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Circular Motion
Acceleration and why weird
things happen to spinning items
As an object moves around a circle, its
direction of motion is constantly
changing.
Therefore its velocity is changing.
Therefore an object moving in a circle is
constantly accelerating.
Recall Angular VelocityRecall that Angular velocity, is the rate of change in
angular displacement. (radians per second.)
Angular velocity in rad/s.t
Example 3: A rope is wrapped many times
around a drum of radius 20 cm. What is the
angular velocity of the drum if it lifts the
bucket to 10 m (remember s is arc length
and is angular displacement) in 5 s?
h = 10 m
R
= 10.0 rad/s
t50 rad
5 s
= 50 rad10 m
0.20 m
s
R
Angular AccelerationAngular acceleration is the rate of change in angular velocity.
(Radians per sec per sec.)
2 Angular acceleration (rad/s )t
Example 5: The block is lifted from rest until the
angular velocity of the drum is 16 rad/s after a time
of 4 s. What is the average angular acceleration?
h = 20 m
R
= 4.00 rad/s2
0
f o f
ort t
2
16 rad/s rad4.00
4 s s
2 Angular acceleration (rad/s )t
Angular and Linear
Acceleration:• Linear Acceleration (Sometimes called Tangential
Acceleration) – acceleration at a radius. How fast
something is speeding up in a straight line.
• A runner in the outside lane has to have more tangential
acceleration to stay along a runner in the inside lane.
a = r
Linear acceleration = angular acceleration x radius
Examples:
R1 = 20 cm R2 =
40 cm
R1
R2
A
B
= 0; f = 20 rad/s
t = 4 s
What is final linear speed at points
A and B?
Consider flat rotating disk:
vAf = R1 Af = (0.2 m)(20 rad/s); vAf = 4 m/s
vAf = R1 Bf = (0.4 m)(20 rad/s); vBf = 8 m/s
v = r
Acceleration Example
R1 = 20 cm
R2 = 40 cm
What is the average angular and linear
acceleration at B?
R1
R2
A
B= 0; f = 20 rad/s
t = 4 s
Consider flat rotating disk:
= 5.00 rad/s2
a = R = (5 rad/s2)(0.4 m) a = 2.00 m/s2
0 20 rad/s
4 s
f
t
Centripetal Acceleration
The acceleration of an object moving in
a circle points toward the center of
the circle.
This is called a centripetal (center
pointing) acceleration.
a
Centripetal Acceleration
The centripetal acceleration depends
on:
The speed of the object.
The radius of the circle.
Acent = v2
r
A Comparison: Linear vs. Angular
0
2
fv vs vt t 0
2
ft t
f o tf ov v at
210 2t t21
0 2s v t at
212f t t
2 2
02 f
2 2
02 fas v v
212fs v t at
Centripetal ForceNewton’s Second Law says that if an
object is accelerating, there must be a net
force on it.
For an object moving in a circle, this is
called the centripetal force.
The centripetal force points toward the
center of the circle.
Centripetal Force
In order to make an object revolve
about an axis, the net force on the
object must pull it toward the center
of the circle.
This force is called a centripetal (center
seeking) force.Fnet
Centripetal Force
Centripetal force on an object depends
on:
The object’s mass - more mass means
more force.
The object’s speed - more speed means
more force.
And…
Centripetal Force
The centripetal force on an object also depends on:
The object’s distance from the axis (radius). If linear velocity is held constant, more
distance requires less force.
If rotational velocity is held constant, more distance requires more force.
Centripetal Force
In symbols:
Fcent=mv2
r= mr 2
“Centrifugal Force”
“Centrifugal force” is a fictitious force -
it is not an interaction between 2
objects, and therefore not a real
force.
Nothing pulls an object away from
the center of the circle.
“Centrifugal Force”
What is erroneously attributed to
“centrifugal force” is actually the action
of the object’s inertia - whatever
velocity it has (speed + direction) it
wants to keep.
If you have a bolt that is stuck, do you want to use a short wrench
or a long wrench? Why?
“Torque”
Torque - A measurement of the tendency
of a force to produce a rotation about an
axis. = F x d
is torqueF is the applied force some distinct form the axis.
d is distance from the axis or pivot point
If torque creates a:
Counterclockwise rotation then it is positive
Clockwise rotation then it is negative.
Center of Gravity
Center of gravity – The point on any
object that acts like the place at which
all the weight is concentrated.
The End