Week 12 – Angular Kinetics
Objectives
• Identify and provide examples the angular
analogues of mass, force, momentum, and
impulse.
• Explain why changes in the configuration of
rotating airborne body can produce changes in the
body’s angular velocity (conservation of
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
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
• Self-study problems
– Sample problems:
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•
•
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#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 Monday, November 28)
– Additional problems, pp 473-474: #1,4,5
– Additional handout problem on moment of inertia
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.5H.7)
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•
•
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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
Instnataneous effect of net torque:
Moment of Inertia Constant
T=I

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)?
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.
k
35
Equivalent systems
k
35
Determining MOI & K
• 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
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
329-331)
– Angular momentum is constant if net impulse is zero
– Sample problem #2, p 462
p
What is angular momentum (L)?
Calculating
Angular
Momentum
Conservation of
AngularMomentum
Conservation of Angular Momentum
Conservation
of angular
momentum
What is angular impulse?
Angular
Impulse:
Mediolateral
axis
Angular Impulse around vertical axis
Impulse-Momentum Relationship
Torque-Angular acceleration
Centripetal &
Centrifugal forces
2
Fc = mv /r