Unit 8 – Rotational Kinematics
8.1 – Rotational Motion and Angular Displacement
The centers of all circular paths define a line, called the axis of rotation.
The angle through which a rigid object rotates about a fixed axis is called the
angular displacement.
A rotating object may rotate either counterclockwise or clockwise, and
standard convention calls a counterclockwise displacement positive and a
clockwise displacement negative.
Definition of Angular displacement
When a rigid body rotates about a fixed axis, the angular displacement is the
angle ∆θ swept out by a line passing through any point on the body and
intersecting the axis of rotation perpendicularly. By convention, the angular
displacement is positive if it is counterclockwise and negative if it is clockwise.
SI Unit of Angular Displacement: radian (rad)
As a disc rotates, the point traces out an arc of length s, which is measured
along a circle of radius r.
𝜃 (in radians) =
Arc length 𝑠
=
Radius
𝑟
The number of radians that corresponds to 360o, or one revolution:
1 revolution = 360° = 2𝜋 rad
8.2 – Angular velocity and Angular Acceleration
Angular Velocity
For rotational motion about a fixed axis, the average angular velocity 𝜔
̅ is the
angular displacement divided by the elapse time during which the
displacement occurs.
Definition of Average Angular Velocity
Average angular velocity =
𝜔
̅=
Angular Displacement
Elapsed time
𝜃 − 𝜃0 ∆𝜃
=
𝑡 − 𝑡0
∆𝑡
SI Unit of Angular Velocity: radian per second (rad/s)
The instantaneous angular velocity ω is the angular velocity that exists at any
given instant. Remember this is a vector quantity so direction does matter.
The magnitude of the instantaneous angular velocity is called the
instantaneous angular speed.
Angular Acceleration
A changing angular velocity means that an angular acceleration is occurring.
Definition of Average Angular Acceleration
Average angular acceleration=
𝛼̅ =
Change in angular velocity
Elapsed time
𝜔 − 𝜔0 ∆𝜔
=
𝑡 − 𝑡0
∆𝑡
SI Unit of Average Angular Acceleration: radian per second squared (
𝑟𝑎𝑑
𝑠2
).
The instantaneous angular acceleration α is the angular acceleration at a given
instant. In this class we will assume the angular acceleration is constant
so 𝛼̅ = 𝛼.
Section: 8.1 & 8.2
Practice Problems: 3, 5
Homework: 1, 2, 4, 6, 7
8.3 – The Equations of Rotational Kinematics
Table 8.1 – The Equations of Kinematics for Rotational and Linear Motion
Rotational Motion
(α=constant)
Linear Motion
(𝑎=constant)
𝜔 = 𝜔0 + 𝛼𝑡
1
𝜃 = (𝜔0 + 𝜔)𝑡
2
1
𝜃 = 𝜔0 𝑡 + 𝛼𝑡 2
2
𝜔2 = 𝜔02 + 2𝛼𝜃
𝑣 = 𝑣0 + 𝑎𝑡
1
𝑥 = (𝑣0 + 𝑣)𝑡
2
1
𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2
2
𝑣 2 = 𝑣02 + 2𝑎𝑥
The equations of rotational kinematics can be used with any self-consistent
set of units for θ, α, ω, ω0, and t.
Section: 8.3
Practice Problems: 19
Homework: 16-18, 20, 21
8.4 – Angular Variables and Tangential Variables
When a rigid body rotates through and angle θ about a fixed axis, any point on
the body moves on a circular arc of length s and a radius r. Such a point has a
tangential velocity (magnitude = vT) and, possibly, a tangential acceleration
(magnitude = aT). The angular and tangential variables are related by the
following equations:
𝑣𝑇 = 𝑟𝜔
(𝜔 in rad/s)
𝜔 − 𝜔0
𝑎𝑇 = 𝑟 (
)
𝑡
𝑎 𝑇 = 𝑟𝛼
(𝛼 in rad/s2)
These equations refer to the magnitudes of the variables involved, without
reference to positive or negative signs, and only radian measure can be used
when applying them.
Section: 8.4
Practice Problems: 28, 33
Homework: 29-32
8.5 – Centripetal Acceleration and Tangential Acceleration
The magnitude ac of the centripetal acceleration of a point on an object
rotating with uniform or non-uniform circular motion can be expressed in
terms of the radial distance r of the point from the axis and the angular speed
ω:
𝑎𝑐 = 𝑟𝜔2
(𝜔 in rad/s)
⃗ that is the vector sum of two
This point experiences a total acceleration 𝒂
perpendicular acceleration components, the centripetal acceleration ac and
the tangential acceleration 𝒂𝑻 :
𝒂 = 𝒂𝒄 + 𝒂 𝑻
8.6 – Rolling Motion
The essence of rolling motion is that there is no slipping at the point where
the object touches the surface upon which it is rolling. As a result, the
tangential speed 𝑣𝑇 of a point on the outer edge of a rolling object, measure
relative to the axis through the center of the object, is equal to the linear
speed 𝑣 with which the object moves parallel to the surface. In other words,
we have
𝑣 = 𝑣𝑇 = 𝑟𝜔
(𝜔 𝑖𝑛 𝑟𝑎𝑑/𝑠)
The magnitudes of the tangential acceleration 𝑎 𝑇 and the linear acceleration
𝑎 of a rolling object are similarly related:
𝑎 = 𝑎 𝑇 = 𝑟𝛼
Section: 8.5 & 8.6
Practice Problems: 39, 47
Homework: 38, 40, 41, 46, 48-50, 52
(𝛼 𝑖𝑛 𝑟𝑎𝑑/𝑠 2 )