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Chapter 7
7.1 Measuring rotational motion
Rotational Quantities
• Rotational motion: motion
of a body that spins about
an axis
– Axis of rotation: the line
about which the rotation
occurs
• Circular motion: motion of
a point on a rotating
object
Rotational Quantities
• Circular Motion
– Direction is constantly changing
– Described as an angle
– All points (except points on the axis) move through
the same angle during any time interval
Circular Motion
• Useful to set a reference
line
• Angles are measured in
radians

s
r
• s= arc length
• r = radius
Angular Motion
• 360o = 2rad
• 180o = rad
(rad) 

180
(deg)
Angular displacement
• Angular dispacement: the angle through which a point line, or
body is rotated in a specified direction and about a specified
axis
 
• Practice:
s
r
– Earth has an equatorial radius of approximately 6380km and
rotates 360o every 24 h.
• What is the angular displacement (in degrees) of a person standing at
the equator for 1.0 h?
• Convert this angular displacement to radians
• What is the arc length traveled by this person?
Angular speed and
acceleration
• Angular speed: The rate at which a body rotates
about an axis, usually expressed in radians per
second

avg 
t
• Angular acceleration: The time rate of change of
angular speed, expressed in radians per second per
second
2  1 
avg 
t2  t1

t
Angular speed and
acceleration
ALL POINTS ON A ROTATING RIGID
OBJECT HAVE THE SAME ANGULAR
SPEED AND ANGULAR
ACCELERATION
Rotational kinematic
equations
Angular kinematics
• Practice
– A barrel is given a downhill rolling start of
1.5 rad/s at the top of a hill. Assume a
constant angular acceleration of 2.9 rad/s
• If the barrel takes 11.5 s to get to the bottom of
the hill, what is the final angular speed of the
barrel?
• What angular displacement does the barrel
experience during the 11.5 s ride?
Homework Assignment
• Page 269: 5 - 12
Chapter 7
7.2 Tangential and Centripetal
Acceleration
Tangential Speed
• Let us look at the relationship between
angular and linear quantities.
• The instantaneous linear speed of an
object directed along the tangent to the
object’s circular path
• Tangent: the line that touches the circle
at one and only one point.
Tangential Speed
• In order for two points at different
distances to have the same angular
displacement, they must travel different
distances
• The object with the larger radius must
have a greater tangential speed
Tangential Speed
v t  r
Tangential Acceleration
• The instantaneous linear acceleration
of an object directed along the tangent
to the object’s circular path
v t
t
r

t
a t  r
Lets do a problem
• A yo-yo has a tangential acceleration of
0.98m/s2 when it is released. The string
is wound around a central shaft of
radius 0.35cm. What is the angular
acceleration of the yo-yo?
Centripetal Acceleration
• Acceleration directed toward the center of a
circular path
• Although an object is moving at a constant
speed, it can still have an acceleration.
• Velocity is a vector, which has both
magnitude and DIRECTION.
• In circular motion, velocity is constantly
changing direction.
Centripetal Acceleration
• vi and vf in the figure to
the right differ only in
direction, not
magnitude
• When the time interval
is very small, vf and vi
will be almost parallel
to each other and
acceleration is directed
towards the center
Centripetal Acceleration
ac 
vt 2
r
a c  r2
Tangential and
centripetal accelerations
• Summary:
– The tangential component of
acceleration is due to changing
speed; the centripetal
component of acceleration is
due to changing direction
• Pythagorean theorem can be
used to find total acceleration
and the inverse tangent
function can be used to find
direction
What’s coming up
•
•
•
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HW: Pg 270, problems 21 - 26
Monday: Section 7.3
Wednesday: Review
Friday: TEST over Chapter 7
Chapter 7
7.3: Causes of Circular Motion
Causes of circular motion
• When an object is in motion,
the inertia of the object tends
to maintain the object’s motion
in a straight-line path.
• In circular motion (I.e. a weight
attached to a string), the string
counteracts this tendency by
exerting a force
• This force is directed along the
length of the string towards
the center of the circle
Force that maintains
circular motion
• According to Newton’s second law
Fc  ma c
or:
Fc 
mv t 2
r
Fc  mr
2
Force that maintains
circular motion
• REMEMBER: The force
that maintains circular
motion acts at right angles
to the motion.
• What happens to a person
in a car(in terms of forces)
when the car makes a
sharp turn.