Circular Motion
Questions for Consideration
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
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How do we measure circular motion?
What is a radian?
What are the angular analogs of linear
motion?
What is centripetal acceleration?
Circular Motion

How do we measure
circular motion?


Typically use radians.
Angle subtended by an
arc equal in length to
the circle’s radius.
Circular Motion



Circumference (distance around) = 2r
 So 2 radians = one full circle.
 So  radians = 180º
To convert rad to deg:
 deg = rad x 180/
 1 rad = 57.3º
To convert deg to rad:
 rad = deg x /180
 1º = 0.0175 rad
Common Angles
Arc Length

Arc Length (s)

Measured in meters along circumference of circle.
Angular Displacement

Angular Displacement ()




Measured in radians.
CCW rotation = +
CW rotation = - 
s = r
+
-
Circular Motion

A wheel with r = 5.00 m
spins counterclockwise
so that an ant resting on
the top travels 12.0
meters along the rim.
Through what angular
displacement did the
wheel rotate?
 s = r
 12.0 m = (5.00 m)()
  = 2.40 rad

Angular Velocity

Angular velocity ()



=

t
Expressed as rad/s.
Can also be given in terms of revolutions
per unit time.

revolutions per minute (rpm)

1 rpm = (2 rad) / (60 s) = 0.105 rad/s
Tangential Velocity

Tangential velocity (v) – the
instantaneous velocity of an
object moving in a circular
path.



Imagine a bucket being swung
around on a rope.
The bucket has a tangential
velocity that is perpendicular
to the rope.
If the rope breaks, the bucket’s
tangential velocity will
become its linear velocity.
Tangential Velocity


Formula for tangential velocity:
 v = r
A child is riding a merry-go-round that is rotating at 30
rpm. How fast is the child moving if she is 2.5 m from the
center?
 Given:
  = 30 rpm
 r = 2.5 m
 Want:
 v=?
Tangential Velocity

First, convert rpm to rad/s:


30 rpm = (30 * 2 rad) / (60 s) = 3.14 rad/s
v = r

v = (2.5 m)(3.1 rad/s) = 7.8 m/s
Tangential Velocity

A satellite moves around the Earth in a
circular orbit with r = 10,000 km. If the
satellite takes 2.76 hours to complete one
orbit, calculate the satellite’s angular and
tangential velocities.

Given:



r = 10,000 km
t = 2.76 hr
Want:


=?
v=?
Tangential Velocity

=/t



 = (2 rad) / (2.76 hr)
 = 2.28 rad/hr
v = r


v = (10,000 km)(2.28 rad/hr)
v = 22,800 km/hr
Centripetal Acceleration

Can something accelerate but maintain a
constant speed?



Yes, if it changes direction.
Acceleration = change in velocity / time
Change in velocity =



speed up,
slow down,
or change direction.
Centripetal Acceleration

Centripetal Acceleration (ac) – causes a
change in direction.



Perpendicular to direction of motion.
Measured in m/s2.
ac =
v2
r
= 2r
Centripetal Acceleration
Centripetal Acceleration

An amusement park ride spins riders around
so fast that they are seemingly stuck to the
walls. If the ride has a radius of 3.50 meters,
what angular velocity (in rpm) is necessary to
create a centripetal acceleration of 20.0 m/s2?

Given:



r = 3.50 m
ac = 20.0 m/s2
Want:

 (in rpm) = ?
Centripetal Acceleration

ac = 2r




20.0 m/s2 = 2(3.50 m)
2 = 5.71 /s2
 = 2.39 rad/s
Now convert to rpm:


Recall that 1 rpm = 0.105 rad/s
(2.39 rad/s) / 0.105 = 22.8 rpm