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Lecture 7

1. Core Concepts

  • Conservative Force (CF):
  • The work it does is independent of the path taken between two points.
  • The work it does over any closed path is zero.

  • Examples: Gravity, Elastic (Spring) Force, Electrostatic Force.

  • Non-Conservative Force (NCF):

  • The work it does depends on the path taken.

  • Examples: Friction, Air Resistance, Tension, an applied push/pull.

  • Potential Energy (U):

  • Energy stored in a system due to its position or configuration.
  • Defined only for conservative forces.

  • The work done by a CF is the negative change in potential energy.

  • Total Mechanical Energy (E):

  • The sum of a system's kinetic and potential energy: E = K + U.

  • Equilibrium & Stability:

  • Equilibrium: A state where the net force is zero (\(F_x = 0\)). This occurs at points where the slope of the potential energy is zero (\(dU/dx = 0\)).
  • Stable Equilibrium: Occurs at a minimum of the potential energy curve (\(d^2U/dx^2 > 0\)). The system returns to this point if slightly displaced.
  • Unstable Equilibrium: Occurs at a maximum of the potential energy curve (\(d^2U/dx^2 < 0\)). The system moves away from this point if slightly displaced.

2. Key Formulas

Formula Description When to Use
\(W_{CF} = -\Delta U = U_1 - U_2\) Work done by a conservative force. 1=initial, 2=final. Relates the work of a CF to the change in U.
\(U_{grav} = mgy\) Gravitational Potential Energy. For objects of mass m at height y near Earth's surface.
\(U_{el} = \frac{1}{2}kx^2\) Elastic Potential Energy. For a spring with constant k displaced by x from equilibrium.
\(K_1 + U_1 = K_2 + U_2\) Conservation of Mechanical Energy. Crucial! Use when only conservative forces (gravity, springs) do work.
\(\frac{1}{2}mv_1^2 + U_1 = \frac{1}{2}mv_2^2 + U_2\) Expanded form of energy conservation. Same as above. The most common formula you'll use from this lecture.
\(W_{other} + W_{CF} = \Delta K\) Work-Energy Theorem. The most general work theorem. Always true.
\(W_{other} = \Delta E = E_2 - E_1\) Work-Energy with NCFs. Use when non-conservative forces (friction, push/pull) are present.
\(F_x = -\frac{dU}{dx}\) Force from Potential Energy (1D). Finds the conservative force component from the potential energy function.

3. Exam Tips & Problem-Solving Strategy

  1. Analyze the Forces: First, identify all forces acting on the object.
  2. Is Mechanical Energy Conserved?
    • Are there any non-conservative forces doing work (friction, air resistance, a person pushing)?
    • YES \(\implies\) Energy is NOT conserved. Use the Work-Energy Theorem: \(W_{other} = \Delta E\).
    • NO \(\implies\) Energy IS conserved. Use Conservation of Energy: \(K_1 + U_1 = K_2 + U_2\). This is usually the easier method!
  3. Define Your System & States:
    • Clearly define your initial (State 1) and final (State 2) points.
    • Write down the known values (\(v_1, y_1, v_2, y_2,\) etc.) for each state.
  4. Set Your Zero Level: Strategically choose where potential energy is zero (e.g., \(y=0\)). The lowest point of motion is often a good choice to avoid negative numbers.
  5. Watch the Signs: The negative sign in \(W_{CF} = -\Delta U\) is critical. Remember: a CF doing positive work (ball falling) decreases potential energy.