Rotational Kinematics - Vernon Kids

filercaliforniaMechanics

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

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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



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