1. Angular Kinetics Review Readings: –Hamill Ch 11 esp pp 382-410 –Kreighbaum pp 318-324, 326-331 –Adrian 33-40 (COM calculations) Homework problem on calculating MOI of lower extremity will be distributed in class

2. Torque and Motion Relationships Relationship between linear and angular motion –displacement, velocity, and acceleration Angular analogue of Newton’s third law (F=ma), the instantaneous effect of a force or torque –Torque = moment of inertia (I) X angular acc (  What is torque? What is moment of inertia ? What is radius of gyration Changing moment of inertia and radius of gyration in the body Calculations using a 3-segment system Homework problem

3. What is torque, or a moment of force? Torque is the turning effect of a force and is the product of force magnitude and moment arm, or perpendicular distance from the force’s line of action to the axis of rotation:

4. Angle of Pull of Muscle & degree of force application Turning component equals Force times sin θ

5. Mechanical Advantage of Elbow Flexors

6. Length of Elbow Flexors as Joint Angle Changes

7. Additional problem #2, p 173: Length-tension, angle of pull combined Sine of

8. Assume force is applied perpendicular to the bar: a. In which position will the exercise be easier? b. If it takes 100 N to move the system at position 2, what will it take to lift it at positions 1 and 3? Hint: use the law of levers: Ff = Rr and solve for F Homework #5 – Musculoskeletal machines: (due Monday, Feb 28) Introductory problems, p 445 - # 4, 8 Additional problems, p 446 - #3 Exercise equipment problem illustrated below:

9. Sample Problem #2, p 433

10. Example of total body torques Torque and impulse about the center of mass

11. What is the COM and why is it important? What is COM (or COG) and why is it important? –It simplifies mechanical analysis of a complicated system –It is the point at which all of the mass of the system may be considered to be located –It is the only point that represents movement of the total system The acceleration of the COM is proportional to the net force and inversely proportional to the mass. –It is the only point that follows a parabolic flight pattern when free of contact with earth –External forces through the COM cause produce only linear motion –External forces not through the COM (eccentric forces) create a torque, or moment, and produce both linear and rotary motion

12. COM/COG Concept and Calculation Method (Adrian pp 33-41) Concept of balancing segmental torques Segmental Calculation of COM –General calculation method –Information needed Proportionate mass of each segment location of COM of each segment

13. Segmental concept of center of mass

15. Hanavan Model used for Segmental Calculation of COM and MOI

16. Segmental concept of center of mass Information needed: 1. Segmental COM location 2. Segmental proportionate mass

18. Instantaneous effect of net torque: Moment of Inertia (MOI) Constant What are torque and MOI? T = I 

19. Instantaneous effect of net torque: Torque is constant

20. Instantaneous effect of net torque: Ang acc ( constant

21. What is Moment of Inertia? Here, r (the radius of rotation) is equal to k (the radius of gyration), but that is not the case with extended bodies It is the resistance of a system to rotational acceleration, and is calculated at follows:

22. What is radius of gyration (k)? An indicator of distribution of mass about the axis. It is the distance from the axis to a point at which all the mass of a system of equal mass would be concentrated to have the MOI equal the original system. It is, then, the average weighted distance of the mass of a system to the axis. Equivalent systems k 35 k

23. Determining MOI & K Simple 3-segment system: –I = 3 m i d i 2 = m 1 d 1 2 + m 2 d 2 2 + m 3 d 3 2 +.......+ m i d i 2 –I = mk 2 ; k = (I/m).5 Irregularly shaped bodies But we can’t measure all of these small masses!

24. Physical pendulum method of determining MOI and K Suspend object at axis Measure mass (m), and distance from axis to COM, r Measure period of oscillation (T) –Moment of inertia (I) = T 2 mr *.248387 m/sec –Radius of gyration (K) = ( I/m).5

25. MOI & K – Geometric Objects

26. Changing I and k in the human body

28. MOI around principal axes of human body in different positions

29. Angular Impulse and Momentum Impulse-momentum relationship - effect of force or torque applied over time –Linear: Ft = mv Rotational: Tt = I  What is angular impulse? Torque X time What is angular momentum? Amount of angular movement: I  Conservation of angular momentum Angular momentum is constant if net impulse is zero

30. Total body torque and angular impulse: Mediolateral axis

31. Angular Impulse around vertical axis

32. What is angular momentum (L)?

33. Example of angular momentum

34. Conservation of Momentum