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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