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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. Torque and Motion Relationships.
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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
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
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:
Angle of Pull of Muscle & degree of force application Turning component equals Force times sin θ
Additional problem #2, p 173: Length-tension, angle of pull combined Sine of
Homework #5 – Musculoskeletal machines: (due Monday, Feb 28) • Introductory problems, p 445 - # 4, 8 • Additional problems, p 446 - #3 • Exercise equipment problem illustrated below: 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
Example of total body torques Torque and impulse about the center of mass
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
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
Hanavan Model used for Segmental Calculation of COM and MOI
Segmental concept of center of mass Information needed: 1. Segmental COMlocation 2. Segmental proportionate mass
Instantaneous effect of net torque: Moment of Inertia (MOI) Constant T = I What are torque and MOI?
What is Moment of Inertia? It is the resistance of a system to rotational acceleration, and is calculated at follows: Here, r (the radius of rotation) is equal to k (the radius of gyration), but that is not the case with extended bodies
What is radius of gyration (k)? k 35 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
Determining MOI & K • Simple 3-segment system: • I = 3mi di2 = m1 d12 + m2 d22+ m3 d32 + . . . . . . .+ mi di2 • I = mk2 ; k = (I/m).5 • Irregularly shaped bodies But we can’t measure all of these small masses!
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) = T2 mr * .248387 m/sec • Radius of gyration (K) = ( I/m).5
MOI around principal axes of human body in different positions
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