From Ch. 5 (circular motion): A mass moving in a circle

has a linear velocity v & a

linear acceleration a.

We’ve just seen that it also

has an angular velocity &

an angular acceleration.

There MUST be relationships between the linear & the angular quantities!

Relations of Angular & Linear Quantities

Connection Between Angular & Linear Quantities

v = (d/dt), d = Rdθ v = R(dθ/dt) = Rω

Depends on R(ω is the same for all points!)

vB = RBω, vA = RAω vB > vA since RB > RA

Objects farther from the axis of rotation will move faster.

Radians!

Every point on a rotating object has an angular velocity ω & a linear velocity v. They are related as:

v = (d/dt), d = Rdθ

v = R(dθ/dt) = Rωv depends on R

ω is the same for all points

Relations Between Angular & Linear Velocity

vB = RBω vA = RAω vB > vA

On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge & another child sits on a lion halfway out from the center.

a. Which child has the greater linear velocity v?

b. Which child has the greater angular velocity ω?

Conceptual Example 10-2 Is the lion faster than the horse?

Relation Between Angular & Linear Acceleration

dv = Rdω

atan = rα

atan depends on r

α is the same for all points

If the angular velocity ω of a rotating object changes, it has a tangential acceleration (in the direction of the motion).

Angular & Linear Acceleration

aR depends on r

ω is the same for all points

in the object

Even if the angular velocity ω is constant, each point on

the object has a radial or centripetal acceleration. From

Ch. 5 discussion, this is to the motion direction.

Total Acceleration

a ---

Two VECTOR components of acceleration

• Tangential: atan= rα

• Radial: aR= rω2

• Total acceleration

= Vector Sum: a = aR+ atan

Angular Velocity & Rotation Frequency

Rotation frequency: f = # revolutions/second (rev/s)

1 rev = 2π rad

ω = 2π f = Angular Frequency 1 rev/s 1 Hz (Hertz)

Period: Time for one revolution.

= (2π/ω)

Translational-Rotational Analogues & Connections

ANALOGUES

Translation Rotation

Displacement x θ

Velocity v ω

Acceleration a α

CONNECTIONS

d = rθ, v = rω

atan= r α

aR = (v2/r) = ω2 r

Correspondence between Linear & Rotational quantities

Example 10-3: Angular & Linear Velocities & Accelerations

A carousel is initially at rest (ω0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s2. At t = 8 s, calculate the following:

a. The angular velocity ω of the carousel.

b. The linear velocity v of a child located r = 2.5 m from the center.

c. The tangential (linear) acceleration atan of that child.

d. The centripetal acceleration aR of the child.

e. The total linear acceleration a of the child.

Example 10-4: Hard Drive

The platter of the hard drive of a computer rotates at frequency f = 7200 rpm (rpm = revolutions per minute = rev/min)

a. Calculate the angular velocity ω (rad/s) of the platter.

b. The reading head of the drive r = 3 cm (= 0.03 m) from the rotation axis. Calculate the linear speed v of the point on the platter just below it.

c. If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is r = 3 cm from the axis?

Example 10-5: Given ω as function of time

A disk of radius R = 3 m rotates at an angular velocity ω = (1.6 + 1.2t) rad/s where t is in seconds.

At t = 2 s, calculate:

a. The angular acceleration.

b. The speed v & the components atan & aR of the acceleration a of a point on the edge of the disk.

Section 10-2: Vector Nature of Angular Quantities

The angular velocity vector points along the axis of rotation, with the direction given by the right-hand rule. If the direction of the rotation axis does not change, the angular acceleration vector points along it as well.

Sect. 10.3: Kinematic Equations• Recall: 1 dimensional kinematic equations for uniform (constant)

acceleration (Ch. 2).

• We’ve just seen analogies between linear & angular quantities:

Displacement & Angular Displacement:

x θ

Velocity & Angular Velocity:

v ωAcceleration & Angular Acceleration:

a α • For α = constant, we can use the same kinematic equations

from Ch. 2 with these replacements!

The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

Example 10-6: Centrifuge Acceleration

A centrifuge rotor is accelerated from rest to frequency f = 20,000 rpm in 30 s.

a. Calculate its average angular acceleration.

b. Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?

Example: Rotating Wheel• A wheel rotates with constant angular acceleration α = 3.5 rad/s2.

It’s angular speed at time t = 0 is ω0 = 2.0 rad/s.

(A) Calculate the angular displacement Δθ it makes after t = 2 s.

Use: Δθ = ω0t + (½)αt2

= (2)(2) + (½)(3)(2)2 = 11.0 rad (630º)

(B) Calculate the number of revolutions it makes in this time.

Convert Δθ from radians to revolutions:

A full circle = 360º = 2π radians = 1 revolution

11.0 rad = 630º = 1.75 rev

(C) Find the angular speed ω after t = 2 s. Use:

ω = ω0 + αt = 2 + (3.5)(2) = 9 rad/s

Example: 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

ro = 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 the inner & outer radii:

ωi = (v/ri) = (1.3)/(2.3 10-2) = 57 rad/s = 5.4 102 rev/min

ωo = (v/ro) = (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