k, Eneg, and Pwe AP Phsics C
Thee ae man diffeent TYPES f Eneg. Eneg is expessed in JOULES (J) 4.19 J = 1 calie Eneg can be expessed me specificall b using the tem ORK() k = The Scala Dt Pduct between ce and Displacement. S that means if u appl a fce n an bject and it cves a displacement u have supplied ENERGY dne ORK n that bject.
Scala Dt Pduct? A pduct is bviusl a esult f multipling numbes. A scala is a quantit with NO DIRECTION. S basicall k is fund b multipling the ce times the displacement and esult is ENERGY, which has n diectin assciated with it. cs displacement vect A dt pduct is basicall a CONSTRAINT n the fmula. In this case it means that and x MUST be paallel. T ensue that the ae paallel we add the csine n the end. = x Aea = Base x Height
k The VERTICAL cmpnent f the fce DOES NOT cause the blck t mve the ight. The eneg impated t the bx is evident b its mtin t the ight. Theefe ONLY the HORIZONTAL COMPONENT f the fce actuall ceates eneg ORK. hen the ORCE and DISPLACEMENT ae in the SAME DIRECTION u get a POSITIVE ORK VALUE. The ANGLE between the fce and displacement is ZERO degees. hat happens when u put this in f the COSINE? hen the ORCE and DISPLACEMENT ae in the OPPOSITE diectin, et still n the same axis, u get a NEGATIVE ORK VALUE. This negative desn't mean the diectin!!!! IT simpl means that the fce and displacement ppse each the. The ANGLE between the fce and displacement in this case is 180 degees. hat happens when u put this in f the COSINE? hen the ORCE and DISPLACEMENT ae PERPENDICULAR, u get NO ORK!!! The ANGLE between the fce and displacement in this case is 90 degees. hat happens when u put this in f the COSINE?
Example cs cs 5 16 cs30 346.4 Nm 346.4 J A bx f mass m =.0 kg is mving ve a fictinal fl ( u k = 0.3) has a fce whse magnitude is = 5 N applied t it at an angle f 30 degees, as shwn t the left. The bx is bseved t mve 16 metes in the hizntal diectin befe falling ff the table. a) Hw much wk des d befe taking the plunge?
Example cnt hat if we had dne this in UNIT VECTOR ntatin? 1.65ˆ i 1.5 ˆj ( x x ) ( (1.65 16) (1.5 346.4 Nm 346.4 J ) 0)
Example cnt n Hw much wk des the ORCE NORMAL d and h? cs Thee is NO ORK since and ae pependicula. N 16 cs90 f 0 J Nte: This negative des nt specif a diectin in this case since ORK is a SCALAR. It simpl means that the fce is invlved in slwing the bject dwn. Hw much wk des the fictinal fce d? f f N ( mg cs cs 0.3((9.8) cs ) cs 5cs30) 16 cs180-34.08 J
hat if the ORCE IS NOT CONSTANT? The functin hee MUST be a ORCE functin with espect t x. Let s lk at a POPULAR fce functin. Net ma Is this functin, with espect t x? NO! Yu can still integate the functin, it simpl needs t be mdified s that it fits the mdel accdingl. dx m ( a) dx ( ma) dx m dv ( ) dx dt dx m ( ) dv dt m v v v dv m v dv
k-eneg Theem dx m ( ) dv dt m v v v dv v m( mv v v ) mv m v dv v v m( ) K Kinetic K Eneg 1 mv Kinetic eneg is the ENERGY f MOTION.
Example =cs A 70 kg base-unne begins t slide int secnd base when mving at a speed f 4.0 m/s. The cefficient f kinetic fictin between his clthes and the eath is 0.70. He slides s that his speed is ze just as he eaches the base (a) Hw much eneg is lst due t fictin acting n the unne? (b) Hw fa des he slide? a) f f f 0 K 1-560 J mv x f 1 560 (70)(4) f 1.17 m cs f n mg (0.70)(70)(9.8) = 480. N (480.) (cs180)
Anthe vaing fce example.. A ball hangs fm a pe attached t a ceiling as shwn. A vaiable fce is applied t the ball s that: is alwas hizntal s magnitude vaies s that the ball mves up the ac at a cnstant speed. The ball s velcit is ve lw Assuming the ball s mass is m, hw much wk des d as it mves fm = 0 t = 1?
Example Cnt Tcs T T cs mg T sin mg ( cs )sin mg d ( mg tan tan ) d Tsin tan d dx d d mg
Example Cnt mg mg tan d d mg d mg( ) d d d U Ptential Eneg U mg mgh mg mg mg mg( ) The eneg f POSITION STORED ENERGY is called Ptential Eneg!
Smething is missing. Suppse the mass was thwn UPARD. Hw much wk des gavit d n the bd as it executes the mtin? gavit gavit gavit gavit mg( mg U 1 Cnside a mass m that mves fm psitin 1 ( 1) t psitin m,(), mving with a cnstant velcit. Hw much wk des gavit d n the bd as it executes the mtin? cs )cs180 gavit gavit gavit gavit mg( mg U cs In bth cases, the negative sign is supplied 1 )cs0
The bttm line.. The amunt f k gavit des n a bd is PATH INDEPENDANT. ce fields that act this wa ae CONSERVATIVE ORCES IELDS. If the abve is tue, the amunt f wk dne n a bd that mves aund a CLOSED PATH in the field will alwas be ZERO RICTION is a nn cnsevative fce. B NON-CONSERVATIVE we mean it DEPENDS n the PATH. If a bd slides up, and then back dwn an incline the ttal wk dne b fictin is NOT ZERO. hen the diectin f mtin eveses, s des the fce and fictin will d NEGATIVE ORK in BOTH diectins.
ce can be fund using the DERIVATIVE x du dx U Since wk is equal t the NEGATIVE change in ptential eneg, the ORCE f an bject is the deivative f the ptential eneg with espect t displacement. Be ve caeful handling the negative sign.
Eneg is CONSERVED! K K K K K U K Eneg befe U ( U U U U K U ) Eneg afte
Example A.0 m pendulum is eleased fm est when the suppt sting is at an angle f 5 degees with the vetical. hat is the speed f the bb at the bttm f the sting? Lcs h L h = L Lcs h = -cs h = 0.187 m E B = E A U O = K mgh = 1/mv gh = 1/v 1.83 = v 1.35 m/s = v
Hw t we measue eneg? One f the things we d eveda is measue hw much eneg we use. The ate at which we use it detemines the amunt we pa t u utilit cmpan. Since ORK is eneg the ate at which wk is dne is efeed t as POER. The unit is eithe Jules pe secnd cmmnl called the ATT. T the left ae seveal vaius vesins f this fmula, including sme vaius Calculus vaiatins.