# Sect. 6-5: Conservative Forces

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Sect. 6-5: Conservative Forces. Sect. 6-5: Conservative Forces. Conservative Force  The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass.  A PE CAN be defined for conservative forces - PowerPoint PPT Presentation

### Text of Sect. 6-5: Conservative Forces

• Sect. 6-5: Conservative Forces

• Sect. 6-5: Conservative Forces Conservative Force The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass. A PE CAN be defined for conservative forcesNon-Conservative Force The work done by that force depends on the path taken between the initial & final positions of the mass. A PE CANNOT be defined for non-conservative forcesThe most common example of a non-conservative force is FRICTION

• Definition: A force is conservative if & only ifthe work done by that force on an object moving from one point to another depends ONLY on the initial & final positions of the object, & is independent of the particular path taken. Example: gravity.y

• Conservative Force: Another definition: A force is conservative if the net work done by the force on an object moving around any closed path is zero.

• Potential Energy: Can only be defined for Conservative Forces! In other words, If a force is Conservative, a PE CAN be defined.lBut,If a force is Non-Conservative, aPE CANNOT be defined!!

• If friction is present, the work done depends not only on the starting & ending points, but also on the path taken. Friction is a non-conservative force!Friction is non-conservative!!!The work done depends on the path!

• If several forces act, (conservative & non-conservative), the total work done is: Wnet = WC + WNC WC work done by conservative forces WNC work done by non-conservative forcesThe work energy principle still holds:Wnet = KE For conservative forces (by the definition of PE):WC = -PE KE = -PE + WNCor: WNC = KE + PE

• In general, WNC = KE + PEThe total work done by allnon-conservative forcesThe total change in KE + The total change in PE

• Sect. 6-6: Mechanical Energy & its Conservation GENERALLY: In any process, total energy is neither created nor destroyed. Energy can be transformed from one form to another & from one object to another, but theTotal Amount Remains Constant. Law of Conservation of Total Energy

• In general, for mechanical systems, we just found:WNC = KE + PEFor the Very Special Case of Conservative Forces Only WNC = 0 KE + PE = 0 The Principle of Conservation of Mechanical EnergyPlease Note!! This is NOT (quite) the same as the Law of Conservation of Total Energy! It is a very special case of this law (where all forces are conservative)

• So, for conservative forces ONLY! In any processKE + PE = 0Conservation of Mechanical EnergyIt is convenient to Define the Mechanical Energy: E KE + PE In any process (conservative forces!): E = 0 = KE + PEOr, E = KE + PE = Constant Conservation of Mechanical EnergyIn any process (conservative forces!), the sum of the KE & the PE is unchanged:That is, the mechanical energy may change from PE to KE or from KE to PE, but

Their Sum Remains Constant.

• Principle of Conservation of Mechanical Energy

If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constantit is conserved.

• Conservation of Mechanical Energy: KE + PE = 0or E = KE + PE = Constant For conservative forces ONLY (gravity, spring, etc.)Suppose that, initially: E = KE1 + PE1, & finally:E = KE2+ PE2. But, E = Constant, so KE1 + PE1 = KE2+ PE2A very powerful method of calculation!!

• Conservation of Mechanical Energy KE + PE = 0 orE = KE + PE = ConstantFor gravitational PE: (PE)grav = mgyE = KE1 + PE1 = KE2+ PE2

()m(v1)2 + mgy1 = ()m(v2)2 + mgy2 y1 = Initial height, v1 = Initial velocity y2 = Final height, v2 = Final velocity

• PE1 = mgh, KE1 = 0PE2 = 0KE2 = ()mv2 KE3 + PE3 = KE2 + PE2 = KE1 + PE1but their sum remains constant!KE1 + PE1 = KE2 + PE2 0 + mgh = ()mv2 + 0v2 = 2ghall PE half KE half U all KE KE1 + PE1 = KE2 + PE2 = KE3 + PE3

• Example 6-8: Falling RockPE onlypart PEpart KEKE onlyv1 = 0y1 = 3.0 mv2 = ?y2 = 1.0 mv3 = ?y3 = 0NOTE!! Always use KE1 + PE1 = KE2 + PE2 = KE3 + PE3 NEVER KE3 = PE3!!!!A very common error! WHY????In general, KE3 PE3!!!Energy buckets are not real!!Speed at y = 1.0 m? Mechanical Energy Conservation! ()m(v1)2 + mgy1 = ()m(v2)2 + mgy2 = ()m(v3)2 + mgy3 (Mass cancels!) y1 = 3.0 m, v1 = 0, y2 = 1.0 m, v2 = ? Result: v2 = 6.3 m/s

• Example 6-9: Roller CoasterMechanical energy conservation! (Frictionless!) ()m(v1)2 + mgy1 = ()m(v2)2 + mgy2 (Mass cancels!) Only height differences matter! Horizontal distance doesnt matter!Speed at the bottom? y1 = 40 m, v1 = 0 y2 = 0 m, v2 = ?Find: v2 = 28 m/sWhat is y when v3 = 14 m/s? Use: ()m(v2)2 + 0 = ()m(v3)2 + mgy3 Find: y3 = 30 mHeight of hill = 40 m. Car starts from rest at top. Calculate: a. Speed of the car at bottom of hill. b. Height at which it will have half this speed. Take y = 0 at bottom of hill.123NOTE!! Always use KE1 + PE1 = KE2 + PE2 = KE3 + PE3 Never KE3 = PE3 !A very common error! WHY????In general, KE3 PE3!!!

• Conceptual Example 6-10: Speeds on 2 Water SlidesFrictionless water slides!Both start here! Both get to the bottom here! Who is traveling faster at the bottom?Who reaches the bottom first?

Demonstration!v = 0, y = hy = 0v = ?

Figure 8-1. Caption: Object of mass m: (a) falls a height h vertically; (b) is raised along an arbitrary two-dimensional path.

Figure 8-2. Caption: (a) A tiny object moves between points 1 and 2 via two different paths, A and B. (b) The object makes a round trip, via path A from point 1 to point 2 and via path B back to point 1.

Figure 8-3. Caption: A crate is pushed at constant speed across a rough floor from position 1 to position 2 via two paths, one straight and one curved. The pushing force FP is always in the direction of motion. (The friction force opposes the motion.) Hence for a constant magnitude pushing force, the work it does is W = FPd, so if d is greater (as for the curved path), then W is greater. The work done does not depend only on points 1 and 2; it also depends on the path taken. ##### Conservation of Energy. Forces Conservative Force - Gives back work done against it. (Gravity & springs) Non-conservative Force - Does not give back work
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