Angular Kinetics Objectives Identify and provide examples of the angular equivalents of mass, force, momentum, and impulse Explain the relationships between the rotational effect of force (torque), rotational inertia, and rotational acceleration Explain and present practical applications of the conservation of angular momentum principle Define centripetal force and explain where and how it acts Solve quantitative problems relating to the factors that cause or modify angular motion

Angular Kinetics Readings & Homework Read Chapter 14 of text Self-study problems –Sample problems: #1, p 459 – angular momentum calculation #2, p 462 – conservation of angular momentum #3, p 466 – angular impulse and change in angular momentum calculation #4, p 469 – Angular analogue of Newton’s law of acceleration –Introductory problems, p 472:#5,6,7,9 Homework problems (due Thursday, April 27) –Additional problems, pp : #1,4,5 –Additional handout problem on moment of inertia

Angular Kinetics Outline Torque and motion relationships Instantaneous effect of net torque on a rotational system Definition of moment of inertia (MOI) and radius of gyration (K) Measuring MOI and K Changing MOI and K in the human body Angular Momentum Conservation of angular momentum Angular momentum and impulse-momentum relationship Sample problems and homework problem handout

Torque and Motion Relationships Angular analogue of Newton’s third law (F=ma), the instantaneous effect of a force or torque (Slides 5-7) –Sample problem #4, p 469 (slide 8) –Torque = moment of inertia (I) X angular acc (  What is torque? What is moment of inertia ?(Slide 9) What is radius of gyration (Slide 10) Calculations using a 3-segment system (Slide 11) Changing moment of inertia and radius of gyration in the body (Slides 13 & 14) Homework problem (handout)

Instnataneous effect of net torque: Moment of Inertia Constant What is torque? T = I 

Instantaneous effect of net torque: Torque is constant What is rotational inertia, Or moment of inertia?

Instantaneous effect of net torque: Ang acc constant

Torque-Angular acceleration

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:

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

Determining MOI & K Simple 3-segment system: –I = 3 m i d i 2 = m 1 d m 2 d m 3 d 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!

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 * m/sec –Radius of gyration (K) = ( I/m).5

Changing I and k in the human body

MOI around principal axes of human body in different positions

Angular Momentum What is angular momentum? (Slide 17) –amount of angular movement: I  –Sample problem #1, p 459 (Slide 18) Conservation of angular momentum (Slides 19-20) –Angular momentum is constant if net impulse is zero –Sample problem #2, p 462 (Slide 21) What is angular impulse? (Slide 22-24) –Torque X time Impulse-momentum relationship concept – the effect of force or torque applied over time –Linear: Ft = mv Rotational: Tt = I  Impulse-momentum relationship problem –Sample problem #3, p 466 (slide 25)

What is angular momentum (L)?

Calculating Angular Momentum

Conservation of AngularMomentum

Conservation of angular momentum

What is angular impulse?

Angular Impulse: Mediolateral axis

Angular Impulse around vertical axis

Impulse-Momentum Relationship

Centripetal & Centrifugal forces F c = mv 2 /r