11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work

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1 MÜHENDİSLİK MEKNİĞİ. HFT İş-Eneji Pwe f a fce: Pwe in he abiliy f a fce d wk F: The fce applied n paicle Q P = F v = Fv cs( θ ) F Q v θ Pah f Q v: The velciy f Q ÖRNEK: İŞ-ENERJİ ω µ k v Calculae he pwe f he ficinal fce d wk n he ginding disk and is pwe d wk n he wk piece. ssume he ginding wheel and cnnecin have a al mass f m. Wha happens he diffeence in he pwe? Çözüm: (a) Pwe f ficin d wk n he ginding wheel

2 N * mg j i ω f = µ k N Kineics: F y = 0 N N mg = 0 N = mg f = µ k mg Kinemaics: v = ω ω P = F v = ( f i) ( vi) = ( µ mgi) ( ω i) = µ ωmg k k f = µ k mg v = v

3 3 (b) Pwe f ficin d wk n wk piece P = F v = ( f i) ( v i) = fv = µ mgv k The diffeence in he pwe beween P and P appeas in he fm f hea ha ends up heaing he sufaces. Ne: The pwe f a fce is elaed he velciy f he paicle i is acing upn. F F v = v sin( θ ) v Fn = F sin( θ ) Pah f Q θ Q v = v v cs( θ ) Pah f Q θ Q F = F cs( θ ) P = Fv cs( θ ) = Fv = F v Wk f a fce: The wk U - f a fce n a paicle ve he ineval f ime fm is he inegal f is pwe ve his ime ineval. F Q v Q v Q v

4 4 U = Pd = F vd = F vd = Fv d Ohe mehds find he wk f a fce ae: U = d v = d F vd d U = F d d = F d U = ( F i + F j + F k) ( dxi + dy j + dzk) x x y z U = F dx + F dy + F dz x y x y y = U F vd U = F ved = ( F e ) vd = v = ve s U = s F ds ÖRNEK : İŞ- ENERJİ F z z z F ds d d θ v Calculae he wk fm ime 0 ime f he cnsan magniude fce F ha changes is diecin by θ = ω. ssume mves wih cnsan speed v. Çözüm:

5 5 U = F vd = F vd = F csθ v d = F cs( ω ) v d 0 0 = F v sin( ω ) ω ÖRNEK 3: İŞ - ENERJİ g F paicle mves unde he cnsan hiznal fce F and gaviy fm pin n he smh cicula ba. Calculae he wk f he fces applied n he paicle. y x mg F N Çözüm:

6 6 Wk f N: N is pependicula he pah a all imes and, as a esul, has n cmpnen angen he pah. Theefe, N des n wk fm. Wk f F and mg: U = F d = ( F i mg j) ( dx i + dy j) = F dx mgdy = F mg Kineic enegy f a paicle: paicle f mass m a each insan in ime has a kineic enegy T given by T = mv The wk-enegy elain: The elain beween he wk dne n a paicle by he fces which ae applied n i and hw is kineic enegy changes fllws fm Newn s secnd law. dv d F = ma F v = ma v F v = ma v F vd = ma vd U = T T d v v = = a v + v a = a v d Ne: The wk dne n a paicle by he esulan fce applied n i ve a given ineval f ime will be equal he change in kineic enegy f he paicle. In he wds, he kineic enegy f a paicle is changed by an amun equal he wk which is dne n he paicle by he esulan fce. ÖRNEK 4: İŞ- ENERJİ T + U = T

7 7 g F The paicle sas fm es a and mves alng he smh cicula ba unde he cnsan hiznal fce F and unde gaviy. Calculae he velciy f he paicle as a funcin f θ. csθ y θ N mg C F x Sluin: Wk f N equals ze since i is pependicula he pah a all he imes. Wk f F and mg. C C U C = F d = ( F i mgj) ( dxi + dyj) sinθ. = F dx 0 csθ mgdy = F sinθ mg(csθ )

8 8 = + 3. U C F sin θ mg( cs θ ) ( ) Wk-Enegy elain: T + U C = TC ( ) T = mv = 0, ( v = 0, sa fm es) () () in () T C = mv F sin θ + mg( cs θ ) = mvc F vc = sin θ + g( cs θ ) m C ÖRNEK 5: İŞ - ENERJİ b k m C g x The paicle sas fm es a. If he ba is smh and he sping has a fee lengh f l, calculae he speed f he paicle as a funcin f x. Sluin:

9 9 F s θ N mg Fis calculae he wk f he fces. N des n wk since i is pependicula he pah f he ba a all imes. Sa fm es v = 0 x C 0 x U = F ds = ( F cs θ + mg) dx = [ k( x + b l ) x + mg ] dx x + b 0 Wk-Enegy elain x = [ kx + kl x + mg ] dx x + b 0 = k x + kl x + b klb + mgx T = mv = 0 TC = mv = mx& s T + U = T kx 0 + kl x + b kl b + mgx = mx&

10 0 kx &x = m m kl x b klb gx m POTNSİYEL ENERJİ Cnsevaive fces: fce F is cnsevaive if i can be epesened as he gadien f a scala funcin. This can wien as F = V whee V is a scala funcin f psiin in space. The negaive sign is nly f cnvenience s ha V can be idenified as he penial enegy. Wk f a cnsevaive fce: Given a cnsevaive fce F, is wk can be calculaed as U = = V ( x F vd = dx d V + y V vd V V V dx dy dz = ( i + j + k) ( i + j + k) d x y z d d d dy V dz + ) d = d z d = [ V ( ) V ( )] Theefe, he wk f a cnsevaive fce is negaive he change in is penial enegy. Ne: Since he wk f a cnsevaive fce des n depend n he pah he paicle akes when ging fm is lcain a ime is lcain a ime, he wk f a cnsevaive fce is pah independen. ÖRNEK : POTNSİYEL ENERJİ Calculae he penial enegy f a cnsan fce F. Sluin: dv d d

11 Since F is cnsan, hen Theefe ae cnsan. V is a funcin f (x,y,z). Ne ha he cnsan f inegain is n a eal cnsan bu is a funcin f y and z. Subsiuin f (4) in () GIVES Subsiuin f (5) in (3) gives Theefe, he penial enegy f a cnsan fce F is ÖRNEK : POTNSİYEL ENERJİ

12 e θ e m θ Calculae he penial enegy f he sping fce. ssume he sping has a fee lengh f. Sluin f -D min e θ e m F = k( ) s In pla cdinaes Theefe, F s V V = e V + θ eθ V k( ) = ( ) = V V 0 = ( ) θ

13 3 Inegaing () gives Subsiuing (3) in (4) gives V = C( ) ( 3 ) dc k( ) = k( ) d = dc d Inegain using w = s ha dw = d gives k ( ) + D = C Theefe, he penial enegy f he sping fce is V = k( ) + D The wk-enegy elain: The wk enegy elain can be ewien ake advanage f he fac ha yu can calculae he wk f cnsevaive fces fm he penial enegy. Le he esulan wk f all fces n a paicle be sepaaed in he wk dne by cnsevaive fces and he wk dne by nncnsevaive fces as fllws U = U. + U = ( V V ) + U Cn nn cn nn cn Inducin f his in he wk enegy elain gives T + U = T T ( V V ) + U = T nn cn T + V + U = T + V nn cn Tal Mechanical Enegy: The al mechanical enegy E is he sum f he kineic enegy and he penial enegy f he cnsevaive fces. E = T + V The wk enegy elain can nw be wien as E + U = E nn cns Cnsevaive sysems: cnsevaive sysem is ne f which he al mechanical enegy E emains cnsan. This can nly happen if he nncnsevaive fces d n wk. Penial enegy f gaviy: The penial enegy f gaviy is given by

14 4 V g = mgh whee h is measued fm an elevain seleced as efeence. m h V g = 0 Penial enegy f a sping: The penial enegy f a sping is given by Vs = k( l l ) = ks whee k is he sping siffness, l is he sping s cuen lengh, and l is he fee lengh f he sping, and s is he sech f he sping. l k F l F l s Fee Sping ÖRNEK 3: POTNSİYEL ENERJİ Seched Sping

15 5 Mass m is eleased fm es a and mves n he smh clla fm unde he acin f he sping and gaviy. Calculae he speed f he paicle befe i his. The sping has an iniial lengh f. Sluin:

16 6 Fce analysis: mg is cnsevaive is cnsevaive N is pependicula he pah a all pins fm, s i des n wk. alance f wk and enegy: Sas fm es a

17 7 Subsiuin in () gives (all fces which d wk ae cnsevaive)

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