Set 4
Circles and Newton
February 3, 2006
Where Are We
• Today
– Quick review of the examination
– we finish one topic from the last chapter – circular
motion
• We then move on to Newton’s Laws
• New WebAssign on board on today’s lecture
material
– Assignment – Read the circular motion stuff and
begin reading Newton’s Laws of Motion
• Next week
– Continue Newton
– Quiz on Friday
• Remember our deal!
Remember from the past …
• Velocity is a vector with magnitude
and direction.
• We can change the velocity in three
ways
– increase the magnitude
– change the direction
– or both
• If any of the components of v change
then there is an acceleration.
Changing Velocity
Dv
a
v2
v2
v1
Uniform Circular Motion
• Uniform circular motion occurs when an
object moves in a circular path with a
constant speed
• An acceleration exists since the direction
of the motion is changing
– This change in velocity is related to an
acceleration
• The velocity vector is always tangent to
the path of the object
Quick Review - Radians
s
q
s
q  Radians
r
Changing Velocity in
Uniform Circular Motion
• The change in the
velocity vector is
due to the change
in direction
• The vector diagram
shows Dv = vf - vi
The acceleration
Dv  vDq
Dv
Dq
v
a
Dt
Dt
rDq  vDt
Dq v

Dt r
v2
a
r
Centripetal
Acceleration
Centripetal Acceleration
• The acceleration is always
perpendicular to the path of the
motion
• The acceleration always points
toward the center of the circle of
motion
• This acceleration is called the
centripetal acceleration
Centripetal Acceleration,
cont
• The magnitude of the centripetal
acceleration vector was shown to be
2
v
aC 
r
• The direction of the centripetal
acceleration vector is always changing, to
stay directed toward the center of the
circle of motion
Period
• The period, T, is the time required
for one complete revolution
• The speed of the particle would be
the circumference of the circle of
motion divided by the period
• Therefore, the period is
2 r
T
v
Tangential Acceleration
• The magnitude of the velocity could
also be changing
• In this case, there would be a
tangential acceleration
Total Acceleration
• The tangential
acceleration causes
the change in the
speed of the
particle
• The radial
acceleration comes
from a change in
the direction of the
velocity vector
Total Acceleration,
equations
• The tangential acceleration:
dv
at 
dt
v2
ar  aC  
r
• The radial acceleration:
• The total acceleration:
– Magnitude
a  a a
2
r
2
t
Total Acceleration, In Terms
of Unit Vectors
• Define the following
unit vectors
rˆ and qˆ
– r lies along the radius
vector
 q is tangent to the
circle
• The total acceleration
2
d
v
v
is
a  at  a r 
qˆ  rˆ
dt
r
A ball on the end of a string is whirled around in a
horizontal circle of radius 0.300 m. The plane of the
circle is 1.20 m above the ground. The string breaks
and the ball lands 2.00 m (horizontally) away from the
point on the ground directly beneath the ball's location
when the string breaks. Find the radial acceleration
of the ball during its circular motion.
r
r
v
12
2
A pendulum with a cord of length r = 1.00 m swings
in a vertical plane (Fig. P4.53). When the pendulum
is in the two horizontal positions = 90.0° and = 270°,
its speed is 5.00 m/s. (a) Find the magnitude of the
radial acceleration and tangential