rk, Energy, and Pwer Physics 1
There are many different TYPES f Energy. Energy is expressed in JOULES (J 419J 4.19 1 calrie Energy can be expressed mre specifically by using the term ORK( rk The Scalar Dt Prduct between rce and Displacement. S that means if yu apply a frce n an bject and it cvers a displacement yu have supplied ENERGY r dne ORK n that bject.
Scalar Dt Prduct? r r r rr r csθ displacement vectr A prduct is bviusly a result f A dt prduct is basically a CONSTRAINT multiplying numbers. A n the frmula. In this case it means that scalar is a quantity with NO and x MUST be parallel. a T ensure suethat DIRECTION. S basically they are parallel we add the csine n the rk is fund by multiplying end. the rce times the displacement and result is ENERGY, which has n directin assciated with it. x Area Base x Height
rk The VERTICAL cmpnent f the frce DOES NOT cause the blck t mve the right. The energy imparted t the bx is evident by its mtin t the right. Therefre ONLY the HORIZONTAL COMPONENT f the frce actually creates energy r ORK. hen the ORCE and DISPLACEMENT are in the SAME DIRECTION yu get a POSITIVE ORK VALUE. The ANGLE between the frce and displacement is ZERO degrees. hat happens when yu put this in fr the COSINE? hen the ORCE and DISPLACEMENT are in the OPPOSITE directin, yet still n the same axis, yu get a NEGATIVE ORK VALUE. This negative desn't mean the directin!!!! IT simply means that the frce and displacement e ppse each ther. The ANGLE between the frce and displacement in this case is 180 degrees. hat happens when yu put this in fr the COSINE? hen the ORCE and DISPLACEMENT are PERPENDICULAR, yu get NO ORK!!! The ANGLE between the frce and displacement in this case is 90 degrees. hat happens when yu put this in fr the COSINE?
Example r r csθ r r cs θ 5 16 cs30 346.4 Nm 346.4 J A bx f mass m.0 kg is mving ver a frictinal flr ( u k 0.3 has a frce whse magnitude is 5 N applied t it at an angle f 30 degrees, as shwn t the left. The bx is bserved t mve 16 meters in the hrizntal directin befre falling ff the table. a Hw much wrk des d befre taking the plunge?
Example cnt hat if we had dne this in UNIT VECTOR ntatin? 1.65ˆ i ( x + 1.5 ˆj r + x ( (1.65 16 346.4 346. 4 Nm J y r y + (1.5 0
Example cnt f n Nte: This negative des nt specify a directin in this case since ORK is a SCALAR. It simply means that the frce is invlved in slwing the bject dwn. Hw much wrk des the ORCE NORMAL d and hy? r r csθ N 16 cs90 0 J There is NO ORK since and r are perpendicular. Hw much wrk des the frictinal frce d? f f µ r r csθθ N r csθ µ ( mg cs θ r cs θ 0.3((9.8 5cs 30 16 cs180-34.08 J
hat if the ORCE IS NOT CONSTANT? The functin here MUST be a ORCE functin with respect t x r r. Let s lk at a POPULAR frce functin. Is this functin, with respect t x? NO! Net ma Net Yu can still integrate the functin, it simply needs t be mdified s that it fits the mdel accrdingly. dx ( ma dx dv m ( a dx m ( dx dt dx m ( dv m dt v ( m v dv v v dv
rk-energy Therem dx m ( dv m v dv dt m v v v dv v v v m( v m( mv mv v K Kinetic Energy K 1 mv Kinetic energy is the ENERGY f MOTION.
Example rcsθ A 70 kg base-runner begins t slide int secnd base when mving at a speed f 4.0 m/s. The cefficient f kinetic frictin between his clthes and the earth is 0.70. He slides s that t his speed is zer just as he reaches the base (a Hw much energy is lst due t frictin acting n the runner? (b Hw far des he slide? a f K f 0 1 mv 1 (70(4 f -560 J 560 x f f 1.17 m r csθ µ mg f n µ (0.70(70(9.8 480. N (480. r(cs180
Anther varying frce example.. A ball hangs frm a rpe attached t a ceiling as shwn. A variable frce is applied t the ball s that: is always hrizntal s magnitude varies s that the ball mves up the arc at a cnstant speed. The ball s velcity is very lw Assuming the ball s mass is m, hw much wrk des d as it mves frm θ 0tθ θ θ 1?
Example Cnt Tcsθ Tsinθ T mg T csθ mg T sinθ mg ( sin θ mg csθ tan θ dr ( mg tan θ dr dy dy tanθ r dx dr
Example Cnt dy mg tanθ dr mg( dr dr mg dy mg mgy mgy y y dy mg y mg( y y U Ptential Energy mgy U y The energy f POSITION r y STORED ENERGY is called Ptential Energy! mgh
Smething is missing. Suppse the mass was thrwn UPARD. Hw much wrk des gravity d n the bdy as it executes the mtin? gravity gravity gravity gravity r r Cnsider a mass m that mves frm psitin 1 ( y1 t psitin m,(y, mving with a cnstant t velcity. Hw much wrk des gravity d n the bdy as it executes the mtin? r r csθ r gravity r csθ gravity gravity gravity gravity mg( y y1cs0 mg y U mg( y1 ycs180 In bth cases, the negative mg y sign is supplied U
The bttm line.. The amunt f rk gravity des n a bdy is PATH INDEPENDANT. rce fields that act this way are CONSERVATIVE ORCES IELDS. If the abve is true, the amunt f wrk dne n a bdy that mves arund a CLOSED PATH in the field will always be ZERO RICTION is a nn cnservative frce. By NON-CONSERVATIVE we mean it DEPENDS n the PATH. If a bdy slides up, and then back dwn an incline the ttal wrk dne by frictin is NOT ZERO. hen the directin f mtin reverses, s des the frce and frictin will d NEGATIVE ORK in BOTH directins.
rce can be fund using the DERIVATIVE x U du dx Since wrk is equal t the NEGATIVE change in ptential energy, the ORCE f an bject is the derivative f the ptential energy with respect t displacement. Be very careful handling the negative sign.
Energy is CONSERVED! K K K K U K ( U U K + U U K Energy + U + U befre Energy after
Example A.0 m pendulum is released frm rest when the supprt string is at an angle f 5 degrees with the vertical. hat is the speed f the bb b at the bttm f the string? θ Lcsθθ hl Lcsθ L h -csθ h 0.187 m h E B E A U O K mgh 1/mv gh 1/v 1.83 v 1.35 m/s v
Hw t we measure energy? One f the things we d everyday is measure hw much energy we use. The rate at which we use it determines the amunt we pay t ur utility cmpany. Since ORK is energy the rate at which wrk is dne is referred t as POER. The unit is either Jules per secnd r cmmnly called the ATT. T the left are several varius versins f this frmula, including sme varius Calculus variatins.