1. Rotational motion

2. Rotational motion
• We have learned about
translational motion up to this
point…
• Now, we will venture into
ROTATIONAL MOTION of a rigid
body
– Rigid body: a body with a
definite shape that does not
change
• Rigid bodies can be analyzed as:
– The translational motion of its
center of mass and the rotational
motion about its center of mass.
• We will concentrate on purely
rotational motion – all points in
the body move in circles.

3. Rotational Motion
C
Last chapter, we talked about an B
object in uniform circular motion:
• Period
• Tangential velocity D
r
• Centripetal Acceleration m
v
• Centripetal force A
Now, we will talk about an object’s
motion as it revolves in a
circle, when it is uniform or non-
uniform!

4. Rotational Motion
• Now we will describe the circular motion of an
object using angles, in terms of its:
– Angular displacement θ [radians]
– Angular velocity ω [rad/sec, rev/sec]
– Angular acceleration α[rad/sec2]

5. Angular Displacement
• Angular Displacement (θ)
is the angle that a rotating
object goes through.
• We measure this in radians
– A fraction of a revolution can
be measured in
degrees, grads or radians
• A degree is 1/360 of a
revolution
• We can convert to radians
using:
– π radians = 180 degrees
– One revolution = 2 radians
…and use factor label method!

6. Example: Angular Displacement
• A rubber stopper is twirled over a student’s head in a
physics lab. Calculate the angular displacement of
the stopper if it travels:
– 30 degrees
– 0.25 revolutions
– 90 degrees
– 1700 degrees
– 12 revolutions

7. Angular distance θ in radians
We can convert from linear distance
(meters) to angular distance (radians) by:
• first converting to radians
• Use equation
θ=s
r
s = arch length in meters (distance)
r = radius of circular path (meters)
θ = angle in radians

8. Example: finding arc length
• What is the angular displacement of a rubber
stopper that is twirled over a physics teacher’s head
at a radius of 0.4 m and it travels 3.0 meters?

9. Example: arc length
Example 8-1 from book: A particular bird’s eye can distinguish
objects that subtend an angle no smaller than about 3x10-4 rad.
How small of an object can the bird just distinguish when flying
at a height of 100 m?
Subtend: The angle formed by an object at a given external point
3 cm

10. Angular velocity ω
In rotational motion, we usually describe
the angular velocity as revolutions per
second (rev/sec, rps), or radians per
second
• You will often have to convert this
number, since it is usually given as a
frequency (revolutions per time frame)
• Conversion from linear velocity:
ω= v
r
v = tangential (linear) velocity (m/s)
r = radius of circular path (meters)
ω = angular velocity (rad/sec)

11. Angular Velocity
Unlike tangential
velocity, the angular
velocity is the same at
every point on a rigid
body, like a wheel

12. Example: angular velocity
Example 8-3: What is the angular and linear speed of a child
sitting 1.2 m from the center of a steadily rotating merry-go-
round that makes one complete revolution in 4.0 seconds?

13. Angular acceleration α
Angular acceleration occurs when the
angular velocity changes over time.
• It acts in the direction of rotation in a
circular motion (NOT the same as
centripetal acceleration)
• In this case, we must also introduce
tangential acceleration (at) since the
tangential velocity is changing
– If there is angular acceleration, there will
also be tangential acceleration
• We can use the following conversion:
α= at
r ar= ω2r
a = tangential (linear) acceleration (m/s) ar = radial (linear) acceleration (m/s)
r = radius of circular path (meters) r = radius of circular path (meters)
2)
ω = angular acceleration (rad/sec ω = angular acceleration (rad/sec2)

14. Example: Angular Acceleration
What is the tangential and angular acceleration of a child seated
1.2 m from the center of a steadily rotating merry-go-round that
makes one complete revolution in 4.0s?

15. Linear & rotational motion equivalents
Now let’s re-write the linear motion equations using our rotational motion values!

16. Examples: kinematic equations
Ex 8-5: A centrifuge rotor is accelerated from rest to 20,000 rpm
in 5.0 min. What is its average angular acceleration?

17. Examples: kinematic equations
Ex 8-5: A centrifuge rotor is accelerated from rest to 20,000 rpm in
5.0 min. (a) What is its average angular acceleration? (b) through
how many revolutions has the centrifuge rotor turned during its
acceleration period? Assume constant angular acceleration.
a. 7.0 rad/s2
b. 50,000 revs