Sect 10 3 Angular Translational Quantities Relations Between

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 = r 2ω2, v 1 = r 1ω1 v 2 > v 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 2ω2, v 1 = r 1ω1 v 2 > v 1

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 = (v 2/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: a --2 ac= rω • Total acceleration = vector sum: a = ac+ at

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 • 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 Translation Rotation Displacement x θ Velocity v ω Acceleration a α CONNECTIONS s = rθ, v = rω at= r α ac = (v 2/r) = ω2 r

Example 10. 2: CD Player • 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