Introduction to Rotational Motion
The motion of a rigid body (an object with a definite shape that
does not change) can be analyzed as the translational motion of its
center of mass AND the rotational motion about its center of mass.
All points on a rotating rigid body move in circles and the center of these circles lie on a line called the axis of rotation.
Axis of Rotation
Axis of Rotation
Angular Displacement
Symbol:
θ
Unit: radian, rad 360° = 2
π
radians1 rad 57.3°
r
= s
Angular Velocity
Symbol: ω (lower case omega) Unit: rad/s
Direction: use right hand rule
Every point on a rotating rigid body has the same angular velocity (ω) However, the farther away an object is from the center of rotation, the
greater its tangential velocity (v)
dt d
=
Angular Velocity Direction and Sign
Typically, we use the convention that counterclockwise rotations are
positive angular velocity (vector is up from right hand rule)
Clockwise rotations are negative
angular velocity (vector is down from right hand rule)
dt d
=
Angular to Linear Velocity Relation
v = r
If an object is rotating with an angular speed of , the linear
velocity for an object at any point is then times the radius r (how far that point is from the axis of
rotation)
Objects farther from the axis are
moving with a greater linear velocity
Angular Acceleration
Symbol: (lower case alpha) Units: rad/s2
𝛼 = 𝑑𝜔 𝑑𝑡
Angular acceleration is the rate of change of an objects angular
velocity.
Radial acceleration vs. tangential acceleration
If an object is spinning with a constant angular speed, a mass towards the rim will have a
centripetal acceleration (also called radial acceleration), but there is no linear acceleration because it’s not speeding up If an object is speeding up (or slowing down), then not only does it have a centripetal
acceleration, but it has a linear acceleration as well
Linear and Angular Equivalent
x is like
v is like
a is like
For uniform accelerations, we can use the motion equations like we did before. Just switch out the variable to generate the
rotational kinematic equations.
2 0
0 2 0 2
0
2 1 2
at t
v x
x
x a
v v
at v
v
+ +
=
+
=
+
= =
0+ t
2=
02+ 2
2 0
0
2
1 t
t
= + + dt
x v d
=
dt d
= dt
v a d
=
dt d
=
More Linear-Angular Relationships
r r r r
a v
r a
r v
r s
2 2
2
( )
=
=
=
=
=
=
Centripetal acceleration in rotational form.
Example 1
A disc with a radius of 2.0 m makes 5 rotations in 8.0 seconds.
What is the angular velocity of the disc?
Point A is 1.0 m from the center and Point B is 2.0 m from the center. How does the angular velocity of point A compare to that of point B?
Example 1
A disc with a radius of 2.0 m makes 5 rotations in 8.0 seconds.
What is the angular velocity of the disc?
𝜔 = ∆𝜃
∆𝑡 = 5(2𝜋 𝑟𝑎𝑑)
8𝑠 = 5𝜋 4
𝑟𝑎𝑑 𝑠
Point A is 1.0 m from the center and Point B is 2.0 m from the center. How does the angular velocity of point A compare to that of point B?
They both have the same angular velocity because they are both rotating at the same rate.
Example 2
A disc with a radius of 2.0 m makes 3 rotations in 1.0 second.
What is the angular velocity of the disc?
Point A is 1.0 m from the center and Point B is 2.0 m from the center.
Determine the linear velocity of point B and the linear velocity of point A.
Example 2
A disc with a radius of 2.0 m makes 3 rotations in 1.0 second.
What is the angular velocity of the disc?
𝜔 = ∆𝜃
∆𝑡 = 3(2𝜋 𝑟𝑎𝑑)
1𝑠 = 6 rad/s
Point A is 1.0 m from the center and Point B is 2.0 m from the center. Determine the linear velocity of point B and the linear
velocity of point A.
𝑣 = 𝑟𝜔 = 1 𝑚 6𝜋 𝑟𝑎𝑑
𝑠 = 6 𝑚 𝑠 𝑣 = 𝑟𝜔 = 2 𝑚 6𝜋 𝑟𝑎𝑑
𝑠 = 12 𝑚/𝑠
Example 3
A disc with a radius of 2.0 m starts from rest and reaches an angular velocity of 4 rad/s in 5 seconds.
What is the angular acceleration of the disc?
Determine the angular displacement of the disc in this time interval.
Example 3
A disc with a radius of 2.0 m starts from rest and reaches an angular velocity of 4 rad/s in 5 seconds.
What is the angular acceleration of the disc?
𝛼 = ∆𝜔
∆𝑡 = 4𝜋
𝑟𝑎𝑑
𝑠 − 𝜋𝑟𝑎𝑑
𝑠
5 𝑠 = 3𝜋
5
𝑟𝑎𝑑 𝑠2
Determine the angular displacement of the disc in this time interval.
𝜃 = 0 + 0 5 + 1
2 (3𝜋
5 )(5)2 𝜃 = 15𝜋
2 radians
Example 4
A disc with a radius of 2.0 m initially rotating at rad/s speeds up and
reaches an angular velocity of 4 rad/s in 6 seconds.
What is the angular acceleration of the disc?
Determine the tangential acceleration for point B at the edge of the disc during this time interval.
Example 4
A disc with a radius of 2.0 m initially rotating at
rad/s speeds up and reaches an angular velocity of 4 rad/s in 6 seconds.
What is the angular acceleration of the disc?
𝛼 = ∆𝜔
∆𝑡 = 4𝜋
𝑟𝑎𝑑
𝑠 − 𝜋𝑟𝑎𝑑
𝑠
6 𝑠 = 𝜋
2
𝑟𝑎𝑑 𝑠2
Determine the tangential acceleration for point B at the edge of the disc during this time
interval.
𝑎 = 𝑟𝛼 = 2𝑚 𝜋
2 = 𝜋 m/s2
Example 5
A disc with a radius of 2.0 m has a
constant angular velocity of 4 rad/s.
What is the angular acceleration of the disc?
Determine the centripetal acceleration for a mass at point A that is 1 m from the center.
Determine the centripetal acceleration for a mass at point B that is 2.0 m from the center.
Example 5
A disc with a radius of 2.0 m has a constant angular velocity of 4 rad/s.
What is the angular acceleration of the disc?
There is no angular acceleration because the disc is maintaining a constant angular velocity.
Determine the centripetal acceleration for a mass at point A that is 1 m from the center.
𝑎 = 𝜔2
𝑟 = (4𝜋)2
(1 𝑚) = 16𝜋2 m/s2
Determine the centripetal acceleration for a mass at point B that is 2.0 m from the center.
𝑎 = 𝜔2
𝑟 = (4𝜋)2
(2 𝑚) = 8𝜋2 m/s2
Example
While sitting on a carousel, a child makes one revolution in 16 seconds. The child is sitting on the horse that is located 3.5 m from the center.
a) calculate the tangential velocity of the child.
b) calculate the angular velocity of the child.
c) if the carousel slows down at a constant rate and comes to a stop in 40 seconds, what is the angular acceleration of the carousel?
d) How many radians did the child rotate while the carousel was slowing down?
a) calculate the tangential velocity of the child.
b) calculate the angular velocity of the child.
c) if the carousel slows down at a constant rate and comes to a stop in 40 seconds, what is the angular acceleration of the carousel?
d) How many radians did the child rotate while the carousel was
slowing down?
m/s 37 . 1
s 16
) m 5 . 3 ( 2 2
)
=
=
= v
T v r
a
rad/s 0.393
) ( m 3.5 m/s
37 . 1
b)
=
=
=
r v
rad/s2
0098 .
0
s 40
rad/s 0.393
- rad/s 0
c)
−
=
=
=
t
rad 7.88
s) 40 )(
rad/s 0098
. 0 2( s) 1 rad/s)(40 393
. 0 (
2 1
d)
2 2
0
2 0
=
− + +
=
+ +
=
t t