Chapter 6: Work and Energy

  • Energy is conserved; can’t be created / destroyed
  • Kinetic Energy describes motion => mv^2
  • Energy on earth originates from the sun
  • Energy on earth is stored thermally and chemically
  • Chemical energy is released by metabolism
  • Energy is stored as potential energy in objects


Conservation of Energy

Energy cannot be created or destroyed, but can be transformed other energies of another form.

  • Dissipation of “heat” – Energy could be lost by transforming into heat


Work

Work done is the dot product (scalar) of the force and the displacement times cos θ of the angle.

  • W = F Δd cos θ = unit J (Joules)
  • Work is only considered when the force applied is not perpendicular to the displacement
  • In a uniform circular motion, the work done is always 0


Kinetic Energy

Work = Σ Force ( ΔDisplacement) = (mass)(acceleration)(Displacement)

W=m(aΔd)=m12(v2fv2i)=12mv2f12mv2iΔK=KfKigiving:K=12mv2W=m(a\Delta d)=m\frac{1}{2}(v_{2f}-v_{2i})=\frac{1}{2}mv_{2f}-\frac{1}{2}mv_{2i}\Rightarrow \Delta K=K_{f}-K_{i} giving: K=\frac{1}{2}mv_{2}

Constant Force \Rightarrow W=(FcosΘ)ΔdW=(F\cos \Theta )\Delta d

Variable Force WF(cosΘ)Δd1+F(cosΘ)Δd....W\approx F(\cos\Theta )\Delta d_{1}+F(\cos\Theta )\Delta d....


Hooke’s Law

W=12kX2withFx(X)(12kx)(x)W=\frac{1}{2}kX_{2}\Rightarrow with F_{x}(X)\Rightarrow (\frac{1}{2}kx)(x)


Gravitational Potential Energy

Wgravity=(FcosΘ)ΔdW_{gravity}=(F\cos\Theta )\Delta d  for Wgravity=mg(hihf)W_{gravity}=mg(h_{i}-h_{f}) with g=9.8m/s2|g|=9.8m/s_{2}

  • Take into consideration with only the parallel force to displacement Take Ugrav=mghWgrav=ΔUgravU_{grav}=mgh\Rightarrow W_{grav} =-\Delta U_{grav}

For an object to fall, ΔUgrav\Delta U_{grav} < 0 for that the final Energy < initial Energy


Elastic Potential Energy

Welastic=12kx1212kxf2W_{elastic}=\frac{1}{2}k x_{12}\frac{1}{2}kx_{f2}

Similar to the gravitational potential energy, this energy is stored depending on distance, x.


Conservative vs Non‐conservative Forces

  1. A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions.
  2. A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point.


Gravitational, Elastic Spring and Electric Force are Conservative

Workdone=WorkConservative+WorkNonconservativeWork_{done}=Work_{Conservative}+Work_{Non-conservative}

  • WorkNonconservative=ΔK+ΔU=(KfKi)+(UfUi)Work_{Non-conservative}=\Delta K+\Delta U=(K_{f}-K_{i})+(U_{f}-U_{i})
  • WNc=(Kf+Uf)(Ki+Ui)W_{Nc}=(K_{f}+U_{f})-(K_{i}+U_{i})
  • E=K+UE=K+U
  • WNc=EfEiW_{Nc}=E_{f}-E_{i}
  • WNc=0W_{Nc}=00=EfEi0=E_{f}-E_{i}Ef=EiE_{f}=E_{i}


Power

Power - the rate of change of energy over time.

  • P = Work / Time = ΔW/ΔtΔW/Δt = Change in energy / time
  • Given Joule/ s = watt (W)
  • Average Power = Force ( Average Velocity)
  • Instantaneous Power = Power = Force (Velocity)


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