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Physics 201, Lecture 17

Today’s Topics

q  Rotation of Rigid Object About A Fixed Axis (Chap. 10.1-10.4)

n  Motion of Extend Object n  Rotational Kinematics:

n  Angular Velocity n  Angular Acceleration

q  Kinetic Energy of a Rotating Object q  Moment of Inertia (More on Thursday)

q  Again and again, hope you have previewed!

Motion Of An Extended Object

q  Extended Object = An object with size and shape A collection of point like objects (particles)

q  Motion of Rigid Object §  Rigid Object: Relative positions of all composing particles are

fixed The shape of the rigid object does not change Ø  Motion of Rigid Object =

Motion of its Center of Mass + Rotation about the Center of Mass

a particle has no size/shape described by m and position

an extended object has mass, shape, and size. Described by m, CM, moments of inertia (this week).

Translational Motion And Rotational Motion q  Translational motion: The orientation of the object is

unchanged during the motion.

q  Rotational Motion: The object moves about an axis or center in circular fashion.

Motion of Rigid Object: Translation + Rotation

+ =

2

Review: Circular Motion q  Circular motion:

§  Angular velocity: ω = dθ/dt §  Linear velocity: v = rω, --- v always perpendicular to r

q  The acceleration has both a tangential

and a centripetal components:

§  The tangential component: at = Δv/Δt = rΔω/Δt = rα

§  The centripetal component: ac = rω2 = v2/r

•  Total acceleration: q  Rotation about a center:

A group of particle together in circular motion

a = aC +at

ac

ω

at

v

Rotation of Rigid Object About A Fixed Axis q  Rotation about fixed axis is the simplest case of rotation Motion is described by change of quantity Angle θ

à When rotating about a fixed axis, all elements on the rigid object are in circular motion with same angular speed: ω =dθ/dt

v

(radian) rs

=θ

sign convention +: counter clockwise

- : clockwise

Quiz: Angular Velocity q  Consider two points on a rigid object that rotates around a fixed

axis as shown. Ø  Which one has larger angular velocity? The Red dot, Blue dot, same. All points have the same angular velocity (ω) Ø  Which one has larger linear velocity? The Red dot, Blue dot, same. v = rω

Angular Velocity And Angular Acceleration

q  Angular Velocity (ω) describes how fast an object rotstes, it has two components: §  Angular speed: and

§  direction of ω: + counter clockwise - clockwise

Ø  All particles of the rigid object have the same angular velocity q  Angular Acceleration (α):

and

Note: the similarity between (θ,ω,α) and (x, v, a)

tave Δ

Δ≡

θω

€

ω ≡Δt→ 0limΔθ

Δt=dθdt

tave Δ

Δ≡

ωα

€

α ≡Δt→ 0limΔω

Δt=dωdt

è Angular velocity ω is a vector! (define direction next page)

è Angular acceleration α is also a vector!

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Direction of Angular Velocity (Right-hand Rule)

q  The direction of angular velocity ω is define by a “right-hand” rule

Practice: Right Hand Rule q  What is the direction of angular velocity of this rotation?

€

ω

Quiz/Practice: Right Hand Rule q  The object rotates about the z axis as shown. Use right hand

rule, what is the direction of its angular momentum?

§  Towards left §  Towards right §  Up §  Down §  Into page §  Out of page

Rotation 1-D motion Angular Velocity: Velocity: Angular Acceleration: Acceleration”

Similarity Between (θ,ω,α) and (x,v,a)

tave Δ

Δ≡

θω

€

ω ≡dθdt

tave Δ

Δ≡

ωα

€

α ≡dωdt

txvave Δ

Δ≡

€

v ≡dxdt

tvaave Δ

Δ≡

€

a ≡ dvdt

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Similarity Between (θ,ω,α) and (x,v,a): Kinematic Relationship

1-D motion

Linear Velocity and Acceleration with Rotation

For rigid object rotating about fixed axis

§ The linear velocity has only tangential component, i.e.

v = vt = rω

§  The linear acceleration can have both tangential and centripetal components:

at = dv/dt = rdω/dt = rα

ac = v2/r = rω2,

Rotational Kinetic Energy q  General Kinetic Energy: KEi = ½ mivi

2 total kinetic energy: KE = Σ ½ mivi2

q  For an object rotating about a fixed axis: vi = ri ω

KE = Σ ½ mivi2 = ½ Σ mi(riωi)2 = ½ (Σ miri

2)ωi2 = ½ Iω2

221 ωIKErot = :Energy Kinetic Rotational

axis Moment of Inertia

q  Moment of Inertia of an object about an axis

(unit of I : kgm2)

Ø  I depends on rotation axis, total mass, and mass distribution.

∑≡ 2iirmI :Inertia ofMoment

€

another form: I ≡ r2dmwhole object∫

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Quick Quiz The picture below shows two different dumbbell shaped objects. Object A has two balls of mass m separated by a distance 2L, and object B has two balls of mass 2m separated by a distance L. Which of the objects has larger moment of inertia for rotations around x-axis?

A.  A. B.  B. C.  They have the same moment of inertia

€

Case A: 2 ×mL2 = 2mL2

Case B: 2× 2m L2⎛

⎝ ⎜ ⎞

⎠ ⎟

2

= mL2

q  Image the hoop is divided into a number of small segments, m1 … q  These segments are equidistant from the axis

Or calculus form: axis

Exercise: Moment of Inertia of a Uniform Ring

€

I = r2dm = R2 dm∫∫ = MR2€

I = miri2∑ = ( mi)∑ R2 = MR2

axis

Exercise: Moment of Inertia of a Uniform Disc

Ø  Area density: α= M/A = M/(πR2) Ø  Mass element at r: dm=αdA= α rdθdr (review some basic geometric calculus of you are in question) q  Now Moment of inertia

I = r2 dmdisc∫

= r2α r dr dθ0

2π

∫0

R

∫

= 2πα r3 dr0

R

∫ =πα2R4 =

12MR2

R

r dA

Moments Of Inertial Of Various Objects

€

I = miri2∑ (= r2dm)∫

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Quick Quiz Compare a thick wall and a thin wall cylinder of same mass and outer radius. Which one has moment of inertia around the axis shown? Thin wall cylinder Thick wall cylinder

R

Quick Quiz Order the following objects, all having the same R and M, according their moments of inertia around there respective axis as shown.

( 1=largest)

1

1 2

3 3

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