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

 There MUST be relations 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!) v2 = r2ω2, v1 = r1ω1 v2 > v1 since r2 > r1

Relation Between Angular & Linear Velocity v = (/t),  = rθ  v = r (θ /t) = rω v : depends on r ω : the same for all points v2 = r2ω2, v1 = r1ω1 v2 > v1

Relation Between Angular & Linear Acceleration In direction of motion: (tangential acceleration) at = (dv/dt), v = rω  at= r (dω/dt) at = rα at: 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) ac = (v2/r) But v = rω  ac= rω2 ac: depends on r ω: the same for all points _______________

Total Acceleration  Two  vector components of acceleration Tangential: at = rα Radial: ac= rω2 Total acceleration = vector sum: a = ac+ at _________________ a ---

Total Acceleration NOTE! The tangential component of the acceleration, at, is due to changing speed The centripetal component of the acceleration, ac, is due to changing direction The total acceleration can be found from these components with standard vector addition:

Relation Between Angular Velocity & 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 Translation Rotation Displacement x θ Velocity v ω Acceleration a α CONNECTIONS s = rθ, v = rω at= r α ac = (v2/r) = ω2 r

Example 10.2: CD Player = 1.8  105 radians = 2.8  104 revolutions 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 ri = 23 mm = 2.3  10-2 m & outer radius rf = 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: ωi = (v/ri) = (1.3)/(2.3  10-2) = 57 rad/s = 5.4  102 rev/min Outer radius: ωf = (v/rf) = (1.3)/(5.8  10-2) = 22 rad/s = 2.1  102 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)(ωi + ωf)t = (½)(57 + 22)(4,473 s) = 1.8  105 radians = 2.8  104 revolutions