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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)
Instantaneous effect of net torque: Torque is constant
What is rotational inertia, Or moment of inertia?
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
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)