Rotary Motion
Physics
Montwood High School
• Rotary motion is the motion of a body
around an internal axis.
– Rotary motion – axis of rotation is inside the
moving object. Ex. Spinning wheel.
– Circular motion – axis of motion is outside the
moving object. Ex. Object on the rim of the
spinning wheel.
• The angles in rotary motion are measured
in radians; linear motion and circular
motion are measured in degrees.
– Calculator will have to be in radian mode
(indicated by rad on the screen)
Radian Definition
• When the arc length
S is equal to the
length of the radius r,
the angle q swept out
by r is 1 radian.
• Any angle q
measured in radians
is defined by the
following:
s
q
r
Radian Definition
• Angular displacement q
– how much of a circle
the object moves
through.
• Because S and r are
measured in meters, the
units cancel and the unit
rad is added (the What
Happens to the Radians
article will have you
substitute rad/m).
360 degrees = 2 radians = 1 revolution (or rotation)
Conversions
• Degrees to Radians
example:
o
60 
2   rad
o
360
 1.0472 rad
 rad
o
60 
o
180
 1.0472 rad
• Radians to Degrees
example:
360 o
1.5 rad 
2   rad
 85 .94
o
o
180
1.5 rad 
 rad
 85 .94 o
Conversions
• Revolutions to
Radians example:
2   rad
3 revs 
1 rev
 18.85 rad
• Radians to
Revolutions example:
1rev
4 rad 
2   rad 
 0.6366 rev
Angular Displacement q
q 
S
r
rad
unit : rad,
m
• Angular displacement
q describes how
much an object has
rotated.
• Counterclockwise
displacements are
considered positive
(as shown).
• Clockwise rotations
are considered
negative.
Angular Speed w
• Angular speed w is
the speed at which
the object is rotating.
• Unit: rad/s; rev/s;
etc.
q
w
t
Angular Speed w
• Angular velocity is a vector
quantity.
• To determine the direction
of the angular velocity
vector, curl fingers of right
hand in the direction of
rotation. Thumb points in
the direction of the angular
velocity vector w.
Angular Speed w
Angular Acceleration a
• Angular acceleration a is the rate at which
the rotational speed changes.
• Unit: rad/s2; rev/s2; etc.
• Increasing rate of rotation
wf > wi, a positive
• Decreasing rate of rotation
wf < wi, a negative
w wf  wi
a

t
t
Angular Speed w Revisited
• For a rotating object,
every point on the object
has the same angular
velocity w and the same
angular acceleration a.
• The tangential (linear)
velocity v or vT of any
point is proportional to its
distance from the axis of
rotation.
• The linear distance d is
equal to the arc length s
of the rotating object.
v  rw
Angular Speed w Revisited
• If a point on the edge of a rotating object
had an angular speed greater than that of
a point near the center, the shape of the
object would be changing.
• For a rotating object to remain rigid, every
point of the object must have the same
angular speed and angular acceleration.
Angular Speed w Revisited
• All skaters have the
same angular velocity
w and the same
angular acceleration
a, however, the linear
or tangential velocity
vT increases from the
center along the
radius to the last
person.
Linear vs. Angular Quantities
Linear vs. Angular Quantities
• Time = t
• Angular
acceleration = a
• Initial angular
velocity = wi
• Final angular
velocity = wf
• Angular
displacement = q
• Radius = r

wi  wf   t
q
2

q  wi  t   0.5  a  t
wf  wi  a  t 
wf  wi  2  a  q
2
2
2

Transferring Rotational Motion
• If two disks are touching
each other as in the figure,
the top disk is driven by a
motor and turns the 2nd
disk (the wheel) by making
use of the friction between
them.
• The relationship between
the diameters D of the two
disks and the number N of
revolutions is:
D1  N1  D2  N 2
Transferring Rotational Motion
• Using two disks to transfer rotational motion is
not very efficient due to slippage that may occur
between disks. The most common way to
prevent disk slippage:
– place teeth on the edge of the disk (b).
– Connect the disks with a belt (c).
• Instead of using disks, we use gears or beltdriven pulleys to transfer the rotational motion.
• Teeth on the gears eliminate slippage and
provides for distance between rotating centers.
Transferring Rotational Motion
Transferring Rotational Motion
D1 N 2

• Rearrange D1  N1  D2  N 2 to get D2 N1
• D1/D2 is the ratio of the diameters of the
disks. If the ratio is 2, this means that D1
has a diameter that is 2 times greater than
D 2.
• N2/N1 is the ratio of the number of
revolutions of the two disks. If the ratio is 2,
the smaller disk makes two revolutions
while the larger disk makes one revolution.
Transferring Rotational Motion
• Pulleys and gears are used to increase or
decrease the angular velocity w of a rotating
shaft or wheel.
• When two gears or pulleys are connected, the
speed at which each turns compared to the
other is inversely proportional to the diameter of
that gear or pulley.
– The larger the diameter of a pulley or gear, the slower
it turns.
– The smaller the diameter of the pulley or gear, the
faster it turns when connected to the larger one.
– http://www.technologystudent.com/gears1/pulley2.htm
– http://www.technologystudent.com/gears1/pulley3.htm
Pulleys, Chains, Gears, Etc.
• The two wheels have different
radii and will turn at different
angular speeds.
• The linear velocity v of the belt is
constant. One part of the belt
cannot travel faster than any
other part of the belt.
• The linear velocity v where the
belt contacts the pulley wheel
must be the same for each
pulley wheel in order to turn both
wheels.
• v1 = v2, so r1·w1 = r2 · w2
Problem Example
On the runway waiting
for take-off, the blades
in a jet engine spin with
an angular velocity of
110 radians per second.
During take-off, the
blades accelerate to an
w f  wi
angular velocity of 330
a
rad/s in just 14 seconds.
t
What was the angular
330rad /s  110rad /s acceleration of the
=
blades?
14s
2
= 16 rad /s
Problem Example
The blades of a blender
spin with an angular
velocity of 375 rad/s
when in “blend” mode.
Pushing the “puree”
button accelerates the
blades with an angular
acceleration of 1740
rad/s2. What is the
angular velocity of the
blades after 7 revs (44
rad)?
Problem Example
wf  wi  2  a  q
2
2
wf  wi  2  a  q
2
2
rad 
rad



wf   375
   2  1740 2  44 rad 
s 
s



rad
wf  541 .98
s
• A device called a stroboscope or strobe light may be used
to measure or check the speed of rotation of a shaft or
other machinery part.
• The stroboscope is used to slow down repeating motion to
be observed more conveniently.
• The light flashes rapidly and the rate of the flash can be
adjusted to coincide with the rotation of a point or points on
the rotating object.
• Knowing the rate of flashing will also then reveal the rate of
rotation.
• A slight variation in the rate of rotation and the flash will
cause the observed point to appear to move either forward
or backward.
• http://en.wikipedia.org/wiki/File:Strobe.gif
• Consider the stroboscope as used in mechanical analysis.
This may be a “strobe light” that is fired at an adjustable
rate. Suppose you are looking at something rotating at 60
revolutions per second: if you view it with a series of short
flashes at 60 times per second, each flash illuminates the
object at the same position in its rotational cycle, so it
appears that the object is stationary. Furthermore, at a
frequency of 60 flashes per second, persistence of vision
smooths out the sequence of flashes so that the perceived
image is continuous.
• If you view the same rotating object at 61 flashes
per second, each flash will illuminate it at a slightly
earlier part of its rotational cycle. Sixty-one flashes
will occur before you see the object in the same
position again, and you will perceive the series of
images as if it is rotating backwards once per
second.
• The same effect occurs if you view the object at 59
flashes per second, except that each flash
illuminates it a little later in its rotational cycle and
so, it seems to be slowly rotating forwards.
• In the case of motion pictures, action is captured
as a rapid series of still images and the same
stroboscopic effect can occur.
• Wagon-wheel effect: Motion-picture cameras
conventionally film at 24 frames per second. Although the
wheels of a vehicle are not likely to be turning at 24
revolutions per second (as that would be extremely fast),
suppose each wheel has twelve spokes and rotates at only
two revolutions per second. Filmed at 24 frames per
second, the spokes in each frame will appear in exactly the
same position. Hence, the wheel will be perceived to be
stationary. In fact, each photographically captured spoke in
any one position will be a different actual spoke in each
successive frame, but since the spokes are close to
identical in shape and color, no difference will be perceived.
• If the wheel rotates a little more slowly than two revolutions
per second, the position of the spokes is seen to fall a little
further behind in each successive frame and therefore the
wheel will seem to be turning backwards.
• If the wheel rotates a little more slowly than two
revolutions per second, the position of the spokes
is seen to fall a little further behind in each
successive frame and therefore the wheel will
seem to be turning backwards.