Week 12 – Angular Kinetics Objectives

• Identify and provide examples the angular analogues of mass, force, momentum, and impulse.

• Explain why changes in the configuration of rotating airborne body can produce changes in the body’s angular velocity (conservation of 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

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• 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 Monday, November 28)– Additional problems, pp 473-474: #1,4,5– Additional handout problem on moment of inertia

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

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 massabout the axis. It is the distance fromthe axis to a point at which all themass of a system of equal masswould be concentrated to have the MOI equal the original system. Itis, then, the average weighted distance of the mass of a systemto the axis.

Equivalent systems

k 35

k 35

Determining MOI & K • Simple 3-segment system:

– I = mi di2 = m1 d1

2 + m2 d22+

m3 d32 + . . . . . . .+ mi di

2

– 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

Changing I and k in the human

body

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 329-331)– Angular momentum is constant if net impulse is zero– Sample problem #2, p 462

What is angular momentum (L)?

Calculating Angular

Momentum

Conservation of AngularMomentum

Conservation of Angular Momentum

Conservationof angular momentum

What is angular impulse?

Angular Impulse:

Mediolateral axis

Angular Impulse around vertical axis

Impulse-Momentum Relationship

Torque-Angular acceleration

Centripetal & Centrifugal forces

Fc = mv2/r