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.