Chapter 9 Notes: Work and Kinetic Energy

Energy Overview

  • Energy lets us "do work."
  • Work is the transfer of energy by a force; it can be positive, negative, or zero.
  • Units of work: Newton x meters = joules (J).
  • Every system has total energy EsysE_{sys}.
  • Energy can transform within a system without loss.
  • Energy can be transferred between a system and its environment through work and heat.
    • Work: Mechanical transfer of energy.
    • Heat: Non-mechanical transfer due to temperature difference.
  • Total energy of a system changes by the amount of work done on it.
  • Isolated system: No energy transferred in or out.

Work and Kinetic Energy

  • Work: W=F⃗⋅Δs⃗=FΔscos⁡θW = \vec{F} \cdot \Delta \vec{s} = F \Delta s \cos{\theta}
  • Only force component in the direction of displacement does work.
  • Work is a scalar quantity.
  • 1 joule (J) = 1 N⋅m
  • Kinetic energy (KE): Energy of motion, K=12mv2K = \frac{1}{2}mv^2
  • Work-Kinetic Energy Theorem: ΔK=W\Delta K = W

Work Done by Multiple Forces

  • Total work: W<em>tot=W</em>1+W<em>2+W</em>3+…W<em>{tot} = W</em>1 + W<em>2 + W</em>3 + \dots
  • Total kinetic energy: K<em>tot=K</em>1+K<em>2+K</em>3+…K<em>{tot} = K</em>1 + K<em>2 + K</em>3 + \dots
  • Change in system energy: ΔE<em>sys=ΔK</em>tot=Wtot\Delta E<em>{sys} = \Delta K</em>{tot} = W_{tot}

Force vs. Position Graphs

  • Work done is the area under the force-versus-position graph.
  • W=∫<em>x</em>initialxfinalF(x)dxW = \int<em>{x</em>{initial}}^{x_{final}} F(x) dx

Restoring Forces and Springs

  • Restoring force: Restores system to equilibrium.
  • Hooke's Law: Fspring=−kΔsF_{spring} = -k \Delta s
    • Δs\Delta s = Spring displacement.
    • k = Spring constant (N/m).
  • Work done by a spring: W=∫<em>x</em>1x<em>2−kxdx=−12k(x</em>22−x12)W = \int<em>{x</em>1}^{x<em>2} -kx dx = -\frac{1}{2}k(x</em>2^2 - x_1^2)
  • W<em>by spring=−12k(Δs</em>f2−Δsi2)W<em>{by \, spring} = -\frac{1}{2}k (\Delta s</em>f^2 - \Delta s_i^2)

Dissipative Forces and Thermal Energy

  • Dissipative forces convert other energy forms to thermal energy (EthE_{th}).
  • ΔE<em>sys=ΔK+ΔE</em>th=Wext\Delta E<em>{sys} = \Delta K + \Delta E</em>{th} = W_{ext}
  • Increase in thermal energy due to friction: ΔE<em>th=f</em>kΔs\Delta E<em>{th} = f</em>k \Delta s

Power

  • Power: Rate of energy transfer.
  • P=dEsysdtP = \frac{dE_{sys}}{dt}
  • Unit: watt (W), 1 W = 1 J/s
  • Power as rate of doing work: P=dWdtP = \frac{dW}{dt}
  • Power delivered to a particle: P=F⃗⋅v⃗=Fvcos⁡θP = \vec{F} \cdot \vec{v} = Fv \cos{\theta}