Transcript Torque and Motion Relationships

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:
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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 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 (
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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)
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
Torque-Angular acceleration
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
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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? (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)
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
Centripetal &
Centrifugal forces
2
Fc = mv /r