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)

Instnataneous effect of net torque: Moment of Inertia Constant What is torque?

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? It is the resistance of a system to rotational acceleration, and is calculated at follows: Here, r (the radius of rotation) is equal to k (the radius of gyration), but that is not the case with extended bodies

What is radius of gyration (k)? 35 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

Determining MOI & K Irregularly shaped bodies Simple 3-segment system: I = 3mi di2 = m1 d12 + m2 d22+ m3 d32 + . . . . . . .+ mi di2 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