Rotational Kinematics
ROTATION
–
turn about an axis external
ex., Earth around the Sun
spin about internal axis
ex., Earth’s daily motion
REVOLUTION
–
O
P
Rotation about point O
Every point on the surface moves
in a circle
O
Angle Measure
θ
r
ℓ
θ
rad
= ℓ/r
1 radian: angle whose arc length
equals its radius
radian is a unitless measure
Radians vs. Degrees
1 revolution = 360
°
θ
rad
= ℓ/r = (2
Π
r)/r = 2
Π
180
°
=
Π
rad
ROTATIONAL (ANGULAR) SPEED
ω
(omega): angle/time
SI unit: radians/second
(at constant speed)
For 1 revolution:
angle =
time =
ω
=
2
Π
T
ω
=
2
Π
f
2
Π
radians
T seconds
LINEAR (TANGENTIAL) SPEED
(assuming a constant speed)
v (speed) = distance /time
For 1 revolution:
v = circumference/period
2
Π
r
T
2
Π
r f
v =
v =
RELATIONSHIP BETWEEN
LINEAR AND ROTATIONAL
SPEEDS
v =
v =
ω
r
2
Π
T
2
Π
r
T
ω
=
Angular speed:
Linear speed:
all points the same
points with larger r
have greater v
Acceleration
Since direction of motion changes,
acceleration towards center
a
rad
=
v
2
/r
=
ω
2
r
If speed changes, a
tan
=
Δ
v/
Δ
t
a
tan
= r(
Δω
/
Δ
t)
= r
α
M
R
Disk rotating at constant rate; makes a complete
rotation in T seconds.
Linear and angular speed of mass M?
Rate of rotation gradually increased.
Coefficient of static friction =
μ
s
.
Speed when mass begins to slip?
If mass is increased, what would
happen to v
max
?
If move mass to greater R, what
happens to max angular speed
before slipping?
Comparing Angular Motion Equations
with Linear Motion Equations
(constant acceleration)
Linear Angular
v = v
0
+ at
ω
=
ω
0
+
α
t
Δ
x = v
0
t + ½ at
2
Δθ
=
ω
0
t + ½
α
t
2
v
2
= v
0
2
+ 2a
Δ
x
ω
2
=
ω
0
2
+ 2
αΔθ
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