Chapter 4 Pham Hong Quang

8/16/2019 Chapter 4 Pham Hong Quang

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Fundamental of PhysicsChapter 4

PETROVIETNAM UNIVERSITY

FUNDAMENTAL SCENCE DEPA!TMENT

Hanoi, August 2011

Pham Hong QuangE-mail: [email protected]


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Chapter 4 Moton of a System of Partcles andof a Rigid Object

Pham Hong Quang Fundamental Science Department 2

4.1 Linear Momentum and Its Conservation

4.2 Conservation of Energy and Momentum

in Collisions

4.3 Center of Mass

4.4 otational Motion

4.! otational Energy

4." otational Inertia

4.# $otal Me%hani%al Energy

4.& Parallel '(is $heorem4.) $or*ue and Momentum

4.1+ Conservation of 'ngular Momentum

4.11 ,or- in otational Motion


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4.1 Lnear Momentum and Its Conservaton


Conservation of Momentum•Law of Conservation of Momentum - Thetotal momentum of an isolated system ofbodies remains constant.

Isolated system - one in which the onlyforces present are those between the objectsof the system.

Momentum before = momentum afterm1vi1 m!vi! = m1v f1 m!v f!


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4.2 Conservaton of Energy and Momentum nCollisions


Momentum is

conserved in all

collisions.

Collisions in which

"inetic ener#y is

conserved as well are

called elastic

collisions$ and those

in which it is not are

called inelastic.f


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%ere we have two

objects collidin#

elastically. &e "now the

masses and the initial

speeds.

'ince both momentum

and "inetic ener#y areconserved$ we can write

two e(uations. This

allows us to solve for the

f d


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2C f d


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'uppose that the masses and initial velocities of

both particles are "nown.*(uations +.1, and +.1+ can be solved for the)nal speeds in terms of the initial speeds becausethere are two e(uations and two un"nowns

It is important to remember that theappropriate si#ns for v 1i and v 2i must be

included in *(uations +.! and +.!1.


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4.3 Center ò Mass


In 0a$ the diver2s motion is pure translation3 in 0b it

is translation plus rotation. There is one point that moves in the same path aparticle would ta"e if subjected to the same force asthe diver. This point is called the center of mass

0CM.


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4.3 Center of Mass


The #eneral motion of an object can be

considered as the sum of the translationalmotion of the CM$ plus rotational$ vibration orother forms of motion about the CM.


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4.3 Center of Mass



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4.3 Center of Mass


The center of #ravity is the

point where the #ravitationalforce can be considered toact. It is the same as thecenter of mass as lon# as the

#ravitational force does notvary amon# di4erent parts ofthe object.

The center of #ravity can be

found e/perimentally bysuspendin# an object fromdi4erent points. The CM neednot be within the actualobject 5 a dou#hnut2s CM is inthe center of the hole.


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4.3 Center of Mas


The total momentum of a system ofparticles is e(ual to the product ofthe total mass and the velocity ofthe center of mass.

The sum of all the forces actin# ona system is e(ual to the total massof the system multiplied by theacceleration of the center of mass


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4.4 Rotational Motion


RotationalMotion

QuantityLinear

Motion

θ Position x

Δθ Displaceent Δ x

ω Velocity v

α Acceleration a

t Tie t

otational and Linear inemati%s


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Centri/etal0or%e


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4.5 Rotational Energy



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4.6 Rotational Inertia


• The rotational inertia for a collection ofparticles is de)ned as

• The rotational "inetic ener#y 6*7 of a ri#idobject rotatin# with an an#ular speed 8about a )/ed a/is and havin# a rotational ofinertia I is

!

i i

i

I m r =∑

Requirement: 8 must be e/pressed in rad9s.SI Unit of Rotational Kinetic Energy: joule

0:


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4.7 Total mechanical Energy


;n object that under#oes combined rotational and

translation motion has two types of "ineticener#y

01a rotational "inetic ener#y due to its rotationabout its center of mass

0!a translational "inetic ener#y due to translation

of its center of mass. The total mechanicalener#y is


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4.8 Parallel Axis Theorem


This theorem will allow us to calculate the

moment of inertia of any rotatin# body around

any a/is$ provided we "now the moment of

inertia about the center of mass.

It basically states that the Moment of Inertia I/

around any a/is


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4.9 Torque and Momentum


$or*ue magnitude anddire%tion

F r ×=τ ⊥⊥ === F r F r F r ""sin"" φ τ


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'ngularmomentummagnitude anddire%tion

⊥⊥ === P r P r P r L ""sin"" φ

ω I P r L =×=


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→→→→

→→→→

=×=×+×= ∑ τ F r dt

P d r P

dt

r d

dt

Ld

→→

= vdt

r d

#=×=×→→→

→

P v P

dt

r d ∑→

→

= F dt

P d

%ere

and

>The net tor(ue actin# on a particle is e(ual to thetime rate of chan#e of its an#ular momentum?.

elation et5een $or*ue and

Momentum


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@ecause

ω I P r L =×=

Then α ϖ τ I dt

d I dt

Ld ===

%ere dt

d ϖ α =

6e%ond 7e5ton La5 in otational

motion


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49TorqueandMomentum


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∧→

=#

z mgbτ

∧→→→

=×=#

z bmgt vmr L

→∧→

== τ # z bmg dt

Ld

6olution

i f l


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4.10 Conservation of Angular Momentum


dt

Ld

ext

→→

=∑τ #=∑ →

ext τ

t cons L tan=→

>if the net e/ternal tor(ueactin# on a system is Aero$the total vector an#ular

momentum of the systemremains constant?

Brom If

Then


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N V A 33

Thank you!

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Chapter 4 Pham Hong Quang