Physcs 0 Lecture 8 Potental Energy and Conservaton Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aovgun.com
Denton o Work W q The work, W, done by a constant orce on an object s dened as the product o the component o the orce along the drecton o dsplacement and the magntude o the dsplacement W º ( F cosq ) Dx n F s the magntude o the orce n Δ x s the magntude o the object s dsplacement!! n q s the angle between F and Dx
Work Done by Multple Forces q I more than one orce acts on an object, then the total work s equal to the algebrac sum o the work done by the ndvdual orces W net = åw by ndvdual orces n Remember work s a scalar, so ths s the algebrac sum W net = W + W + W = ( F cosq ) Dr g N F
Knetc Energy and Work q Knetc energy assocated wth the moton o an object KE = mv q Scalar quantty wth the same unt as work q Work s related to knetc energy mv - mv0 = ( Fnet cos q ) Dx x = ò F x dr Wnet = KE - KE = DKE Unts: N-m or J
Work done by a Gravtatonal Force q Gravtatonal Force n Magntude: mg n Drecton: downwards to the Earth s center q Work done by Gravtatonal Force!! W = FD rcosq = F Dr W net = mv - mv 0 W g = mgdr cosq
Potental Energy q Potental energy s assocated wth the poston o the object q Gravtatonal Potental Energy s the energy assocated wth the relatve poston o an object n space near the Earth s surace q The gravtatonal potental energy n n n n PE º mgy m s the mass o an object g s the acceleraton o gravty y s the vertcal poston o the mass relatve the surace o the Earth SI unt: joule (J)
Extended Work-Energy Theorem q The work-energy theorem can be extended to nclude potental energy: Wnet = KE - KE = DKE W = PE - PE grav ty q I we only have gravtatonal orce, then KE - KE = PE - KE + PE = PE + q The sum o the knetc energy and the gravtatonal potental energy remans constant at all tme and hence s a conserved quantty PE KE W net = W gravty
Extended Work-Energy Theorem q We denote the total mechancal energy by q Snce E = KE + PE KE + PE = PE + KE q The total mechancal energy s conserved and remans the same at all tmes mv + mgy = mv + mgy
Sprng Force q Involves the sprng constant, k q Hooke s Law gves the orce! F! = -kd n F s n the opposte drecton o dsplacement d, always back towards the equlbrum pont. n k depends on how the sprng was ormed, the materal t s made rom, thckness o the wre, etc. Unt: N/m.
Potental Energy n a Sprng q Elastc Potental Energy: n SI unt: Joule (J) n related to the work requred to compress a sprng rom ts equlbrum poston to some nal, arbtrary, poston x q Work done by the sprng W s = ò x x (-kx) dx = kx - kx W = PE - s s PE s PEs = kx
Extended Work-Energy Theorem q The work-energy theorem can be extended to nclude potental energy: W = KE - KE = DKE W = PE - PE grav ty net q I we nclude gravtatonal orce and sprng orce, then W = W + W net gravty ( KE KE ) + ( PE - PE ) + ( PE - PE ) = W = PE - - s s s s s PE s 0 KE + PE + PE = PE + KE + s KE s
Extended Work-Energy Theorem q We denote the total mechancal energy by E = KE + PE + PE s q Snce ( KE + PE + PE ) = ( KE + PE + PE ) s s q The total mechancal energy s conserved and remans the same at all tmes mv + mgy + kx = mv + mgy + kx
Ex: A block projected up a nclne q A 0.5-kg block rests on a horzontal, rctonless surace. The block s pressed back aganst a sprng havng a constant o k = 65 N/m, compressng the sprng by 0.0 cm to pont A. Then the block s released. q (a) Fnd the maxmum dstance d the block travels up the rctonless nclne θ = 30. q (b) How ast s the block gong when halway to ts maxmum heght?
a) A block projected up a q Pont A (ntal state): q Pont B (nal state): nclne v = 0, y = 0, x = -0cm = -0. m v 0, y = h = d snq, x = 0 mv + mgy + kx = mv + mgy + = kx d = = 0.5(65N / m)( -0.m) (0.5kg)(9.8m / s )sn 30 =.8m kx mg snq! kx = mgy = mgd snq
b) A block projected up a q Pont A (ntal state): q Pont B (nal state): k v = x - gh m =... =.5m / s nclne v v = 0, y = 0, x = -0cm = -0. m?, y = h / = d snq /, x = = mv + mgy + kx = mv + mgy + kx h k kx = mv + mg( ) x = v + gh m! h = d sn q = (.8m)sn30 = 0. 64m 0
Types o Forces q Conservatve orces n Work and energy assocated wth the orce can be recovered n Examples: Gravty, Sprng Force, EM orces q Non conservatve orces n The orces are generally dsspatve and work done aganst t cannot easly be recovered n Examples: Knetc rcton, ar drag orces, normal orces, tenson orces, appled orces
Conservatve Forces q A orce s conservatve the work t does on an object movng between two ponts s ndependent o the path the objects take between the ponts n The work depends only upon the ntal and nal postons o the object n Any conservatve orce can have a potental energy uncton assocated wth t n Work done by gravty Wg = PE - PE = mgy - mgy n Work done by sprng orce Ws = PEs - PEs = kx - kx
Non conservatve Forces q A orce s non conservatve the work t does on an object depends on the path taken by the object between ts nal and startng ponts. n The work depends upon the movement path n For a non-conservatve orce, potental energy can NOT be dened n Work done by a non conservatve orce!! W F d = - d W nc å +å k = otherorces n It s generally dsspatve. The dspersal o energy takes the orm o heat or sound
Extended Work-Energy Theorem q The work-energy theorem can be wrtten as: n n W nc represents the work done by non conservatve orces W c represents the work done by conservatve orces q Any work done by conservatve orces can be accounted or by changes n potental energy W = PE - PE n n Gravty work Sprng orce work Wnet = KE - KE = DKE W = W + W g net nc W = PE - PE = mgy - mgy Ws = PE - PE = kx - c c kx
Extended Work-Energy Theorem q Any work done by conservatve orces can be accounted or by changes n potental energy W = PE - PE = -( PE - PE ) = -DPE q Mechancal energy ncludes knetc and potental energy E W nc c = DKE + DPE = KE + PE W nc = ( KE + PE ) - ( KE + PE ) = KE + PE = ( KE - KE ) + ( PE - PE ) PE W = E - nc mv mgy g + s = + + E kx
Problem-Solvng Strategy q Dene the system to see t ncludes non-conservatve orces (especally rcton, drag orce ) q Wthout non-conservatve orces q Wth non-conservatve orces q Select the locaton o zero potental energy n Do not change ths locaton whle solvng the problem q Identy two ponts the object o nterest moves between n n mv + mgy + kx = mv + mgy + One pont should be where normaton s gven The other pont should be where you want to nd out somethng kx W nc = ( KE + PE ) - ( KE + PE ) - d + åwotherorce s = ( mv + mgy + kx ) - ( mv + mgy + kx )
Ex: Conservaton o Mechancal Energy q A block o mass m = 0.40 kg sldes across a horzontal rctonless counter wth a speed o v = 0.50 m/s. It runs nto and compresses a sprng o sprng constant k = 750 N/m. When the block s momentarly stopped by the sprng, by what dstance d s the sprng compressed? W nc = ( KE + PE ) - ( KE + PE ) mv + mgy + kx = mv + mgy + kx 0 + 0 + kd = mv + 0 + 0 0 + 0 + kd = mv + 0 + 0 m d = v =. 5cm k
Changes n Mechancal Energy or conservatve orces q A 3-kg crate sldes down a ramp. The ramp s m n length and nclned at an angle o 30 as shown. The crate starts rom rest at the top. The surace rcton can be neglgble. Use energy methods to determne the speed o the crate at the bottom o the ramp. - d + åwotherorce s = ( mv + mgy + kx ) - ( mv + mgy + kx ) ( mv + mgy + kx ) = ( mv + mgy + kx ) d = (! m, y = d sn 30 = 0.5m, v = y 0, v =? = mv + 0 + 0) = (0 + mgy + 0) 0 v = gy = 3.m / s
Changes n Mechancal Energy or Nonconservatve orces q A 3-kg crate sldes down a ramp. The ramp s m n length and nclned at an angle o 30 as shown. The crate starts rom rest at the top. The surace n contact have a coecent o knetc rcton o 0.5. Use energy methods to determne the speed o the crate at the bottom o the ramp. - d + åwotherorce s = ( mv + mgy + kx ) - ( mv + mgy + - µ k Nd + 0 = ( mv + 0 + 0) - (0 + mgy + 0)! k µ = 0.5, d = m, y = d sn 30 = 0.5m, N =? k N - mg cos q = 0 - µ kdmg cosq = mv - mgy v = g( y - µ kd cosq ) =.7m / s kx ) N
Changes n Mechancal Energy or Nonconservatve orces q A 3-kg crate sldes down a ramp. The ramp s m n length and nclned at an angle o 30 as shown. The crate starts rom rest at the top. The surace n contact have a coecent o knetc rcton o 0.5. How ar does the crate slde on the horzontal loor t contnues to experence a rcton orce. - d + åwotherorce s = ( mv + mgy + kx ) - ( mv + mgy + - µ k Nx + 0 = (0 + 0 + 0) - ( mv + 0 + 0) µ k = 0.5, =.7m / s, N =? v N - mg - µ mgx = - = 0 k mv v x = µ g =. 5 k m kx )
Ex 3: Block-Sprng Collson q A block havng a mass o 0.8 kg s gven an ntal velocty v A =. m/s to the rght and colldes wth a sprng whose mass s neglgble and whose orce constant s k = 50 N/m as shown n gure. Assumng the surace to be rctonless, calculate the maxmum compresson o the sprng ater the collson. mv + mgy + kx = mv + mgy + mv 0 0 max + + = mva + 0 + 0 m 0.8kg xmax = va = (.m / s) = 0. 5m k 50N / m kx
Block-Sprng Collson q A block havng a mass o 0.8 kg s gven an ntal velocty v A =. m/s to the rght and colldes wth a sprng whose mass s neglgble and whose orce constant s k = 50 N/m as shown n gure. Suppose a constant orce o knetc rcton acts between the block and the surace, wth µ k = 0.5, what s the maxmum compresson x c n the sprng. - d + åwotherorce s = ( mv + mgy + kx ) - ( mv + mgy + - µ k Nd + 0 = (0 + 0 + kxc ) - ( mva N = mg and d = kx c - mva = -µ kmgx 5x 3.9x - 0.58 = 0 c + c c x c x c = 0. 093m + 0 + 0) kx )
Conservaton o Energy q Energy s conserved n Ths means that energy cannot be created nor destroyed n I the total amount o energy n a system changes, t can only be due to the act that energy has crossed the boundary o the system by some method o energy transer
Ways to Transer Energy Into or Out o A System q Work transers by applyng a orce and causng a dsplacement o the pont o applcaton o the orce q Mechancal Waves allow a dsturbance to propagate through a medum q Heat s drven by a temperature derence between two regons n space q Matter Transer matter physcally crosses the boundary o the system, carryng energy wth t q Electrcal Transmsson transer s by electrc current q Electromagnetc Radaton energy s transerred by electromagnetc waves
Ex 4: Connected Blocks n q Moton Two blocks are connected by a lght strng that passes over a rctonless pulley. The block o mass m les on a horzontal surace and s connected to a sprng o orce constant k. The system s released rom rest when the sprng s unstretched. I the hangng block o mass m alls a dstance h beore comng to rest, calculate the coecent o knetc rcton between the block o mass m and the surace. - d + åwotherorces = DKE + DPE DPE = DPE - µ g + DPE s = (0 - m 0 kx k Nx + = -mgh + N = mg and x = gh) + ( kx - h - µ k m gh = - m gh + kh µ k = m g 0) m g - kh
Power q Work does not depend on tme nterval q The rate at whch energy s transerred s mportant n the desgn and use o practcal devce q The tme rate o energy transer s called power q The average power s gven by W P = Dt n when the method o energy transer s work
Instantaneous Power q Power s the tme rate o energy transer. Power s vald or any means o energy transer q Other expresson W FDx P = = = Dt Dt q A more general denton o nstantaneous power W dw!! dr!! P = lm = = F = F v D t 0 Dt dt dt!! P = F v = Fvcosq Fv
Unts o Power qthe SI unt o power s called the watt n watt = joule / second = kg. m / s 3 qa unt o power n the US Customary system s horsepower n hp = 550 t. lb/s = 746 W qunts o power can also be used to express unts o work or energy n kwh = (000 W)(3600 s) = 3.6 x0 6 J
Power Delvered by an Elevator Motor q A 000-kg elevator carres a maxmum load o 800 kg. A constant rctonal orce o 4000 N retards ts moton upward. What mnmum power must the motor delver to lt the ully loaded elevator at a constant speed o 3 m/s? F = net, y ma y T - - Mg = 0 T = + Mg =.6 0 4 N P = Fv = (.6 0 4 = 6.48 0 W P = 64.8kW = 86. 9hp 4 N)(3m / s)
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