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In the study of linear motion, the
important concepts in kinematics are
displacement ∆𝑥, velocity v, and
acceleration a. Each of these concepts
has its analog in rotational motion:
angular displacement ∆𝜃, angular
velocity 𝜔, and angular acceleration 𝛼.

We will consider mainly the rotation
of rigid objects. A rigid object is an
object with a definite shape that
doesn’t change, so that the particles
composing it stay in fixed positions
relative to one another.

In a purely rotational
motion, all points in the
object move in circles,
such as the point P in
the rotating wheel. The
center of these circles
all lie on a line called
the axis of rotation.

Every point in an object
rotating about a fixed axis moves
in a circle whose center is on the
axis and whose radius, r, the
distance of that point from the
axis of rotation. A straight line
drawn from the axis to any point
sweeps out the same angle 𝜃 in
the same time.

Point P moves through an
angle 𝜃 when it travels the
distance l measured along
the circumference of its
circular path. In this topic,
we will use radian as the
unit of angular measure. It is
a dimensionless unit and it
relates to degrees.

The numbers 2𝜋 rad, 𝜋 rad,
𝜋
2
rad
correspond to angles 360o, 180o, 90o,
respectively.
The angle 𝜃 subtended by an arc length l
along a circle of radius r, measured in radians
counterclockwise from the x-axis, is
𝜃 =
𝑙
𝑟

After a time interval ∆𝑡 has
elapsed, the line OP has moved
through the angle 𝜃 with respect to
the fixed reference line. The angle
𝜃, measured in radians (SI), is called
the angular position. Thus, the
object’s angular displacement is
the difference in its final and initial
angles:
∆𝜃 = 𝜃𝑓 − 𝜃𝑖

Note that we use angular variables to
describe the rotating object because each point
on the disc undergoes the same angular
displacement in any given time interval.
Then, we can rewrite the formula to be
∆𝜃 =
∆𝑙
𝑟
Where ∆𝑙 is a displacement along the circular arc.

The average angular
velocity (rad/s in SI unit)
of a rotating rigid object
during the time interval is
the angular displacement
divided by the time
elapsed:
ഥ
𝜔 =
𝜃𝑓 − 𝜃𝑖
𝑡𝑓 − 𝑡𝑖
=
∆𝜃
∆t

For very short time intervals, the
average angular velocity approaches
the instantaneous angular velocity,
just as in the translational case.
𝜔 = lim
∆t→0
∆𝜃
∆t

We take the angular velocity to be
positive when the object rotates in
counterclockwise motion and negative
when it is rotating clockwise. Also,
when the angular velocity is constant,
the instantaneous angular velocity is
equal to the average angular velocity.

Angular acceleration (rad/s2 in SI) is
defined as the change in angular
velocity divided by the time required to
make this change.
ത
𝛼 =
𝜔𝑓 − 𝜔𝑖
𝑡𝑓 − 𝑡𝑖
=
∆𝜔
∆t

Relating translational velocity
𝑣 =
∆𝑙
∆𝑡
to angular velocity
∆𝜃 =
∆𝑙
𝑟
, we have
𝑣 =
∆𝑙
∆𝑡
=
𝑟∆𝜃
∆𝑡
= 𝑟𝜔
Relating it to the centripetal
acceleration,
𝑎𝐶 =
𝑣2
𝑟
=
(𝑟𝜔)2
𝑟
= 𝜔2
𝑟

Thus, although 𝜔 is the
same for every point in
the rotating object at
any instant, v is greater
for points farther from
the axis of rotation.

If the angular velocity changes, then there
must be an angular acceleration pointing
tangent to the circular path. This is the equal
to the tangential acceleration mention in the
previous topic.
𝑎𝑡𝑎𝑛 =
∆𝑣
∆𝑡
=
𝑟∆𝜔
∆𝑡
= 𝑟𝛼

The total linear acceleration of a
point is now
a = atan + aC
𝑎 = 𝑎𝑇
2 + 𝑎𝐶
2 = 𝑟 𝛼2 + 𝜔4

Example #1:
A carousel is initially at rest. At t = 0, it
is given a constant angular acceleration of
0.060 rad/s2, which increases its angular
velocity for 8.0 s. At t = 8.0 s, determine the
following quantities:
a.The angular velocity of the carousel
b.The translational velocity of a child
located 2.5 m from the center
c.The tangential acceleration of that child
d.The centripetal acceleration of the child
e.The total linear acceleration of the child

We can relate the angular velocity to the frequency of
rotation. The frequency is the number of complete
revolutions per second.
1
𝑟𝑒𝑣
𝑠
= 2𝜋
𝑟𝑎𝑑
𝑠
= 2𝜋 1
𝑟𝑎𝑑
𝑠
Thus, the general formula, relating frequency and
angular velocity would be
𝑓 =
1
2𝜋
(𝜔)
𝜔 = 2𝜋𝑓

The unit for frequency, revolutions per second (rev/s),
is given the special name the hertz (Hz). That is
1 Hz = 1 rev/s = 1 s-1
The time required for one complete revolution is called
the period (T), and it is related to the frequency by
𝑇 =
1
𝑓
Thus, if a particle rotates at a frequency of three
revolutions per second, then the period of each
revolution is 𝑇 =
1
3
𝑠.

Example #2:
The platter of the hard drive of a computer rotates
at 7200 rpm (revolutions per minute = rev/min).
a.What is the angular velocity of the platter?
b.If the reading head of the drive is located 3.00 cm from
the axis of rotation, what is the linear/translational
speed of the point on the platter just below it?
c.If a single bit requires 0.50 μm of length along the
direction of motion, how many bits per second can the
writing head write when it is 3.00 cm from the axis?

Translational Equations Angular Equations
Equation Missing Quantity Equation Missing Quantity
𝑣 = 𝑣𝑜 + 𝑎𝑡 𝑥 𝑎𝑛𝑑 𝑥𝑜 𝜔 = 𝜔𝑜 + 𝛼𝑡 𝜃 𝑎𝑛𝑑 𝜃𝑜
𝑥 = 𝑥𝑜 +
𝑣𝑜 + 𝑣
2
𝑡
𝑎
𝜃 = 𝜃𝑜 +
𝜔𝑜 + 𝜔
2
𝑡
𝛼
𝑥 = 𝑥𝑜 + 𝑣𝑜𝑡 +
1
2
𝑎𝑡2 𝑣
𝜃 = 𝜃𝑜 + 𝜔𝑜𝑡 +
1
2
𝛼𝑡2 𝜔
𝑥 = 𝑥𝑜 + 𝑣𝑡 −
1
2
𝑎𝑡2 𝑣𝑜
𝜃 = 𝜃𝑜 + 𝜔𝑡 −
1
2
𝛼𝑡2 𝜔𝑜
𝑣2
= 𝑣𝑜
2
+ 2𝑎(𝑥 − 𝑥𝑜) 𝑡 𝜔2
= 𝜔𝑜
2
+ 2𝛼(𝜃 − 𝜃𝑜) 𝑡
Constant Angular Acceleration (𝜶 = ഥ
𝜶)
The angular equations for constant angular acceleration are
analogous to the translational equations.

Example #3:
A centrifuge rotor is accelerated from
rest to 20,000 rpm in 30 s.
a.What is its average angular acceleration?
b.Through how many revolutions has the
centrifuge rotor turned during its
acceleration period, assuming constant
angular acceleration?

Rolling Motion (Without Slipping)
Rolling without slipping is readily
analyzed and depends on static friction
between the rolling object and the
ground. The friction is static because
the rolling object’s point of contact with
the ground is at rest at each moment.
Rolling without slipping involves
both rotation and translation.
𝑣 = 𝑟𝜔

Example #4:
A bicycle slows down uniformly from vo = 8.40 m/s to rest
over a distance of 115m. Each wheel and tire has an overall
diameter of 68.0 cm. Determine the angular velocity of the
wheels at the initial instant (t = 0); the total number of
revolutions each wheel rotates before coming to rest; the
angular acceleration of the wheel; and the time it took to
come to a stop.

Example #5:
A wheel rotates with a constant angular
acceleration of 3.50 rad/s2. If the angular speed
of the wheel is 2.00 rad/s at t = 0,
a.Through what angle does the wheel rotate
between t = 0 and t = 2.00 s, in terms of radians?
b.How many revolutions made from the
preceding time interval?
c.What is the angular speed of the wheel at t =
2.00 s?

Example #6:
An airplane propeller slows from an
initial angular speed of 12.5 rev/s to a
final angular speed of 5.00 rev/s. During
this process, the propeller rotates through
21.0 revolutions. Find the angular
acceleration of the propeller, assuming it
is constant.

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