10-01-08LectPHY2048

# 10-01-08LectPHY2048 - Energy methods(continued Last time we...

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Energy methods (continued) Last time we developed the following ideas: WF d r G G Energy (an abstract quantity that takes different, calculable) forms, is a conserved quantity. The energy of pure translation (no rotation) is kinetic energy, K defined as, 2 1 Km v 2 The energy transferred to (or from) a body is called work and (in its most general form) is given by, Where is the force acting on the body and is the infinitesimal displacement along the path through which the body moves. dr G F G (body’s mass, m , and speed v )

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And the work-kinetic energy theorem that relates the work (energy transferred to a body) to its change of kinetic energy, WK = Δ From the definition of work as we can deduce some general behaviors and specialize to particular circumstances. WF d r G G If the force is generated by multiple distinct sources , e.g. gravity, designated and two other applied forces designated and such that, then F G g F G a F G b F G gab FF F F = ++ G GGG d r( F F F ) d r =⋅ = + +⋅ ∫∫ GG G G G G gabg a b d r F d r F d r W W W + + = + + ∫∫∫ G The work is the sum of the work from distinct sources.
And from the work-kinetic energy theorem since, gab WWW K + += Δ Hence if say W b is not known but the others are we can determine W b by solving for it, WK = Δ b ga W W = Δ− Example – A felled, 300 kg tree is dragged in a straight line by a tractor from rest to a speed of 3 m/s over a distance of 10 m. The constant force applied by the tractor was 2010 N. What was the work done by friction? Let the work done by the tractor be W T and that done by friction be W fr then, Tf r fi WW K KK + =Δ =

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Since the tree started from rest v i = 0 making K i = 0. Tf rf WW K + = 2 fr f T 1 Wm v W 2 =− Now TT WF d r =⋅ G G The tractor pulls in a straight line so its force is exactly along the displacement i.e. 0 T T F dr F drcos F drcos0 F dr ⋅= φ = = G G Moreover the force F T is constant so it comes out of the integral d T 0 d r F d ==
Hence, 2 fr f T 1 Wm v F d 2 =− 2 fr 1m W (300kg)(3 ) (2010 N)(10 m) 2s fr W 1350J 20100J 18750J = So of the total energy (20,100 J) that the tractor transferred to the tree, 1350 J were required to accelerate it from rest to it’s velocity of 3 m/s, while the major fraction (18,750 J) was “lost” to friction. In the most general sense that energy did not vanish from the universe, it still exists but in the form of heat.

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10-01-08LectPHY2048 - Energy methods(continued Last time we...

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