Discusses angular motion kinetics and dynamics. **More good stuff available at: www.wsautter.com and http://www.youtube.com/results?search_query=wnsautter&aq=f

1. Copyright Sautter 2003

2. Angular Motion
• Angular motion involves rotation or circular motion. Some
elements of circular motion have already been discussed
and we will review them here.
• Circular motion (rotation) can be measured using linear
units or angular units. Angular units refer to revolutions,
degrees or radians.
• The properties of circular motion include displacement,
velocity and acceleration. When applied to rotation the
values become angular displacement, angular velocity or
angular acceleration. Additionally, angular motion can be
measured using frequencies and periods or rotation.
• The Greek letters theta (), omega () and alpha () are
used to represent angular displacement, angular velocity
and angular acceleration

3. 



4. AVERAGE =  /  t = (2 + 1) / 2
 = o t + ½ t2
i = o + t
i = ½ (i
2 - o
2) / 
s =  r
Vlinear =  r
alinear =  r
f = 1/ T
T = 1 / f
1 revolution = 360 degrees = 2  radians
 = 2  f
 = 2  / T

5. Moment of Inertia
• All states of motion are subject to the laws of inertia, that is
tend to remain at the same rate and in the same directional
orientation.
• In the case of rotational motion, the angular velocity tends to
remain unchanged and the plane of rotation persists.
• As you will recall, outside forces can change inertial
conditions. In rotation, outside torques must be applied to
change an objects rotational inertia.
• Torque, as you remember, is a force applied perpendicularly
to the center of rotation.
• τ = F x r

6. Moment of Inertia
• The tendency of a body to resist changes in its linear
state of motion is measured by its mass.
• The tendency of a body to resist changes in its
rotational state of motion is measured by its moment of
inertia.
• Moment of inertia involves not just the mass of a
rotating object but also the distribution of the mass
within the object.
• τ = F x r, recall that F = ma, therefore:
• τ = ma x r, since a =  r, τ = m  r x r =(mr2 )
• I = mr2 and τ = I

7. Moment of Inertia
• τ = I
• Note the similarity to F = ma for linear motion.
Instead of a applied force (F) and applied torque (τ) is
necessary to provide acceleration.
• Instead of mass (m), the moment of inertia (I)
determines the resulting acceleration.
• Instead of linear acceleration (a), angular acceleration
() results from the application of a torque.
• Although, the moment of inertia in its simplest form is
given as mass times radius squared (mr2), in more
complex bodies the value of I must be found by
calculus methods (integration) or experimental
means.

8. The Laws of Motion for Rotating Bodies
(A summary)
• First Law – A body which is rotating tends to keep
rotating at the same rate and in the same plane unless
acted on by an outside torque.
• Second Law –
Torque = Moment of Inertia x angular acceleration
(τ = I)
• Third Law – for every torque there must be an equal
but opposite torque.

9. Moment of Inertia
• When the moment of inertia is found by experiment
a simplifying technique similar to the center of mass
concept is used.
• Remember, the center of mass of an object is a point
where all the mass of the body could be concentrated
to give the same inertial properties as the actual
mass distribution of the body.
• When describing rotational motion, radius of
gyration is used in place of the center of mass
concept.
• The radius of gyration of a body is the radius of a
thin ring of a mass equal to the mass of the body
which would give the same rotational characteristics
as the actual body.

10. Sphere
I = 2/5 mr2
Cylinder
I = 1/2 mr2
Thin Ring
I = mr2
Thin Rod
I = 1/12 mr2
Rotational
Axis
Rotational
Axis
Rotational
Axis
Rotational
Axis
I for any object of mass m and radius of gyration rg
I = mrg
2

11. 11
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