equal and opposite torque exerted by the second body on the first
Newtonrsquos Laws of Motion Angular Analogues
Centripetal ForceCentripetal force (Fc)is what keeps objects moving along a curved path (ie centripetal force produces the centripetal component of acceleration)
Fc = mac = m(v2 r)
and since v = r
then Fc also equals m r 2
Centripetal Force (FCP in this figure) Summarybull Moment of inertia (I) or angular inertia depends on
mass and mass location relative to the axis of rotation
bull Angular momentum (H = I ω) is conserved in the absence of external torques
bull Angular impulse is required to alter H
bull For every torque an equal and opposite torque will exist
bull Centripetal force is required for an object to rotate
5
Practicehellip
Answer each configuration has ~70 units of angular momentum
frontal A 727 kg shot makes seven complete revolutions during its 25 second flight If its radius of gyration is 254 cm what is its angular momentum
a 00825 kgmiddotm2sb 738 kgmiddotm2sc 4616 kgmiddotm2sd None of the above
A 727 kg shot makes seven complete revolutions during its 25 second flight Its radius of gyration is 254 cm
What would happen to ω if the ball had more m while conserving H
It would decrease
What would happen to H if ω and k were increased
It would increase
A 727 kg hammer on a 1 m wire is released with a linear velocity of 28 ms What reaction force is exerted on the thrower by the hammer at the instant before release
a 05 Nb 1000 Nc 1604 Nd None of the above
3
Transfer of Angular MomentumEarly in the dive H is concentrated in upper body while later in the dive H is concentrated in the lower body
HTOTAL (constant)
HLOWER (changes)
HUPPER (changes)
How does the transfer of H between body segments affect the COG trajectory
Center of gravity still follows a parabolic path while the jumper is airborne
Newtonrsquos Second Law A net torque produces angular acceleration (also a change for H) of a body that is directly proportional to the magnitude of torque in the same direction as the torque and inversely proportional to the bodyrsquos moment of inertia
ΣT = I (compare with ΣF = m a)
Newtonrsquos Laws of Motion Angular Analogues
Change in Angular Momentum
bull How does someone change H To help answer this question recall how someone changes L
bull Linear and angular impulse
ndash Linear Impulse = force time = F t
ndash Angular Impulse = torque time = T t
bull Impulse-momentum relationships
ndash Linear Ft = M F t = (mv)2 ndash (mv)1
ndash Angular Tt = H T t = (I )2 ndash (I )1
Practice ProblemsAs she initiates a twisting jump Mikirsquos angular momentum (about her long axis) increases from 0 to 50 kgmiddotm2s in 025 s During this time her moment of inertia (about her long axis) is 22 kgmiddotm2
1How large must the angular impulse have been Answer 50 Nmmiddots
2How large is the corresponding torque Answer 200 Nm
3How fast is Mikirsquos angular velocity (about her long axis) at the end of the 025 s Answer 227 rads
Miki Ando is one of a few female skaters to have landed a quadruple
jump (a salchow) in competition She first completed the jump at the 2002
ISU Junior Grand Prix Final in the Netherlands at age 15
Practice ProblemsIn preparation to rotate four times in the air during a single jump Miki must increase ω about her long axis
1 Can Miki manipulate I (about her long axis) If so how
2 Is it beneficial for Miki to manipulate I (about her long axis) If so why
3 What might the overall effect of an increased or decreased I (about her long axis) be on her final score
Japanese figure skater Miki Ando is the only female skater to have landed a
quadruple jump (a salchow) in competition She first completed the
jump at the 2002 ISU Junior Grand Prix Final in the Netherlands at age 15
4
Another Practice Problem Related to Law 2
The Answer
Newtonrsquos Third Law
For every torque exerted by one body on another there is an equal and opposite torque exerted by the second body on the first
Newtonrsquos Laws of Motion Angular Analogues
Centripetal ForceCentripetal force (Fc)is what keeps objects moving along a curved path (ie centripetal force produces the centripetal component of acceleration)
Fc = mac = m(v2 r)
and since v = r
then Fc also equals m r 2
Centripetal Force (FCP in this figure) Summarybull Moment of inertia (I) or angular inertia depends on
mass and mass location relative to the axis of rotation
bull Angular momentum (H = I ω) is conserved in the absence of external torques
bull Angular impulse is required to alter H
bull For every torque an equal and opposite torque will exist
bull Centripetal force is required for an object to rotate
5
Practicehellip
Answer each configuration has ~70 units of angular momentum
frontal A 727 kg shot makes seven complete revolutions during its 25 second flight If its radius of gyration is 254 cm what is its angular momentum
a 00825 kgmiddotm2sb 738 kgmiddotm2sc 4616 kgmiddotm2sd None of the above
A 727 kg shot makes seven complete revolutions during its 25 second flight Its radius of gyration is 254 cm
What would happen to ω if the ball had more m while conserving H
It would decrease
What would happen to H if ω and k were increased
It would increase
A 727 kg hammer on a 1 m wire is released with a linear velocity of 28 ms What reaction force is exerted on the thrower by the hammer at the instant before release
a 05 Nb 1000 Nc 1604 Nd None of the above
4
Another Practice Problem Related to Law 2
The Answer
Newtonrsquos Third Law
For every torque exerted by one body on another there is an equal and opposite torque exerted by the second body on the first
Newtonrsquos Laws of Motion Angular Analogues
Centripetal ForceCentripetal force (Fc)is what keeps objects moving along a curved path (ie centripetal force produces the centripetal component of acceleration)
Fc = mac = m(v2 r)
and since v = r
then Fc also equals m r 2
Centripetal Force (FCP in this figure) Summarybull Moment of inertia (I) or angular inertia depends on
mass and mass location relative to the axis of rotation
bull Angular momentum (H = I ω) is conserved in the absence of external torques
bull Angular impulse is required to alter H
bull For every torque an equal and opposite torque will exist
bull Centripetal force is required for an object to rotate
5
Practicehellip
Answer each configuration has ~70 units of angular momentum
frontal A 727 kg shot makes seven complete revolutions during its 25 second flight If its radius of gyration is 254 cm what is its angular momentum
a 00825 kgmiddotm2sb 738 kgmiddotm2sc 4616 kgmiddotm2sd None of the above
A 727 kg shot makes seven complete revolutions during its 25 second flight Its radius of gyration is 254 cm
What would happen to ω if the ball had more m while conserving H
It would decrease
What would happen to H if ω and k were increased
It would increase
A 727 kg hammer on a 1 m wire is released with a linear velocity of 28 ms What reaction force is exerted on the thrower by the hammer at the instant before release
a 05 Nb 1000 Nc 1604 Nd None of the above
5
Practicehellip
Answer each configuration has ~70 units of angular momentum
frontal A 727 kg shot makes seven complete revolutions during its 25 second flight If its radius of gyration is 254 cm what is its angular momentum
a 00825 kgmiddotm2sb 738 kgmiddotm2sc 4616 kgmiddotm2sd None of the above
A 727 kg shot makes seven complete revolutions during its 25 second flight Its radius of gyration is 254 cm
What would happen to ω if the ball had more m while conserving H
It would decrease
What would happen to H if ω and k were increased
It would increase
A 727 kg hammer on a 1 m wire is released with a linear velocity of 28 ms What reaction force is exerted on the thrower by the hammer at the instant before release
a 05 Nb 1000 Nc 1604 Nd None of the above