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 473-474: #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 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

MOI & K – Geometric Objects

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 impulse?

Angular Impulse:

Mediolateral axis

Angular Impulse around vertical axis

What is angular momentum (L)?

Conservation of AngularMomentum

Conservation of Angular Momentum

Centripetal & Centrifugal forces

Fc = mv2/r