Week 12 – Angular Kinetics Objectives Identify the angular analogues of mass, force, momentum, and impulse. Explain why changes in the configuration of rotating airborne body can produce chagnes in the body’s angular velocity Identify and provide examples of the angular analogues of Newton’s laws of motion Define centripetal force and explain where and how it acts Solve quantitative problems relating to the factors that cause or modify angular motion

Week 12 Angular Kinetics Read Chapter 14 of text Reference to figures in this presentation refer to the former text by Kreighbaum, which is on reserve Problems –Homework problem – to be handed out in class –Introductory problems, p 472:#5,6,7,9 –Additional problems, pp : #1,4,5 –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

Torque and Motion Relationships Relationship between linear and angular motion –displacement, velocity, and acceleration (Fig H.1, p 315) Angular analogue of Newton’s third law (F=ma), the instantaneous effect of a force or torque Sample problem #4, p 469 –Torque = moment of inertia (I) X angular acc (  (Fig H.5- H.7) What is torque? What is moment of inertia ?(Fig H.3, p 319) What is radius of gyration (Fig H.4, p 320) Changing moment of inertia and radius of gyration in the body (Figures H.8 and H.9, p 323 and 324) Calculations using a 3-segment system Homework problem

Relationship between linear and angular motion (kinematics) a = r 

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

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

MOI & K – Geometric Objects

Changing I and k in the human body

MOI around principal axes of human body in different positions

Angular Momentum What is angular momentum? (Fig I.4, p 329) –amount of angular movement: I  –Sample problem #1, p 459 Impulse-momentum relationship - effect of force or torque applied over time –Linear: Ft = mv Rotational: Tt = I  What is angular impulse? (Fig I.1, I.2, I.3, p 327-8) –Torque X time –Sample problem #3, p 466 Conservation of angular momentum (Fig I.4, I.5, I.6 p ) –Angular momentum is constant if net impulse is zero –Sample problem #2, p 462

What is angular impulse?

Angular Impulse: Mediolateral axis

Angular Impulse around vertical axis

What is angular momentum (L)?

Conservation of AngularMomentum

Centripetal & Centrifugal forces F c = mv 2 /r