PHY 101 1nd Edition Lecture 14 Outline of Last Lecture I. EnergyII. 5.1 WorkIII. Work Done by Force of Kinetic FrictionIV. 5.2 Kinetic Energy, Work – Energy Theorem Outline of Current Lecture I. 5.2 continuedII. Conservative vs. Nonconservative ForcesIII. 5.3 Gravitational Potential EnergyIV. Gravitational Potential Energy & WorkV. Conservation of Mechanical Energy VI. 5.5 Energy Conservation: Work – Energy Theorem Revisted Current Lecture5.2 Continued- Ex.- A car of mass 1000kg is traveling at 35m/s. To avoid collision with another car, which is atrest, the driver slams on the break. The friction force on the car is 8x103 N.- 1. What minimum distance should the brake be applied to avoid the collision?- Analysis:o Initial KE = (1/2)mv2o Final KE = 0o The change in KE = work done by the friction forceo M =1000kgo Vi = 35m/s, vf = 0o Fk = 8x103 N- Solution:o Kef – Kei = 0 – (1/2)mv2These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.o Work done by friction force is Wg = - FkΔx- Using work – energy theorem: Wf = Kef – Keio –FkΔx = -(1/2)mv2o  Δx = (1/2)mv2/fk = (0.5 x 1000 x 352)( 8x103) = 76.6mo ** minimum separation to avoid collision**- 2. If the initial distance between the 2 cars is 30m, what is the speed of the 1st car at impact?o Wf = KEf – Keio - FkΔx = -(1/2)mvi2 - (1/2)mvf2o -FkΔx + (1/2)mvf2= -(1/2)mvi2 o (-8x103)(30) + (1/2)(1000)(352) = (1/2)(1000)vf2- Solve for vf2o vf2 = 745 = (745)(1/2) = 27.3 m/so *deadly impact*Conservative vs. Nonconservative Forces- There are generally two kinds of forces:o One is called conservative: eg. Gravitational forceo The other is called nonconservative: eg. Friction force- Work done against a conservative force is transferred into another form of mechanical energy: potential energy, which can be transformed back into kinetic energy- A force is conservative if the work it does on an object moving between two points is independent of the path the objects take between the points- Work done by nonconservative force does depend on the path- However, the work – energy theorem always holdso Wnet = Wc + Wnc = KEf – Keio Wc: work done by the conservative forceo Wnc: work done by the nonconservative forceo Ex. Conservative force:  gravitational force, electric forceo Ex. Nonconservative force  frictionGravitational Potential Energy- When taking the slide, did you ever wonder where “energy” came from?o You start from rest (zero kinetic energy)o You gains speed while sliding down – gain kinetic energy- Gravitational potential energy- While you’re climbing up, you do work against the gravitational force. The energy is “stored” in the potential energy- When you slide down, the potential energy is then converted into kinetic energy – you gain speed (neglect the friction of the slide)- Two kinds of mechanical friction:o Kinetic energy we discussed: 1/2mv2o Potential energy- Lifting an object up: you do positive work, potential energy increases (the gravitation force does negative work)- While the object falls down: gravitational force does the positive work, the potential energy is converted into kinetic energy- Gravitational potential energy (relative to the reference point where y = 0)o PE = mg·y- The higher the position of the object, the greater the potential energy- Work done by the gravitational force while moving from an initial position yi to a final position vf:o Wg = -ΔPE ΔPE = PEf – Pei = mgΔyo If PEi > PEf, the object falls, gravitational force does positive worko If Pei < PEf, the object rises, gravitational force does negative workGravitational Potential Energy & Work- Wg = -(PEf – PEi) = ΔPE- The work done by the gravitational force is equal to the negative of the change in gravitational potential energy- If the gravitational force is the only force acting on the object, from the work – energy theorem:o Wg = ΔKEo We have, - ΔPE = ΔKE, or ΔKE + ΔPE = 0Conservation of Mechanical Energy – When Gravitational Force is the Only Force That Does The Work- ΔKE + ΔPE = 0- This is the conservation of mechanical energy- ΔKE + ΔPE = (KEf – KEi) + (PEf – PEi) = 0- Rewrite the equation by moving initial values to the right and final values to the left- KEf + PEf = KEi +PEi- The sum of the kinetic energy and the gravitational potential energy remains constant at all times. In other words, the mechanical energy is a conserved quantity- This is why the gravitational force is called a conservative forceExample: Platform Diver- A diver of mass m drops from rest from a platform 10m above the water- Calculate the speed of the diver when he is 5m above the water- We could use several methods to solve this problem:o Kinematic equationso Newton’s second lawo Conservation of mechanical energy KEf + PEf = KEi + PEio Solution: from the conservation of mechanical energy (1/2)mv2 + mgyf = (1/2)mv2 + mgyi Vi = 0  (1/2)v2 + mgyf = gyi  (1/2)vf2 = gyi – gyf = g(yi – yf)  vf2 = 2g(yi – yf) vf = (2g(yi – yf))1/2 = (2x9.8x(10-5))1/2 = 9.9m/s5.5 Energy Conservation: Work – Energy Theorem Revisited- in addition to the gravitational force, if there are nonconservative forces involved, the work – energy theorem becomes (as discussed before)o Wnet = Wg + Wnc = KEf – KEio Wnc: work done by the nonconservative force- Since Wg = PEi – Pef, we haveo Wnc = (KEf – KEi) – Wg = (KEf – KEi) – (PEi – PEf)o Wnc = (KEf – KEi) + (PEf – PEi)o Wnc = ΔKE + ΔPE = ΔEmecho ΔEmech = KE +