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.) $or*ue and Momentum
4.1+ Conservation of 'ngular Momentum
4.11 ,or- in otational Motion
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4.1 Lnear Momentum and Its Conservaton
Pham Hong Quang Fundamental Science Department 3
Conservation of Momentum•Law of Conservation of Momentum - Thetotal momentum of an isolated system ofbodies remains constant.
Isolated system - one in which the onlyforces present are those between the objectsof the system.
Momentum before = momentum afterm1vi1 m!vi! = m1v f1 m!v f!
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4.2 Conservaton of Energy and Momentum nCollisions
Pham Hong Quang Fundamental Science Department 4
Momentum is
conserved in all
collisions.
Collisions in which
"inetic ener#y is
conserved as well are
called elastic
collisions$ and those
in which it is not are
called inelastic.f
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4.2 Conservaton of Energy and Momentum nCollisions
Pham Hong Quang Fundamental Science Department 5
%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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4.2 Conservaton of Energy and Momentum nCollisions
Pham Hong Quang Fundamental Science Department 6
2C f d
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4.2 Conservaton of Energy and Momentum nCollisions
Pham Hong Quang Fundamental Science Department 7
'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
Pham Hong Quang Fundamental Science Department 9
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
Pham Hong Quang Fundamental Science Department 10
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
Pham Hong Quang Fundamental Science Department 11
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4.3 Center of Mass
Pham Hong Quang Fundamental Science Department 12
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
Pham Hong Quang Fundamental Science Department 13
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
Pham Hong Quang Fundamental Science Department 14
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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4.4 Rotational Motion
Pham Hong Quang Fundamental Science Department 15
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4.4 Rotational Motion
Pham Hong Quang Fundamental Science Department 16
Centri/etal0or%e
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4.5 Rotational Energy
Pham Hong Quang Fundamental Science Department 17
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4.6 Rotational Inertia
Pham Hong Quang Fundamental Science Department 18
• 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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4.6 Rotational Inertia
Pham Hong Quang Fundamental Science Department 19
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4.6 Rotational Inertia
Pham Hong Quang Fundamental Science Department 20
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4.6 Rotational Inertia
Pham Hong Quang Fundamental Science Department 21
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4.7 Total mechanical Energy
Pham Hong Quang Fundamental Science Department 22
;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
Pham Hong Quang Fundamental Science Department 23
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
Pham Hong Quang Fundamental Science Department 24
$or*ue magnitude anddire%tion
F r ×=τ ⊥⊥ === F r F r F r ""sin"" φ τ
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4.9 Torque and Momentum
Pham Hong Quang Fundamental Science Department 25
'ngularmomentummagnitude anddire%tion
⊥⊥ === P r P r P r L ""sin"" φ
ω I P r L =×=
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4.9 Torque and Momentum
Pham Hong Quang Fundamental Science Department 26
→→→→
→→→→
=×=×+×= ∑ 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 $or*ue and
Momentum
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4.9 Torque and Momentum
Pham Hong Quang Fundamental Science Department 27
@ecause
ω I P r L =×=
Then α ϖ τ I dt
d I dt
Ld ===
%ere dt
d ϖ α =
6e%ond 7e5ton La5 in otational
motion
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49TorqueandMomentum
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4.9 Torque and Momentum
Pham Hong Quang Fundamental Science Department 29
∧→
=#
z mgbτ
∧→→→
=×=#
z bmgt vmr L
→∧→
== τ # z bmg dt
Ld
6olution
i f l
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4.10 Conservation of Angular Momentum
Pham Hong Quang Fundamental Science Department 30
dt
Ld
ext
→→
=∑τ #=∑ →
ext τ
t cons L tan=→
>if the net e/ternal tor(ueactin# on a system is Aero$the total vector an#ular
momentum of the systemremains constant?
Brom If
Then
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N V A 33
Thank you!