Post on 28-Mar-2018
transcript
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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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