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## Chapter 7:Potential Energy and Energy Conservation

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**Chapter 7:Potential Energy and Energy Conservation**• Conservative/Nonconservative Forces Work along a path (Path integral) Work around any closed path (Path integral) • Potential Energy Mechanical Energy Conservation Energy Conservation**y**Work Done bythe Gravitational Force (I) Near the Earth’s surface l (Path integral) Energy Conservation**y**Work Done bythe Gravitational Force (II) Near the Earth’s surface (Path integral) dl Energy Conservation**Wg < 0 if y2 > y1**Wg > 0 if y2 < y1 The work done by the gravitational force depends only on the initial and final positions.. Work Done bythe Gravitational Force (III) Energy Conservation**Wg(ABCA)**=Wg(AB) + Wg(BC) + Wg(CA) =mg(y1 – y2) + 0 + mg(y2- y1) = 0 Work Done bythe Gravitational Force (IV) C B dl A Energy Conservation**Wg = 0 for a closed path**The gravitational force is a conservative force. Work Done bythe Gravitational Force (V) Energy Conservation**Work Done by Ff (I)**(Path integral) - μmg L L depends on the path. LB Path B Path A LA Energy Conservation**The work done by the friction force**depends on the path length. The friction force: (a) is a non-conservative force; (b) decreases mechanical energy of the system. Work Done by Ff (II) Wf = 0 (any closed path) Energy Conservation**Example 1**A 1000-kg roller-coaster car moves from point A, to point B and then to point C. What is its gravitational potential energy at B and C relative to point A? Energy Conservation**Wg(ABC) =Wg(AB) + Wg(BC)**=mg(yA- yB) + mg(yB - yC) = mg(yA - yC) Wg(AC) = Ug(yA) – Ug(yC) y B A dl B C A Energy Conservation**Work-Energy Theorem Conservation ofMechanical Energy**(K+U) Wconservative (AC)= UA – UC If Wnet = Wconservative , then KC – KA = Wnet (AC)= UA – UC KC + UC = KA + UA Energy Conservation**Example 1**X h/2 U K X Energy Conservation**Work Done by FS using Uel(x)**y x WS = Uel(xi ) – Uel(xf ) where Uel(x) = (1/2) k x2 Energy Conservation**Glossary**• K: Energy associated with the motion of an object. • U: Energy stored in a system of objects • Can either do work or be converted to K. • Q: Thermal Energy (Internal Energy) The energy of atoms and molecules that make up a body. Energy Conservation**Work Done by Fg using Ug(h)**Wg = U(hi ) – U(hf ) where: Ug(h) = m g h (near the Earth’s surface) h1 h3 h4 = 0 h2 Energy Conservation**Example 2**A roller coaster sliding without friction along a circular vertical loop (radius R) is to remain on the track at all times. Find the minimum release height h. A C B Energy Conservation**Example 2 (cont’d)**A roller coaster sliding without friction along a circular vertical loop (radius R) is to remain on the track at all times. Find the minimum release height h. A v C (2) mv2/R = mg FN = 0 mg (1) UA = UC + KC B Energy Conservation**Example 4**Circular Motion & WT = 0 A C vC = ? B vB = ? Energy Conservation**Work Done by FG using UG(r)**WG = UG(ri ) - UG(rf ) where: UG(r)= - GmME / r (dl)r = (dl)f Energy Conservation**Work Done by FS using Uel(x)**y x WS = Uel(xi ) – Uel(xf ) where Uel(x) = (1/2) k x2 Energy Conservation**(4) W-E Theorem to**find v2(= 1.93 m/s). (1) F.B.D. (2) W by each force (3) Wnet Example 2 v2 = ? FN FP v1 = 0 motion d=5m Ff Fg mk= 0.100 Wg = Ug2 – Ug1 or Wg = m g d (FP)x = FP /cosq Energy Conservation**Example 3**mk = ? vf = 0 d Energy Conservation**Example 5**How much work must the satellite’s engines perform to move its satellite (mass m = 300 kg) from a circular orbit of radius rA = 8000 km about the Earth to another circular orbit of radius rC = 3 rA? vA = ? vC = ? Energy Conservation**y**Gravitational Potential Energy Near the Earth’s surface dl Energy Conservation