# Physics 207, Lecture 13, Oct. 15. Energy

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1 Physcs 07 Lecture 3 Physcs 07, Lecture 3, Oct. 5 Goals: Chapter 0 Understand the relatonshp between moton and energy Dene Potental Energy n a Hooke s Law sprng Deelop and explot conseraton o energy prncple n problem solng Chapter Understand the relatonshp between orce, dsplacement and work Assgnment: HW6 due Wednesday, Oct. For Monday: Read all o Chapter Physcs 07: Lecture 3, Pg Energy I only conserate orces are present, the total energy (sum o potental, U, and knetc energes, K) ) o a system s consered For an object n a gratatonal eld ½ m y + mgy = ½ m y + mgy K ½ m U mgy E mech = K + U E mech = K + U = constant K and U may change, but E mech = K + U remans a xed alue. E mech s called mechancal energy Physcs 07: Lecture 3, Pg Page

2 Physcs 07 Lecture 3 Example o a conserate system: The smple pendulum. Suppose we release a mass m rom rest a dstance h aboe ts lowest possble pont. What s the maxmum speed o the mass and where does ths happen? To what heght h does t rse on the other sde? m h h Physcs 07: Lecture 3, Pg 3 Example: The smple pendulum. What s the maxmum speed o the mass and where does ths happen? E = K + U = constant and so K s maxmum when U s a mnmum. y y=h y=0 Physcs 07: Lecture 3, Pg 4 Page

3 Physcs 07 Lecture 3 Example: The smple pendulum. What s the maxmum speed o the mass and where does ths happen? E = K + U = constant and so K s maxmum when U s a mnmum E = mgh at top E = mgh = ½ m at bottom o the swng y y=h y=0 h Physcs 07: Lecture 3, Pg 5 Example: The smple pendulum. To what heght h does t rse on the other sde? E = K + U = constant and so when U s maxmum agan (when K = 0) t wll be at ts hghest pont. E = mgh = mgh or h = h y y=h =h y=0 Physcs 07: Lecture 3, Pg 6 Page 3

4 Physcs 07 Lecture 3 Example The Loop-the-Loop agan To complete the loop the loop, how hgh do we hae to let the release the car? Condton or completng the loop the loop: Crcular moton at the top o the loop (a c = / R) Use act that E = U + K = constant! y=0 U b =mgh U=mgR h? Car has mass m R Recall that g s the source o the centrpetal acceleraton and N just goes to zero s the lmtng case. Also recall the mnmum speed at the top s = Physcs 07: Lecture 3, Pg 7 gr Example The Loop-the-Loop agan Use E = K + U = constant mgh + 0 = mg R + ½ m mgh = mg R + ½ mgr = 5/ mgr = gr h = 5/ R h? R Physcs 07: Lecture 3, Pg 8 Page 4

5 Physcs 07 Lecture 3 What speed wll the skateboarder reach halway down the hll there s no rcton and the skateboarder starts at rest? Assume we can treat the skateboarder as a pont Assume zero o gratatonal U s at bottom o the hll R=0 m m = 5 kg R=0 m Example Skateboard y=0 Physcs 07: Lecture 3, Pg 9 What speed wll the skateboarder reach halway down the hll there s no rcton and the skateboarder starts at rest? Assume we can treat the skateboarder as pont Assume zero o gratatonal U s at bottom o the hll R=0 m m = 5 kg R=0 m Example Skateboard Use E = K + U = constant E beore = E ater 0 + m g R = ½ m + mgr (-sn 30 ) mgr/ = ½ m gr = = (gr) ½ = (0 x 0) ½ = 0 m/s Physcs 07: Lecture 3, Pg 0 Page 5

6 Physcs 07 Lecture 3 Potental Energy, Energy Transer and Path A ball o mass m, ntally at rest, s released and ollows three derence paths. All suraces are rctonless. Ball s dropped. Ball sldes down a straght nclne 3. Ball sldes down a cured nclne Ater traelng a ertcal dstance h, how do the three speeds compare? 3 h (A) > > 3 (B) 3 > > (C) 3 = = (D) Can t tell Physcs 07: Lecture 3, Pg Potental Energy, Energy Transer and Path A ball o mass m, ntally at rest, s released and ollows three derence paths. All suraces are rctonless. The ball s dropped. The ball sldes down a straght nclne 3. The ball sldes down a cured nclne Ater traelng a ertcal dstance h, how do the three speeds compare? 3 h (A) > > 3 (B) 3 > > (C) 3 = = (D) Can t tell Physcs 07: Lecture 3, Pg Page 6

7 Physcs 07 Lecture 3 Example Skateboard Now what s the normal orce on the skate boarder? N.. R=0 m m = 5 kg R=0 m 60 mg Σ F r = ma r = m / R = N mg cos 60 N = m /R + mg cos 60 N = 5 00 / (0.87) N = =470 Newtons Physcs 07: Lecture 3, Pg 3 Elastc s. Inelastc Collsons A collson s sad to be elastc when energy as well as momentum s consered beore and ater the collson. K beore = K ater Carts colldng wth a perect sprng, bllard balls, etc. Physcs 07: Lecture 3, Pg 4 Page 7

8 Physcs 07 Lecture 3 Elastc s. Inelastc Collsons A collson s sad to be nelastc when energy s not consered beore and ater the collson, but momentum s consered. K beore K ater Car crashes, collsons where objects stck together, etc. Physcs 07: Lecture 3, Pg 5 Inelastc collson n -D: Example A block o mass M s ntally at rest on a rctonless horzontal surace. A bullet o mass m s red at the block wth a muzzle elocty (speed). The bullet lodges n the block, and the block ends up wth a speed V. What s the ntal energy o the system? What s the nal energy o the system? Is energy consered? V x beore ater Physcs 07: Lecture 3, Pg 6 Page 8

9 Physcs 07 Lecture 3 Inelastc collson n -D: Example What s the momentum o the bullet wth speed? What s the ntal energy o the system? m r m r r = m What s the nal energy o the system? ( m + M )V Is momentum consered (yes)? m + M 0 = ( m + M )V Is energy consered? Examne E beore -E ater m [( m + M )V]V = m m ( m) m + M m = m ( m + M ) No! V beore ater x Physcs 07: Lecture 3, Pg 7 Example Fully Elastc Collson Suppose I hae dentcal bumper cars. One s motonless and the other s approachng t wth elocty. I they collde elastcally, what s the nal elocty o each car? Identcal means m = m = m Intally Green = and Red = 0 COM m + 0 = m + m = + COE ½ m = ½ m + ½ m = + = ( + ) = + + = 0 Soln : = 0 and = Soln : = 0 and = Physcs 07: Lecture 3, Pg 8 Page 9

10 Physcs 07 Lecture 3 Varable orce deces: Hooke s Law Sprngs Sprngs are eerywhere, (probe mcroscopes, DNA, an eecte nteracton between atoms) F Rest or equlbrum poston In ths sprng, the magntude o the orce ncreases as the sprng s urther compressed (a dsplacement). Hooke s Law, F s = - k s s s s the amount the sprng s stretched or compressed rom t restng poston. Physcs 07: Lecture 3, Pg 9 Hooke s Law Sprng For a sprng we know that F x = -k s. F(x) s s s relaxed poston -k s F = - k s F = - k s Physcs 07: Lecture 3, Pg 0 Page 0

11 Physcs 07 Lecture 3 Home Exercse Hooke s Law 8 m 9 m What s the sprng constant k? 50 kg F sprng ΣF = 0 = F s mg = k s - mg Use k = mg/ s = 5 N / 0.0 m (A) 50 N/m (B) 00 N/m (C) 400 N/m (D) 500 N/m mg Physcs 07: Lecture 3, Pg F-s relaton or a sngle DNA molecule Physcs 07: Lecture 3, Pg Page

12 Physcs 07 Lecture 3 Measurement technque: optcal tweezers Physcs 07: Lecture 3, Pg 3 Force s. Energy or a Hooke s Law sprng F = - k (x x equlbrum ) F = ma = m d/dt = m (d/dx dx/dt) = m d/dx = m d/dx So - k (x x equlbrum ) dx = m d Let u = x x eq. & du = dx x m ku du= m d x kx + kx = x ku x = m m m kx + m = kx + m Physcs 07: Lecture 3, Pg 4 Page

13 Physcs 07 Lecture 3 Energy or a Hooke s Law sprng kx + m = kx + m Assocate ½ kx wth the potental energy o the sprng m U s + K = U + s K Hooke s Law sprngs are conserate so the mechancal energy s constant Physcs 07: Lecture 3, Pg 5 In general: Energy dagrams Energy Ball allng E mech K U Energy Sprng/Mass system E mech K U y s Physcs 07: Lecture 3, Pg 6 Page 3

14 Physcs 07 Lecture 3 Energy dagrams Sprng/Mass/Graty system Force sprng alone y -mg sprng & graty m Energy E mech K K U g U sprng U Total y Physcs 07: Lecture 3, Pg 7 Equlbrum Example Sprng: F x = 0 => du / dx = 0 or x=0 The sprng s n equlbrum poston In general: du / dx = 0 equlbrum or ANY uncton establshes U U stable equlbrum unstable equlbrum Physcs 07: Lecture 3, Pg 8 Page 4

15 Physcs 07 Lecture 3 Comment on Energy Conseraton We hae seen that the total knetc energy o a system undergong an nelastc collson s not consered. Mechancal energy s lost: Heat (rcton) Bendng o metal and deormaton Knetc energy s not consered by these non-conserate orces occurrng durng the collson! Momentum along a specc drecton s consered when there are no external orces actng n ths drecton. In general, easer to satsy conseraton o momentum than energy conseraton. Physcs 07: Lecture 3, Pg 9 Physcs 07, Lecture 3, Oct. 5 Assgnment: HW6 due Wednesday For Monday: Read all o chapter Physcs 07: Lecture 3, Pg 30 Page 5

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