More regarding Moment of Inertia More regarding moment of...

5
1 Angular Kinetics of Human Movement Angular analogues to of Newton’s 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 = mk 2 mass, and the position of the mass, relative to the axis of rotation, both affect I More regarding Moment of Inertia A relatively practical approach: I AXIS = (m BODY )(k 2 ) 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 inertia Body segments have different axes of rotation and a corresponding I for each plane of motion The whole human body, rotating free of external force, also has a I that is relative to one of the three cardinal axes Applications: Why 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? Applications: Tuck vs. Layout for a diver or gymnast A runner's leg during swing phase tuck layout hip

Transcript of More regarding Moment of Inertia More regarding moment of...

Page 1: More regarding Moment of Inertia More regarding moment of inertiabiomech.byu.edu/Portals/83/docs/exsc365/chapter07/... · 2014-02-27 · More regarding moment of inertia • Body

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

Page 2: More regarding Moment of Inertia More regarding moment of inertiabiomech.byu.edu/Portals/83/docs/exsc365/chapter07/... · 2014-02-27 · More regarding moment of inertia • Body

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

Page 3: More regarding Moment of Inertia More regarding moment of inertiabiomech.byu.edu/Portals/83/docs/exsc365/chapter07/... · 2014-02-27 · More regarding moment of inertia • Body

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

Page 4: More regarding Moment of Inertia More regarding moment of inertiabiomech.byu.edu/Portals/83/docs/exsc365/chapter07/... · 2014-02-27 · More regarding moment of inertia • Body

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

Page 5: More regarding Moment of Inertia More regarding moment of inertiabiomech.byu.edu/Portals/83/docs/exsc365/chapter07/... · 2014-02-27 · More regarding moment of inertia • Body

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