Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee
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DYNAMICS - The branch branch of mech mechani anics cs that that deals deals with with bodi bodies es in in motio motion n BRANCHES OF DYNAMICS KINEMATICS - The geom geometr etry y of motio motion. n. This This ter term m is used used to to dene dene the the motio motion n of a particle or body without consideration of the forces causing the moti motion on.. It is esse essent ntia iall lly y a trea treatm tmen entt of the the relat elatio ions ns betw betwee een n displacement; velocity and acceleration. KINETICS - The bran branch ch of mech mechani anics cs that that rela relates tes the the forc force e acting acting on on the body body to its mass and acceleration. NEWTON’S LAW OF MOTION 1 A body body at rest rest will rem remai ain n to be at rest est or in motion motion will will rema remain in in motion along a straight path unless acted upon by an unbalanced force.
2 A par arti ticl cle e acte acted d upon upon by an unbal nbalan ance ced d for force syst system em has has an acceleration in line with and directly proportional to the resultant of the force system and inversely proportional to its mass. kF
a
M
or ! "a
# In every every action$ action$ ther there e is always always an an e%ual e%ual and oppos opposite ite reacti reaction. on. KINEMATICS MOTION OF BODIES I Translation - The moti motion on of of a rigid rigid body body in whic which h a straig straight ht line line passi passing ng thr through ough any two of its particle always remain to be parallel to its initial position.
II -
&otation The The moti motion on of rigi rigid d body body in whic which h the the part partic icle les s move move in cir circula cularr paths with their centers on a 'ed straight line called the a'is of rotation.
III (lane "otion - The moti motion on of rigid rigid body body in which which all all parti particle cles s in the the body body remai remain n at a constant distance from the 'ed reference plane. TRANSLATION: )lements* + distance
v velocity
Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee
g acceleration due to gravity ,.1 ms2 #2.2 fts 2 A acceleration
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/o initial velocity t time
RECTILINEAR TRANSLATION
0niform "otion constant acceleration s=vt /ariable Acceleration ds=vdt dv=adt vdv=ads where a maybe a function of velocity v$ time t$ or distance s$ and v maybe a function of time t or distance s. Constant Acceleration Free Falling Body (Vo =o, s=h) V= Vo + at v= t " S= Vot + ! at #= ! t" V"= Vo" + "as v"="# FREE FALLIN$ BODIES R%&t'(')%a* Mot'o) 't# Va*'a,(% A&&%(%*at'o) dS v dT
dV
a
dT
ads vdv
CASE I T#% d's.(a&%/%)t 's ') t%*/s o0 t#% t'/% .
+ ft; solve for v 3 a (&456)". 7etermine the velocity and acceleration of a body after # sec. If the motion is dened by the relation + 8t 9 :t # dS v dT
dV
a
v8 9 12t 2 8 9 12#2 v11# fts
dT
9 2:t 2:# a<2 ftsec2
CASE II T#% a&&%(%*at'o) 's %1.*%ss%d ') t%*/s o0 t#% t'/% aft; solve for the v 3 s dV
a
dT
; dvadt
Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee v
∫ dV vo
s
t
=
∫¿ adT
dsvdT
∫ dS so
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t
=
∫¿ vdT
2ROBLEM. 7etermine the velocity and displacement of body after 2 sec if ft
the motion is dened by the relation; a2t. a sec2 3 tse tsec c
if it is
=nown that s :ft and v2 fts when t1 sec. ANS: 3 0t4s%&
CASE III T#% v%(o&'t5 's ') t%*/s o0 t'/% /ft; solve for a 3 s a dvdt s
∫ dS so
t
=
∫¿ vdT
2ROBLEM The velocity of an automobile starting from rest is given by 90 t
dsdt ( t + 10) ftsec. 7etermine the acceleration after an interval of 1 sec ftsec2. ANS: ""3 0t4s%& "
FREELY FALLIN$ BODIES 1. A stone is thrown vertically into the air from a tower # m high > the same instant that the second stone is thrown upward from the ground; vs2 ms$ v s2# ms ; when 3 where will the stones be at the same level or height from the ground?
+1-+2# +2 /ot 9 @ gt 2 #t @,.1 t2 +2#t :.,8 t2 +1 /ot - @ gt2
Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee
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2t :.,8 t2 +1-+2# #t :.,8 t2 - 2t :.,8 t 2 # 1t# t=6 s%& +2 /o# - :.,8# 2 S"= 7389 / 2. A man standing > window 8m tall watches a falling stone pass by the window in .# sec. !rom how high above the top of the window way the stone released? +2 /ot 9 @ gt 2 8 /o.# 9 @ ,.1.# 2 /o18.2 ms + 8m t.#s
/22gh 18.222,.1h #=;; /
T#% C%)t*'.%ta( Fo*&% a)d D'*%&t'o) C#a)%
Any obBect moving in a circle or along a circular path e'periences a centripetal force. That is$ there is some physical force pushing or pulling the obBect towards the center of the circle. This is the centripetal force re%uirement. The word centripetal is merely an adBective used to describe the direction of the force. Ce are not introducing a new type of force but rather describing the direction of the net force acting upon the obBect that moves in the circle. Chatever the obBect$ if it moves in a circle$ there is some force acting upon it to cause it to deviate from its straight-line path$ accelerate inwards and move along a circular path. Three such e'amples of centripetal force are shown below.
As a car ma=es a turn$ the force of friction acting upon the turned wheels of the car pro provid vides cen centrip tripet etal al force re%uired for circular motion.
As a buc=et of water is tied to a string and spun in a cir circle$ cle$ the the tens tensio ion n for force actin cting g upo upon the the buc=et provides the centripetal force re%uired for circular motion.
As the moon orbits the )arth$ the force of gravity acting upon the moon provides the cent centri ripe peta tall for force re%uired for
Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee
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circular motion.
There are three mathematical %uantities that will be of primary interest to us as we analyDe the motion of obBects in circles. These three %uantities are speed$ acceleration and force. The speed of an obBect moving in a circle is given by the following e%uation.
The acceleration of an obBect moving in a circle can be determined by either two of the following e%uations.
The e%uation on the right above is derived from the e%uation on the left by the substitution of the e'pression for speed. The net force !net acting upon an obBect moving in circular motion is directed inwards. Chile there may by more than one force acting upon the obBect$ the vector sum of all of them should add up to the net force. In general$ the inward force is larger than the outward force if any such that the outward force cancels and the unbalanced force is in the direction of the center of the circle. The net force is related to the acceleration of the obBect as is always the case and is thus given by the following three e%uations*
The e%uations in the middle above and on the right above are derived from the e%uation on the left by the substitution of the e'pressions for acceleration. +ample (roblem E1 A ,-=g car moving at 1 ms ta=es a turn around a circle with a radius of 28. m. 7etermine the acceleration and the net force acting upon the car. The solution of this problem begins with the identication of the =nown and re%uested information. Fnown Information* & 28. m m , =g v 1. ms &e%uested Information*
Lecture 4 - Mech 351 Statics & Dynamics Dynamics of Rigid Bodies Engr. En gr. Re Reji ji e . Ma Magn gnay ayee
a ????
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!net ????
To To determine the acceleration of the car$ use the e%uation a v 2 &. The solution is as follows* a v2 & a 1. ms 2 28. m a 1 m2s2 28. m a : ms 2 To To determine the net force acting upon the car$ use the e%uation ! net mGa. The solution is as follows. !net m G a !net , =g G : ms 2 !net #H +ample (roblem E2 A ,8-=g halfbac= halfbac= ma=es a turn turn on the football football eld. The halfbac= sweeps sweeps out a path that is a portion of a circle with a radius of 12-meters. The half halfba bac= c= ma= ma=es a %uar %uarte terr of a tur turn arou around nd the the circl circle e in 2.1 2.1 seco second nds. s. 7etermine the speed$ acceleration and net force acting upon the halfbac=. The solution of this problem begins with the identication of the =nown and re%uested information. Fnown Information* &e%uested Information* m ,8. =g v ???? & 12. m a ???? Traveled Traveled 1:-th of the !net ???? circumference circumference in 2.1 s
To To determine the speed of the halfbac=$ use the e%uation e %uation v d t where the d is one-fourth of the circumference and the time is 2.1 s. The solution is as follows* vdt v .28 G 2 G pi G & t v .28 G 2 G #.1: G 12. m 2.1 s v .,< ms To To determine the acceleration of the halfbac=$ use the e%uation a v 2 &. The solution is as follows* follows* 2 av & a .,< ms 2 12. m a .8 m2s2 12. m a H.<1 ms2 To To determine the net force acting upon the halfbac=$ use the e%uation ! net mGa. The solution is as follows. !net mJa !net ,8. =gJH.<1 ms 2 !net H#< E1%*&'s%s: 1. A 6incoln Kontinental and a Lugo are ma=ing a turn. The 6incoln is four times more massive than the Lugo. If they ma=e the turn at the same spee speed$ d$ then then how how do the the cent centri ripe peta tall for forces ces acti acting ng upon upon the the two two cars cars compare. )'plain.
2. The KaBun KliMhanger at Nreat America is a ride in which occupants line the perimeter of a cylinder and spin in a circle at a high rate of turning. Chen the cylinder begins spinning very rapidly$ the Ooor is removed from under the ridersP feet. Chat aMect does a doubling in speed have upon the centripetal force? )'plain. #. 7etermine the centripetal force acting upon a :-=g child who ma=es 1 revolutions around the KliMhanger in 2,.# seconds. The radius of the barrel barrel is 2., meters. WORK AND ENER$Y Chen a force acts upon an obBect to cause a displacement of the obBect$ it is said that o*< was was done done upon upon the the obBe obBect ct.. Ther There e are are thr three =ey ingredients to wor= - force$ displacement$ and cause. In order for a force to %ualify as having done work on on an obBect$ there must be a displacement and and the the for force must cause the displa displacem cement ent.. There There are are severa severall good good e'amples of wor= that can be observed in everyday life - a horse pulling a plow through the eld$ a father pushing a grocery cart down the aisle of a groc grocer ery y stor store$ e$ a fres freshm hman an lifti lifting ng a bac= bac=pa pac= c= full full of boo= boo=s s upon upon her her
should shoulder$ er$ a weight weightlif lifter ter liftin lifting g a barbell barbell above above his head$ head$ an 4lympi 4lympian an launching the shot-put$ etc. In each case described here there is a force e'erted e'erted upon an obBect to cause that obBect to be displaced. &ead the following ve statements and determine whether or not they represent e'amples of wor=.
Stat%/%)t
A)s%* 't# E1.(a)at'o)
A teac teache herr app applies lies a for force to a wall all and becom ecomes es e'hausted.
A boo= falls oM a table and free falls to the ground.
A waiter carries a tray full of meals above his head by one arm straight across the room at constant speed. A roc=et accelerates through space.
"athematically$ wor= can be e'pressed by the following e%uation.
where F is the force$ d is the displacement$ and and the the angl angle e t#%ta is dene ened d as the the angle between the force and the displacement vector. The angle measure is dened as the angle between the force and the displacement. T To o gather an idea of itPs meaning$ consider the following three scenarios. )'ts o0 Wo*< Chenev Chenever er a new %uant %uantity ity is intro introduc duced ed in physic physics$ s$ the stand standar ard d metric units associated with that %uantity are discussed. In the case of wor= and also energy$ the standard metric unit is the >o?(% abbreviated >. 4ne Qoule is e%uivalent to one ewton of force causing a displacement of one meter. In other words$ T#% >o?(% 's t#% ?)'t o0 o*< >o?(% = N%to) @ /%t%* >=N@/
In fact$ any unit of force times any unit of displacement is e%uivalent to a unit of wor=. +ome nonstandard units for wor= are shown below. otice that when analyDed$ each set of units is e%uivalent to a force unit times a displacement unit.
E1a/.(% 2*o,(%/ 1. Apply the wor= e%uation to determine the amount of wor= done by the applied force in each of the three situations described below.
2. 5en Travlun carries a 2- suitcase up three Oights of stairs a height of 1. m and then pushes it with a horiDontal force of 8. at a constant speed of .8 ms for a horiDontal distance of #8. meters. Row much wor= does 5en do on his suitcase during this entire motion?
#. A force of 8 acts on the bloc= at the angle shown in the diagram. The bloc= moves a horiDontal distance of #. m. Row much wor= is done by the applied force?
:. Row much wor= is done by an applied force to lift a 18-ewton bloc= #. meters vertically at a constant speed?
2ot%)t'a( %)%*5 is the stored energy of position possessed by an obBect.
$*av'tat'o)a( 2ot%)t'a( E)%*5
Nravitational potential energy is the energy stored in an obBect as the result of its vertical position or height. The energy is stored as the resul esultt of the the grav gravit itat atio iona nall attr attrac acti tion on of the the )art )arth h for for the the obBe obBect ct.. The The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an obBect. "ore massive obBects have greater gravit gravitati ationa onall potent potential ial energ energy y. There There is also also a direct direct relat relation ion betwee between n gravitational potential energy and the height of an obBect. The higher that an obBect is elevated$ the greater the gravitational potential energy. energy. These relationships are e'pressed e'pressed by the following e%uation* 2E*av = /ass #%'#t 2E*av = / @ #
In the above e%uation$ / represents the mass of the obBect$ # represents the height of the obBect and represents the gravitational eld strength ,. =g on )arth - sometimes referred to as the acceleration of gravity.
E(ast'& 2ot%)t'a( E)%*5
The second form of potential energy that we will discuss is elastic potential energy. E(ast'& .ot%)t'a( %)%*5 is the energy stored in elastic materials as the result of their stretching or compressing. )lastic potential ener energy gy can can be stor stored ed in rubb rubber er band bands$ s$ bung bungee ee chor chords ds$$ tram trampo polin lines es$$ springs$ an arrow drawn into a bow$ etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the devi device ce - the the mor more str stretch etch$$ the the mor more stored energy.
+prings are a special instance of a device device that that can store store elasti elastic c potent potential ial energy due to either compression or stretching. A force is re%uired to compress a spring; the more compression there is$ the more force that is re%uired to compress it further. !or certain springs$ the amount of force is direct directly ly propo proport rtion ional al to the amount amount of stretc stretch h or compr compress ession ion '; '; the constant of proportionality is =nown as the spring constant =.
La If a spring is not +uch +uch spri spring ngs s are are said said to foll follow ow Hoo<%s La stretched or compressed$ then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The e%uilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy$ the e%uilibrium position could could be called called the Dero-p Dero-pote otenti ntial al energ energy y positio position. n. There There is a specia speciall e%uation for springs that relates the amount of elastic potential energy to the amount amount of stretc stretch h or compr compress ession ion and the sprin spring g consta constant. nt. The e%uation is
E1a/.(% 2*o,(%/s
1. A cart is loaded with a bric= and pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is #. =g and the height of the seat top is .:8 meters$ then what is the potential energy of the loaded cart at the height of the seat-top?
2. If a force of 1:.< is used to drag the loaded cart from previous %uestion along the incline for a distance of ., meters$ then how much wor= is done on the loaded cart?
K')%t'& E)%*5 K')%t'& %)%*5 is the energy of motion. An obBect that has motion whether it is vertical or horiDontal motion - has =inetic energy. There are many forms of =inetic energy - vibrational the energy due to vibrational motion$ rotational the energy due to rotational motion$ and translational the energy due to motion from one location to another. To =eep matters simple$ we will focus upon translational =inetic energy. The amount of translational =inetic energy from here on$ the phrase =inetic energy will refer to translational =inetic energy that an obBect has depends upon two variables* the mass m of the obBect and the speed v of the obBect. The follow following ing e%uati e%uation on is used used to repr represe esent nt the =inetic =inetic energy energy F) F) of an obBect.
where / mass of obBect v speed of obBect This e%uation reveals that the =inetic energy of an obBect is directly di rectly proportional to the s%uare of its speed. That means that for a twofold increase in speed$ the =inetic energy will increase by a factor of four. !or !or a threefold increase in speed$ the =inetic energy will increase by a factor of nine. And for a fourfold increase in speed$ the =inetic energy will increase by a factor of si'teen. The =inetic energy is dependent upon the s%uare of the speed. As it is often said$ an e%uation is not merely a recipe for alge algebr brai aic c prob proble lem m solv solvin ing$ g$ but but also also a guid guide e to thin thin=i =ing ng abou aboutt the the relationship between %uantities. Fine Fineti tic c ener energy gy is a scalar %uantity; %uantity; it does not have a direction. 0nli=e velocity velocity$$ acceleration acceleration$$ force force$$ and momentum momentum$$ the =inetic energy of an obBect obBect is comple completel tely y descri described bed by magnit magnitude ude alone. alone. 6i=e 6i=e wor= wor= and and potent potential ial energ energy$ y$ the stand standar ard d metric metric unit unit of measur measureme ement nt for =ineti =inetic c energy is the Qoule. As might be implied by the above e%uation$ 1 Qoule is e%uivalent to 1 =gJmsS2.
E1a/.(% 2*o,(%/s 1. 7etermine the =inetic energy of a H28-=g roller coaster car that is moving with a speed of 1.# ms.
2. If the roller coaster car in the above problem were moving with twice the speed$ then what would be its new =inetic energy?
#. "issy "issy 7iwate 7iwater$ r$ the former former platfo platform rm diver diver for the &ingli &ingling ng 5roth 5rotherP erPs s Kircus$ had a =inetic energy of 12 Q Bust prior to hitting the buc=et of water. water. If "issyPs mass is : =g$ then what is her speed?
:. A ,-=g compact car moving at H mihr has appro'imately #2 Qoules of =inetic energy. energy. )stimate its new =inetic energy if it is moving at # mihr.
2o%* 2o%* is the the rate rate at whic which h wor= wor= is done done.. It is the the wor= wor=t tim ime e rati ratio o. "athematically$ it is computed using the following e%uation.
The standard standard metric unit of power is the Watt . As is implied by the e%uation for power$ a unit of power is e%uivalent to a unit of wor= divided by a unit unit of time time.. Thus Thus$$ a Catt att is e%ui e%uiva vale lent nt to a Qoul Qoule ese seco cond nd.. !or histor historica icall reasons easons$$ the horsepower is occasionally used to describe the power delivered by a machine. 4ne horsepower is e%uivalent to appro'imately <8 Catts.
This new e%uation for power reveals reveals that a powerful machine is both strong big force and fast big velocity. A powerful car engine is strong and and fast fast.. A powe powerf rful ul piec piece e of far farm e%ui e%uipm pmen entt is str strong ong and and fast fast.. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A machine that is strong enough to apply a big
for force to caus cause e a disp displa lace ceme ment nt in a smal smalll moun mountt of time time i.e i.e.$ .$ a big big velocity is a powerful machine.
E1a/.(% 2*o,(%/s:
1. Two physics students$ Cill . Andable and 5en (umpiniron$ are in the weightlifting room. Cill lifts the 1-pound barbell over his head 1 times in one minute; 5en lifts the 1-pound barbell over his head 1 times in 1 second seconds. s. Chich Chich stude student nt does does the most most wor=? wor=? Chich Chich stud studen entt deli delive vers rs the the most most powe power? r? )'plain your answers.
2. 7uring a physics lab$ Qac= and Qill ran up a hill. Qac= is twice as massive as Qill; yet Qill ascends the same distance in half the time. Cho did the most wor=? Cho delivered the most power? )'plain your answers.
#. A tired s%uirrel mass of appro'imately appro'imately 1 =g does push-ups by applying a force to elevate its center-of-mass by 8 cm in order to do a mere .8 Qoule of wor=. If the tired s%uirrel s%uirrel does all this wor= in 2 seconds$ then determine its power.
:. Lour householdPs monthly electric bill is often e'pressed in =ilowatthours. 4ne <'(oatt-#o?* is the amount of energy delivered by the Oow of l =ilowatt of electricity el ectricity for one hour. hour. 0se conversion factors to show how many Boules of energy you get when you buy 1 =ilowatt-hour of electricity. electricity.
Mo/%)t?/ Mo/%)t?/ is a commonly used term in sports. A team that has the momentum is on the moe and is going to ta=e some eMort to stop. A team that has a lot of momentum is really on the moe and is going to be hard to stop. "omentum is a physics term; it refers to the %uantity of motion motion that an obBect obBect has. A sports sports team that that is on the moe has the moment entum. If an obBect is in motion on the moe then it has momentum.
"omentum can be dened as Umass in motion.U All obBects have mass; so if an obBect is moving$ then it has momentum - it has its mass in motion. The amount of momentum that an obBect has is dependent upon two variables* how much stu! is is moving and how fast the stu! is is moving. "omentum depends upon the variables mass mass and and velocity velocity.. In terms of an e%uation$ the momentum of an obBect is e%ual to the mass of the obBect times the velocity of the obBect. Mo/%)t?/ = /ass v%(o&'t5 In physics$ the symbol for the %uantity momentum is the lower case UpU. Thus$ the above e%uation e%uation can be rewritten rewritten as .=/v wher where / is the the mass mass and v is the velocity. The e%uation illustrates that mome moment ntum um is dir directl ectly y prop propor orti tion onal al to an obBec obBectP tPs s mass mass and and direc directl tly y proportional to the obBectPs velocity. The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the =gGms. Chile the =gGms is the standard metric unit of momentum$ there are a variety of other units that are acceptable though not conventional units of momentum. )'amples include =gGmihr$ =gG=mhr$ and gGcms. In each of these e'amples$ a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with with the e%uation for momentum. momentum. E1%*&'s%s: )'pr 'press ess your your unde underrstand tandin ing g of the conc concep eptt momentum by answering the following %uestions. 1. 7etermine the momentum of a ... a. H-=g halfbac= moving eastward at , ms. b. 1-=g car moving northward at 2 ms. c. :-=g freshman moving southward at 2 ms.
and and
mathe athem matic atics s
of
2. A car possesses 2 units of momentum. Chat would be the carPs new momentum if ...
a. its velocity was doubled. b. its velocity was tripled. c. its mass was doubled by adding more passengers passengers and a greater load d. both its velocity was doubled and its mass was doubled. #. A halfbac= m H =g$ a tight end m , =g$ and a lineman m 12 =g are are runni running ng down down the footbal footballl eld. eld. Konsid Konsider er their their tic=er tape patterns below. patterns below.
Kompare the velocities of these three players. Row many times greater are the velocity of the halfbac= and the velocity of the tight end than the velocity of the lineman? Chich player has the greatest momentum? )'plain.
