lecture 3. mechanical force
TRANSCRIPT
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Mechanical ForceMechanical Force
Equilibrium
Prof. Dr hab. Zbigniew DunajskiProf. Dr hab. Zbigniew Dunajski
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Translational Equilibrium First Condition of Equilibrium
The net eternal force must be !ero
This is a necessar"# but not sufficient#
condition to ensure that an object is incom$lete mechanical equilibrium This is a statement of translational
equilibrium
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%ector
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Module
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&ddition
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Product of %ectors
Work = force x distance
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Friction
The kinetic frictional force is exerted on the upper body by the
stationary lower body. The upper body is moving with velocity
and is pressed together with the lower body by a normal force . It
may also be acted upon by an additional non-normal external force
Fext .
is called the coefficient of
kinetic friction.
is thecoefficient of static friction. Generally we find that
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TThe larger the force is, the larger thehe larger the force is, the larger the
accelerationacceleration
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Torque
The door is free to rotate about an ais through '
There are three factors that determine theeffecti%eness of the force in o$ening the door( The magnitude of the force The position of the a$$lication of the force The angle at which the force is a$$lied
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Torque# contTorque# cont
Torque# τ# is the tendenc" of aforce to rotate an object about
some ais τ ) r F
τ is the torque F is the force
s"mbol is the *reek tau r is the length of the $osition %ector
+, unit is -.m
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Lever
- F r 1 + Fc r 2 + R.
0 = 0
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e%er &rm
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/ight 0and /ule Point the fingers
in the direction of
the $osition %ector Curl the fingerstoward the force%ector
The thumb $ointsin the direction ofthe torque
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Eam$lea. & man a$$lies a force
as shown. Find the
torque on the doorrelati%e to the hinges.
b. +u$$ose a wedge is$laced 1.23 m from
the hinges. 4hatforce must the wedgeeert so that the doorwill not o$en.
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*eneral Definition of
Torque Taking the angle into account leads to a
more general definition of torque(
τ = r F sin θ F is the force r is the $osition %ector θ is the angle between the force and the
$osition %ector
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Equilibrium Eam$le
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&is of /otation ,f the object is in equilibrium# it does not
matter where "ou $ut the ais ofrotation! for calculating the net torque 'ften the nature of the $roblem will suggest a
con%enient location for the ais 5usuall" toeliminate a torque6
4hen sol%ing a $roblem# "ou must s$ecif" an
ais of rotation 'nce "ou ha%e chosen an ais# "ou must maintain
that choice throughout the $roblem
The fulcrum does matter# but the origin! selected for le%er arms will not
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/otational Equilibrium To ensure mechanical equilibrium#
"ou need to ensure rotational
equilibrium as well as translational The +econd Condition of
Equilibrium states The net eternal torque must be !ero
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Center of *ra%it" The force of gra%it" acting on an
object must be considered
,n finding the torque $roduced b"the force of gra%it"# all of theweight of the object can be
considered to be concentrated at asingle $oint# the center of gra%it"
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Calculating the Center of
*ra%it" The object is
di%ided u$ into alarge number of%er" small $articlesof weight 5mi g6
Each $article will
ha%e a set ofcoordinatesindicating itslocation 5#"6
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Find "our center of gra%it"
Consider a $erson with L ) 178 cm and
weight w ) 712 -. a"ing on a board withweight w b ) 9: -# a scale has a force
reading of F ) 823 -. Find the $erson"scenter of gra%it".
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Eam$le of a Free ;od"
Diagram 5Forearm6
,solate the object to be anal"!ed
Draw the free bod" diagram for that object ,nclude all the eternal forces acting on the object
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-ewton"s +econd aw for a
/otating 'bject
The angular acceleration is directl"$ro$ortional to the net torque
The angular acceleration isin%ersel" $ro$ortional to themoment of inertia of the object
H k ’ L
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Hooke’s Law
k ’
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HookeHooke’’s Laws Law
Force !"
#xtension m"
$radient gives
the spring
constant
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+tress# +train < 0ooke=s aw
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Takes into accountthe thickness andlength of a wire.
>oung Modulus E )stress ? strain.
@nits are - mAB or Pa E can be worked out
from the gradient ofthe stressAstraingra$h.
%tress
&a"
%train
!ot always
easy to see
Elastic
limit
Ultimate
tensile
stress
%naps
Young ModulusYoung Modulus
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Fig '.()* p.))'
%lide (+
Eam$le of a Free ;od"
Diagram 5;eam6 The free bod"
diagram includes
the directions ofthe forces
The weights actthrough the
centers of gra%it"of their objects
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Eam$le of a Free ;od"
Diagram 5adder6
The free bod" diagram shows the normal force
and the force of static friction acting on theladder at the ground The last diagram shows the le%er arms for the
forces
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Total Energ" of a +"stem Conser%ation of Mechanical Energ"
/emember# this is for conser%ati%eforces# no dissi$ati%e forces such as
friction can be $resent Potential energies of an" other
conser%ati%e forces could be added
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Energ" Methods & ball of mass M
and radius /
starts from rest.Determine itslinear s$eed atthe bottom of the
incline# assumingit rolls withoutsli$$ing.
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More &ngular Momentum ,f the net torque is !ero# the angular
momentum remains constant
Conservation of Angular Momentum states( The angular momentum of as"stem is conser%ed when the neteternal torque acting on the s"stems is
!ero. That is# when
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Conser%ation of &ngular
Momentum# Eam$le 4ith hands and
feet drawn closer
to the bod"# theskater"s angulars$eed increases is conser%ed# ,
decreases# ω increases