Sect. 10.3: Angular & Translational Quantities.
Relations Between Them

• From circular motion:
– A mass moving in a circle has a translational (linear) velocity v & a translational
(linear) acceleration a .
– We’ve just seen that it also has an angular velocity and an angular acceleration.

There MUST be relations between the translational
& the angular quantities!

Connection Between Angular & Linear Quantities
Radians!
 v = (  
/  t),  
= r  θ
 v = r(  θ/  t) = rω v = rω
 Depends on r
(ω is the same for all points!) v
2 v
2
= r
> v
2
1
ω
2
, v
1
= r
1
ω
1 since r
2
> r
1

Relation Between Angular & Linear Velocity
• v = (
  /
 t),
  = r
 θ
 v = r (
 θ /  t) = rω v : depends on r
ω
: the same for all points v
2
= r
2
ω
2 v
2
, v
1
> v
1
= r
1
ω
1

Relation Between Angular & Linear
Acceleration
In direction of motion:
( tangential acceleration )
_________________ a t
= (dv/dt), v = rω
 a t
= r (dω/dt) a t
= rα a t
: depends on r
α
: the same for all points

Angular & Linear Acceleration
From circular motion: there is also an acceleration
 to
_______________ motion direction
( radial or centripetal acceleration ) a c
= (v 2 /r)
But v = rω
 a c
= r ω 2 a c
: depends on r
ω
: the same for all points


Two
 vector components of acceleration
•
Tangential:
_________________ a t
= rα
•
Radial: a c
= rω 2
•
Total acceleration
= vector sum: a = a c
+ a t a  ---

NOTE!
• The tangential component of the acceleration, a t
, is due to changing speed
• The centripetal component of the acceleration, a c
, is due to changing direction
• The total acceleration can be found from these components with standard vector addition: a
 a t
2  a r
2  r
2

2  r
2

4  r

2  
4

• Rotation frequency: f = # revolutions / second (rev/s)
1 rev = 2π rad
 f = (ω/2π) or
ω = 2π f = angular frequency
1 rev/s

1 Hz (Hertz)
• Period: Time for one revolution.

T = (1/f) = (2π/ω)

Translational-Rotational Analogues &
Connections
ANALOGUES
Displacement
Translation Rotation x
θ
Velocity
Acceleration v a
ω
α
CONNECTIONS s = rθ, v = r ω a t
= r α a c
= (v 2 /r) = ω 2 r

• Consider a CD player playing a CD. For the player to read a CD, the angular speed ω must vary to keep the tangential speed constant ( v = ωr ).
A CD has inner radius r f r i
= 23 mm = 2.3

10 -2 m & outer radius
= 58 mm = 5.8

10 -2 m . The tangential speed at the outer radius is v = 1.3 m/s .
(A) Find angular speed in rev/min at inner radius:
ω i
= (v/r
Outer radius: i
ω
) = (1.3)/(2.3 f
= (v/r f

10 -2
) = (1.3)/(5.8
) = 57 rad/s = 5.4

10 -2

) = 22 rad/s = 2.1
10

2
10 rev/min
2 rev/min
•
(B) Standard playing time for a CD is 74 min, 33 s ( = 4,473 s ). How many revolutions does the disk make in that time?
 θ = (½)(ω i
+ ω f
)t = (½)(57 + 22)(4,473 s)
= 1.8

10 5 radians = 2.8

10 4 revolutions