Chapter 8: Potential Energy and Conservation of Energy

Chapter 8: Potential Energy and Conservation of Energy

Conservative and Nonconservative Forces

Overview of Forces

  • Two general kinds of forces are classified in mechanics:

    • Conservative Forces:

    • The work and energy associated with these forces can be recovered.

    • The work done by conservative forces is stored in the form of potential energy that can be converted back into kinetic energy at a later time.

    • Nonconservative Forces:

    • These forces are typically dissipative, meaning the work done against them cannot easily be recovered, ultimately converting mechanical energy into other forms.

Characteristics of Forces

  • Work Done by a Conservative Force:

    • The work done by a conservative force is path independent.

    • Examples of conservative forces include:

    • Gravity

    • Spring force

  • Work Done by a Nonconservative Force:

    • The work done by a nonconservative force is path dependent.

    • Example: Friction force, drag force.

  • For conservative forces, the work done moving an object around a closed path is zero, unlike nonconservative forces.

Work Done by Gravity

  • Gravitational Force is considered constant near the Earth's surface.

  • Example: Consider a falling rock;

    • The net force on the rock can be expressed as:
      F_{net, y} = -mg

    • The net work done by gravity on the rock can be defined as:

    • In terms of displacement from a reference height, W_{g}=-mg\Delta y

      • where: \Delta y=y-y_0

    • Here, since y<y_0,\Delta y<0 which makes W_{g} positive when force and displacement are in the same direction.

Potential Energy

  • Definition: Potential energy is associated with the position of the object within a system.

  • For every conservative force, a corresponding potential energy function can be defined.

    • Choosing y_0=0 as our reference height for zero potential energy U = 0, allows us to express potential energy as:
      \Delta U=-W_{conservative}

  • Gravitational Potential Energy:

    • \Delta U_{g}=-W_{g}=-\left(-mg\Delta y\right)=mg\Delta y=mg\left(y-y_0\right)

    • or U_{g}=mgy

  • This defines gravitational potential energy as the energy due to an object's position in a gravitational field, particularly that of Earth at a given height.

Conservation of Mechanical Energy

  • For conservative forces, the principle of conservation of mechanical energy states:

    • The total mechanical energy remains constant:
      Mechanical~Energy = Kinetic~Energy + Potential~Energy = Constant

  • Expressing this through net work, we have:

    • W_{net}=K-K_0=\Delta K  

    • Additionally, it holds that:
      \Delta U=-W_{net}

    • This can be rearranged to show that:
      W_{net}=\Delta K=-\Delta U

    • This gives:
      \Delta K=-\Delta U or \Delta K+\Delta U=0   

  • Thus, overall energy conservation can also be summarized as:
    E=K+U=K_0+U_0= constant

  • This upholds the Conservation of Mechanical Energy principle.

Elastic Potential Energy

  • Defined as the energy stored in a stretched or compressed spring, related to the work done when stretching/compressing it.

  • Selecting x_0 = 0 (the equilibrium position) as the reference point U = 0 gives:

    • \Delta U_{s}=-W_{s}=-\left(\frac12kx_0^2-\frac12kx^2\right)=\frac12kx^2-\frac12kx_0^2

    • Therefore:
      U_{s}=\frac12kx^2

  • Spring Potential Energy is the energy stored in the spring due to its deformation (either compression or extension).

Conservation of Energy Including a Spring

  • The potential energy of the spring is factored into both sides of the conservation of energy equation.

  • This leads to the same problem-solving strategies for mechanical systems, where:
    \left(KE+PE_{g}+PE_{s}\right)_{i}=\left(KE+PE_{g}+PE_{s}\right)_{f}

Work Done by Nonconservative Forces

  • When nonconservative forces are present in a mechanical system, total mechanical energy is not conserved:

    • W_{total}=W_{c}+W_{nc}=-\Delta U+W_{nc}=\Delta K

    • Positive work done by nonconservative forces increases the total mechanical energy of the system; negative work decreases the mechanical energy, converting it to other forms.

  • The work can be expressed as:

    • W_{nc}=\Delta U+\Delta K=\Delta E\rightarrow W_{nc}=E_{f}-E_{i}

    • Rearranging gives us:

    • E_{f}=E_{i}+W_{nc}

    • Nonconservative work can be either positive (e.g., running) or negative (e.g., due to friction).

Potential Energy Curves

  • A potential energy curve represents the potential energy (U) as a function of position.

  • The curve can portray various physical scenarios and can be visualized similar to the profile of hills or roller coasters, representing the change in gravitational potential energy with position, denoted as:

    • \Delta U\left(x\right)=-W=-F\left(x\right)\Delta x

    • where: F\left(x\right)=-\frac{\Delta U\left(x\right)}{\Delta x}