rotational kinematics chapter 8. expectations after chapter 8, students will: understand and apply...
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Rotational Kinematics
Chapter 8
Expectations
After Chapter 8, students will: understand and apply the rotational versions of
the kinematic equations. be able to mathematically associate tangential
variables with corresponding angular ones understand and apply the concept of total
acceleration in rotational motion state and use the principle of rolling motion
A Brief Review from Chapter 5
Angular displacement:
Units: radians (rad)
S
rrS
A Brief Review from Chapter 5
Average angular
velocity:
units: rad/s
or: degrees/s, rev/min, etc.
x
r
v
r
v
t
Angular Acceleration
Average angular acceleration:
units: rad/s2
or: degrees/s2, rev/min2, etc.
0
0
ttt
Rotational Kinematic Equations
Definition of average angular velocity:
t
tt
2
12
1
0
0
Rotational Kinematic Equations
Definition of average angular acceleration:
t
tt
0
0
0
Rotational Kinematic Equations
A previous result:
2
0
2000
0
0
2
1
22
1
2
1
2
1
tt
tttt
t
t
Rotational Kinematic EquationsSolve definition of average acceleration for t:
Substitute into a previous result:
00
t
t
2
2
2
1
2
1
20
2
20
200
000
t
Comparison: Kinematic Equations Rotational Linear
( = constant) (a = constant)
2
2
12
1
20
2
20
0
0
tt
t
t
axvv
attvx
tvvx
atvv
2
2
12
1
20
2
20
0
0
Comparison: Kinematic EquationsSame equations, (some) different variables
Position, displacement: x
Time: t t
Velocity, speed: v
Acceleration: a
Average angular velocity is the angular displacement divided by the time interval in which it occurred.
x
r
v
rvr
v
t
rtv
rx
tvx
TT
T
T
) (small
Angular and Tangential Velocity
From the definition of linear acceleration:
From the definition of angular acceleration:
Combining:
Angular and Tangential Acceleration
t
rt
rr
t
vva TTT
000
t0
raT
From chapter 5: But:
Substituting:
Angular Velocity, Centripetal Acceleration
r
va TC
2
rvT
222
r
r
raC
2raC
The tangential and centripetal accelerations are vector components of the total acceleration.
Total Acceleration
C
T
TC
a
a
aaa
tan
22
When a circular, cylindrical, or spherical object rolls without slipping over a surface:
Rolling Motion: Velocity
rv linear speed of axle
wheel radius
angular speed of wheel
When a circular, cylindrical, or spherical object rolls without slipping over a surface:
Rolling Motion: Acceleration
ra linear acceleration of axle
wheel radius
angular acceleration of wheel
Angular displacement, , is not a vector quantity. the reason: addition of angular displacements is not commutative. Where you end up depends on the order in which the angular displacements (rotations) occur.
Angular Vectors
Angular velocity, , and angular acceleration, , are vectors.
Magnitudes: and
Directions: Parallel to the axis of rotation, and in the direction given by the right-hand rule:
Angular Vectors
t
t
Right-hand rule direction for :
Angular Vectors
Right-hand rule direction for :
Also parallel to axis of rotation Same direction as change in vector
Same direction as if is increasing in magnitude Opposite direction from if is decreasing in magnitude
Angular Vectors