Lesson 5:
Describing Motion:
Angular Kinematics
Learning Objectives
After finishing this lesson, you should be able to:
• Define the following terms: rigid body, axis of
rotation, angular position, angular displacement,
angular velocity, and angular acceleration.
• Write equations for the following concepts:
angular displacement, angular velocity, and
acceleration.
Learning Objectives (continued)
After finishing this lesson, you should be able to:
• Convert angular velocity to linear velocity.
• Convert angular acceleration to tangential and
centripetal acceleration.
• Identify angular velocity on an angular positiontime curve.
Learning Objectives (continued)
After finishing this lesson, you should be able to:
• Identify angular acceleration on an angular
velocity-time curve.
• Given angular displacement and time data,
calculate angular velocity and angular
acceleration.
Section Question
Is the motion of the wheel
the same as the motion of a
body previously explored?
© gdvcom/Shutterstock.
Rigid Bodies
• Rigid Body
– _____________________________________
– Example:
• _____________________
Left: © Roman Gorielov/Shutterstock; Right: © Joe Belanger/Shutterstock.
Axis of Rotation
• __________________________
____________________
Frame of Reference and
Axis of Rotation
• Rotations occur in a plane about
an axis
– For the cardinal planes:
• ______________
(Anterior/Posterior
axis) = rotation in the xy
plane (frontal)
• _____________ (polar axis)
= rotation in the
xz plane (transverse)
• ___________
(medial/lateral axis) =
rotation in the yz plane
(sagittal)
Sign Conventions
_________________________
• __________________________________
– Can be measured in:
• Degrees (°)
• Radians (rads)
• 1 radian equals
• Approximately
• 57.3°(180°/π)
________________________
• __________________________
__________________________
– Change in angular
position between two
time periods of interest
Dq = q ¢¢- q ¢
Goniometer
Angular Displacements
• Two Scenarios:
• Both Scenarios:
– X to A to B
– Scenario 1: B to C
counterclockwise
– Scenario 2 : B to C
clockwise
© Kletr/Shutterstock.
Angular Displacements
Angular Velocity (ω)
• _________________– How fast a body is
rotating
– Quantity: scalar
• ____________________
___ (ω)
– How fast a body
is rotating in a
particular
direction
– Quantity: vector
– Typical unit: °/s
or rad/s
Angular Velocity
Angular speed
How fast a body is rotating
– Quantity: scalar
Note: Angular Velocity (ω)
• Quantity: vector
Angular distance/ speed scalar
Angular position/ displacement/ velocity vector
Angular acceleration vector
4.1.6 Angular Acceleration (α)
• ___________________________________________________
___________________________________
• Time rate of change of angular velocity
• Quantity: _____________
• Typical unit: °/s2 or rad/s2
Angular Acceleration
Comparing Linear and
Angular Kinematics
• ____________________
– A distance from the origin
• __________________
– A change in position
• __________________
– Amount of displacement in a given time
• ___________________
– Change in velocity in a given time
Comparing Linear and Angular
Kinematics
Relating Angular Kinematics
to Linear Kinematics
• Determining the angular velocity of the wheel is
interesting, but what does it mean?
• For a given angular velocity, just how fast is the
wheelchair actually going?
Relating Angular Kinematics
to Linear Kinematics
© Glen Jones/Shutterstock;
© Jeff Thrower/Shutterstock;
© Chris Mole/Shutterstock
© Mark Herreid/Shutterstock.
© Diego Barbieri/Shutterstock.
© Neale Cousland/Shutterstock.
Relation Between Linear and
Angular Position
The Relation between Linear
and Angular Velocity
• The linear velocity of a
point on a rotating
body is determined
by: v = l r ´ w
– ___________________________
_________________________
– ___________________________
__________________________
Components and Resultants
• The velocity vector will always be
perpendicular to the rigid body
– Tangent to the arc
– Use trig functions to calculate the resultant
vector
© Ljupco Smokovski/Shutterstock.
4.2.2 The Relation Between Linear
and Angular Acceleration
• For each angular velocity, there are two
linear accelerations to consider:
–____________________
• Change in magnitude of the linear
velocity
–______________________
• Change in direction of the linear
velocity
Relationships Among Radius of a
Circle,
Arc Length, and Angle Measured in
Radians
Tangential Acceleration
• Component of ________________ tangential to
______________ path
• Mathematically: aT = 𝛂 r
• Where
– aT = instantaneous tangential acceleration
– 𝛂 = instantaneous angular acceleration (must be in
rad/s2)
– r = radius
Centripetal (Radial) Acceleration
• ________________________: occurs when an object
slows down, speeds up, or changes direction
• The vT of an object moving on a circular path at
a constant 𝛚 changes
_____________________________________
– If released at different points on the path, object
will travel _____________ in different directions
(continued)
Tangential and Centripetal
Accelerations