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 473-474: #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 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? (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

Conservationof angular momentum

What is angular impulse?

Angular Impulse:

Mediolateral axis

Angular Impulse around vertical axis

Impulse-Momentum Relationship

Centripetal & Centrifugal forces

Fc = mv2/r