# Rotational Kinematics and Dynamics

Mechanics

Nov 14, 2013 (4 years and 5 months ago)

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Rotational Kinematics

and Dynamics

AP Physics

Lecture Notes

Angular Kinematics

q

r

s

r
s

q
Angular Kinematics

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
Angular Kinematics

Average angular Acceleration

t

Instantaneous angular Acceleration

t
lim

0
t

2
s
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