Resistance to Acceleration. More regarding moment of inertia. More regarding Moment of Inertia. Applications: Applications:

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Angular Kinetics of Human Movement Angular analogues to of Newton s Laws of Motion 1.

An external torque will cause angular acceleration (ΣT = Iα) 3.

acceleration: inertia, or mass Resistance to angular acceleration: angular inertia, or moment of inertia (I) What

I = mk 2 mass, and the position of the mass, relative to the axis of rotation, both affect I More regarding Moment of

experimentally determined length that applies to the whole object k will change as the position of the mass, relative

segments have different axes of rotation and a corresponding I for each plane of motion The whole human body, rotating

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? Applications: Tuck vs. Layout for a diver or gymnast Why might the Big Bertha driver result in better drives? tuck layout A runner's leg during swing phase hip 1

2 (H) Momentum: For linear motion: L = mv For angular motion: H = I Or:H = (mk 2 ) Factors that affect angular momentum (H): mass of the object (m) distribution of mass relative to axis or rotation (k) angular velocity of the object ( ) Newton s Laws of Motion: Angular Analogues Newton s 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. Units for angular momentum: kg m 2 s 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 ) A 60 kg diver is in a layout position with radius of gyration of 0.5 m as he leaves the board with an angular velocity of 4 rad/s. What is the diver s angular velocity when he assumes a tuck position, reducing his/her radius of gyration to 0.25 m? k = 0.5 m k = 0.25 m First, find H when diver leaves the board: H = mk 2 H = (60 kg)(0.5 m) 2 (4 rad/s) = 60 kg m 2 /s H is constant, so now find when k is reduced to 0.25 m: Transfer of Total body angular momentum is constant while the body is freely suspended in the air (no external torques).. but, you can: Transfer angular velocity from one part of the body to another Change the total body axis of rotation 60 kg m 2 /s = (60 kg)(0.25 m) 2 = 16 rad/s 2

Transfer of Early in the dive, H is concentrated in upper body, while later in the dive H is concentrated in the lower body How does the transfer of H between body segments affect the COG trajectory?

3 Transfer of Early in the dive, H is concentrated in upper body, while later in the dive H is concentrated in the lower body How does the transfer of H between body segments affect the COG trajectory? H TOTAL (constant) H LOWER (changes) H UPPER (changes) Center of gravity still follows a parabolic path while the jumper is airborne Newton s Laws of Motion: Angular Analogues Newton s 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 body s moment of inertia ΣT = I (compare with ΣF = m a) Change in How does someone change H? To help answer this question, recall how someone changes L? Linear and angular impulse Linear Impulse = force time = F t Angular Impulse = torque time = T t Impulse-momentum relationships Linear: Ft = M F t = (mv) 2 (mv) 1 Angular: Tt = H T t = (I ) 2 (I ) 1 Practice Problems As she initiates a twisting jump, Miki s angular momentum (about her long axis) increases from 0 to 50 kg m 2 /s in 0.25 s. During this time, her moment of inertia (about her long axis) is 2.2 kg m 2. 1.How large must the angular impulse have been? Answer: 50 Nm s 2.How large is the corresponding torque? Answer: 200 Nm. 3.How fast is Miki s angular velocity (about her long axis) at the end of the 0.25 s? Answer 22.7 rad/s 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 Problems In 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. 3

Another Practice Problem Related to Law #2 The Answer Newton s Laws of Motion: Angular Analogues Newton s Third Law: For every torque exerted by one body on another, there is an equal and opposite

exerted by the second body on the first Centripetal Force Centripetal force (F c )is what keeps objects moving along a curved path (i.e., centripetal force produces the centripetal component of

4 Another Practice Problem Related to Law #2 The Answer Newton s Laws of Motion: Angular Analogues Newton s 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 Centripetal Force Centripetal force (F c )is what keeps objects moving along a curved path (i.e., centripetal force produces the centripetal component of acceleration) F c = ma c = m(v 2 / r), and since v = r, then F c also equals m r 2 Centripetal Force (F CP in this figure) Summary Moment of inertia (I), or angular inertia, depends on mass and mass location relative to the axis of rotation Angular momentum (H = I ω) is conserved in the absence of external torques Angular impulse is required to alter H For every torque, an equal and opposite torque will exist Centripetal force is required for an object to rotate 4

Practice frontal Answer: each configuration has ~70 units of

27 kg shot makes seven complete revolutions during its 2.

54 cm, what is its angular momentum? a. 0.0825 kg m 2 /s b. 7.

5 second flight. Its radius of gyration is 2.54 cm.

5 Practice frontal Answer: each configuration has ~70 units of angular momentum A 7.27 kg shot makes seven complete revolutions during its 2.5 second flight. If its radius of gyration is 2.54 cm, what is its angular momentum? a kg m 2 /s b kg m 2 /s c kg m 2 /s d. None of the above A 7.27 kg shot makes seven complete revolutions during its 2.5 second flight. Its radius of gyration is 2.54 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 7.27 kg hammer on a 1 m wire is released with a linear velocity of 28 m/s. What reaction force is exerted on the thrower by the hammer at the instant before release? a. 0.5 N b N c N d. None of the above 5

Physics 130: Questions to study for midterm #1 from Chapter 8

Physics 130: Questions to study for midterm #1 from Chapter 8 1. If the beaters on a mixer make 800 revolutions in 5 minutes, what is the average rotational speed of the beaters? a. 2.67 rev/min b. 16.8