Rotational Kinematics and Dynamics

filercaliforniaMechanics

Nov 14, 2013 (3 years and 7 months ago)

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Rotational Kinematics

and Dynamics

AP Physics

Lecture Notes

Angular Kinematics

q

r

s

r
s

q
Angular Kinematics

radian (rad)

1
q
2
q
q

Angular Displacement

1
2
q

q

q

Angular Kinematics

Average angular Speed

t

q



Instantaneous angular Speed

dt
d
t
lim

0
t
q


q





s
rad
Angular Kinematics

Average angular Acceleration

t





Instantaneous angular Acceleration

t
lim

0
t







2
s
rad
Angular Kinematics

dt
d


Quantity Linear Angular


Angular Kinematics

Acceleration a



V敬潣楴礠†††††††††瘠††††††††††††


䑩獰污捥浥湴n††††⁸††††††††††††
q

at
v
v
i
f


t

i
f






Linear Equations

Angular Equations

Angular Kinematics

2
2
1
i
i
f
at
t
v
x
x



ax
2
v
v
2
i
2
f


q





2

2
i
2
f
2
2
1
i
i
f
t
t





q

q

t

2
v
v
x
f
i








t

2

f
i










q

Acceleration a




V敬潣楴礠†††††††††瘠




䑩獰污捥浥湴n††††⁸†


q

Angular and Linear Relationships

dt
ds
v

2
r


Angular Kinematics

q

d

r
ds


q
dt
d
dt
dv
a
t



d

r
dv



dt
d
r
v
a
2
r

dt
d
r
v
q

dt
ds
v



r
v
dt
d
r
v
q

dt
d
r
a
t


dt
dv
a
t



r
a
t
dt
d
r
a
t




r
r
2


r
v
a
2
r

Rotational Energy

m
1

m
2

r

v

v

2
i
i
i
2
1
v
m


2
i
2
i
i
2
1
R
r
m
K











I
r
m
i
2
i
i


Moment of

Inertia

Rotational

Kinetic Energy



r
v
2
2
2
1
2
1
2
1
L
v
m
v
m
K


2
2
i
i
i
2
1
L
r
m
K



2
i
i
i
2
1
v
m


2
2
1
R
I
K


2
i
2
i
i
2
1
R
r
m
K











Moment of Inertia





i
2
i
i
lim

0
m
r
m
I
Calculating moment of inertia



dm
r
I
2
Parallel
-
axis Theorem

2
mL
3
1
I





2
m

1.5
kg

2.5
3
1
I

2
m

kg

1.88
I

2
2
mL
4
1
mL
12
1
I


I = ?

L = 1.5 m

m = 2.5 kg

Example

2
12
1
cm
mL
I

D

L

2
cm
mD
I
I


2
cm
mD
I
I


2
L
D

2
2
2
L
m
mL
12
1
I








2
cm
mD
I
I


Torque

F
r





q


sin
rF
F

r

q

q

r sin(
q
)




rF
Newton’s 2nd Law for Rotation

r

m

F

a

ma
F

Fr




r
a






mr
2




I
Newton’s 2nd law

for rotation

r
F


I
mr
2






r
m
r
ma
F




I






mr
2
Rotational Energy

ds

r

F

d
q

q

dt
d
θ

dt
dW






r

θ
Fsin
d
θ
r

ds

P
dt
dW

ds
F
dW


d
θ

dW




d
θ
r

θ
Fsin
dW





P


ds

in
s
F
dW
q



q
dt
d
Rotational Energy

d
θ

dW



dt
d
d
d
I






q
q









f
i
d

I
W
2
i
2
1
2
f
2
1
I
I
W





d
d






q
q
f
i
2
2
I
W




R
2
2
1
K
I



I




dt
d
I


q



d
d
I

Rotational

Work
-
Energy

Theorem




d

I
dW
R
i
R
f
R
K
K
K
W