Angular Kinetics Review

• Source: Chapter 12 of Basic Biomechanics by Susan Hall

• Reference to figures in this presentation refer to the former text by Kreighbaum, which is on reserve

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

– 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• 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

• What is angular momentum? (Fig I.4, p 329)– amount of angular movement: I

• Conservation of angular momentum (Fig I.4, I.5, I.6 p 329-331)– Angular momentum is constant if net impulse is zero

What is angular impulse?

Angular Impulse:

Mediolateral axis

Angular Impulse around vertical axis

What is angular momentum (L)?

Conservation of Momentum

Conservation of Momentum