Chapter 8: Rotational Motion

• Topic of Chapter: Objects rotating– First, rotating, without translating.

– Then, rotating AND translating together.

• Assumption: Rigid Body– Definite shape. Does not deform or change shape.

• Rigid Body motion = Translational motion of center of mass (everything done up to now) + Rotational motion about an axis through center of mass. Can treat the two parts of motion separately.

COURSE THEME: NEWTON’S LAWS OF MOTION!

• Chs. 4 - 7: Methods to analyze the dynamics of objects in

TRANSLATIONAL MOTION. Newton’s Laws! – Chs. 4 & 5: Newton’s Laws using Forces

– Ch. 6: Newton’s Laws using Energy & Work

– Ch. 7: Newton’s Laws using Momentum.

NOW• Ch. 8: Methods to analyze dynamics of objects in

ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! – First, Rotational Language. Analogues of each translational

concept we already know!

– Then, Newton’s Laws in Rotational Language.

A rigid body is an extended object whose size, shape, & distribution of mass don’t change as the object moves and rotates. Example: a CD

Rigid Body Rotation

Three Basic Types of Rigid Body Motion

Pure Rotational MotionAll points in the object movein circles about the rotation

axis (through the Center of Mass)

Reference Line

The axis of rotation is through O & is

to the picture. All points move in circles about O

r

In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time.

r

r

Sect. 8-1: Angular Quantities

• Description of rotational

motion: Need concepts:

Angular Displacement

Angular Velocity, Angular Acceleration

• Defined in direct analogy to linear quantities.

• Obey similar relationships!

Positive Rotation! r

• Rigid object rotation:– Each point (P) moves

in a circle with the

same center!

• Look at OP: When P

(at radius R) travels an

arc length ℓ, OP sweeps

out angle θ.

θ Angular Displacement of the object

Reference Line

r

• θ Angular Displacement• Commonly, measure θ in degrees.• Math of rotation: Easier if

θ is measured in Radians

• 1 Radian Angle swept out

when the arc length = radius

• When R, θ 1 Radian

• θ in Radians is defined as:

θ = ratio of 2 lengths (dimensionless)

θ MUST be in radians for this to be valid!

Reference Line

r

• θ in Radians for a circle of radius r, arc length is defined as: θ (/r)

• Conversion between radians & degrees:

θ for a full circle = 360º = (/r) radians

Arc length for a full circle = 2πr

θ for a full circle = 360º = 2π radians

Or 1 radian (rad) = (360/2π)º 57.3º

Or 1º = (2π/360) rad 0.017 rad– In doing problems in this chapter, put your

calculators in RADIAN MODE!!!!

Example 8-2: θ 310-4 rad = ? º

r = 100 m, = ?

a) θ = (310-4 rad)

[(360/2π)º/rad] = 0.017º

b) = rθ = (100) (310-4)

= 0.03 m = 3 cm

θ MUST be in radians in part b!

Angular Displacement

Average Angular Velocity =

angular displacement θ = θ2 – θ1

(rad) divided by time t:

(Lower case Greek omega, NOT w!)

Instantaneous Angular Velocity

(Units = rad/s) The SAME for all points

in the object! Valid ONLY if θ is in rad!

Angular Velocity(Analogous to linear velocity!)

• Average Angular Acceleration = change in angular velocity ω = ω2 – ω1 divided by time t:

(Lower case Greek alpha!)

• Instantaneous Angular Acceleration = limit of α as t, ω 0

(Units = rad/s2)

The SAME for all points in body! Valid ONLY for θ in rad & ω in rad/s!

Angular Acceleration(Analogous to linear acceleration!)

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

Δθ

Δ

r

Connection Between Angular & Linear Quantities

v = (/t), = rθ v = r(θ/t) = rω

Radians!

v = rω Depends on r(ω is the same for all points!)

vB = rBωB, vA = rAωA vB > vA since rB > rA

Summary: Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related as:

Relation Between Angular & Linear Acceleration

In direction of motion:(Tangential acceleration!)

atan= (v/t), v = rω

atan= r (ω/t)

atan= rα

atan : depends on r

α : the same for all points

_____________

Angular & Linear AccelerationFrom Ch. 5: there is also

an acceleration to the

motion direction (radial or

centripetal acceleration)

aR = (v2/r)

But v = rω

aR= rω2

aR: depends on r

ω: the same for all points

_____________

Total Acceleration Two vector components

of acceleration

• Tangential:

atan= rα

• Radial:

aR= rω2

• Total acceleration

= vector sum:

a = aR+ atan

_____________

a ---

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 & ConnectionsANALOGUES

Translation Rotation

Displacement x θ

Velocity v ω

Acceleration a α

CONNECTIONS

= rθ, v = rω

atan= r α

aR = (v2/r) = ω2 r

Correspondence between Linear & Rotational quantities

On a rotating 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 translational velocity v?

b. Which child has the greater angular velocity ω?

Conceptual Example 8-3: Is the lion faster than the horse?

Example 8-4: Angular & Linear Velocities & Accelerations

A merry-go-round 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. 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 8-5: 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?