Kinetic vs. Potential Energy

Kinetic vs. Potential Energy: Physics 111 Notes

Theory & Concepts

• Mechanical Energy: The total energy (kinetic + potential) in a system.

  • Kinetic Energy (KE): Energy associated with motion.

  • Parameters:

    • Velocity ($v$)

    • Mass ($m$)

  • Formula:
    $KE = rac{1}{2} mv^2$

  • Potential Energy (PE): Energy stored based on an object's position.

  • Parameters:

    • Height ($h$) for gravitational potential energy.

    • Spring compression for elastic potential energy.

  • Types of Potential Energy:

    • Gravitational Potential Energy (GPE):

    • Formula:
      $PE_g = mgh$

    • Elastic Potential Energy (SPE):

    • Stored in springs.

    • Formula: $PE_s = rac{1}{2} kx^2$

      • Where $k$ is the spring constant, $x$ is the displacement from the equilibrium position.

    • Electric Potential Energy: Related to positions of charged particles.

• Work-Energy Theorem: Connects work done on an object to its kinetic energy change.

  • Formula:
    $W<em>{net} = riangle KE = KE</em>f - KE_i$

• Non-conservative Forces: Forces like friction which transform mechanical energy into other forms (e.g., heat).

  • Include non-conservative work in the theorem:
    $W<em>{net} + W</em>{c} = riangle KE$

  • Where $W_c$ is work done by non-conservative forces.
    } mv^2 $</p></li></ul></li></ul><h4 id="f652910d-1dce-4908-b244-987c7b60d789" data-toc-id="f652910d-1dce-4908-b244-987c7b60d789" collapsed="false" seolevelmigrated="true">Work-Energy Theorem</h4><ul><li><p><strong>Scalar Quantity</strong>:</p><ul><li><p>Kinetic energy is measured in Joules (J).</p></li><li><p><strong>Work-Energy Theorem Statement</strong>:</p></li><li><p>The net work on an object is equal to the change in the object's kinetic energy.</p></li><li><p>Important to consider nonconservative forces: <br>$ W{net} + W{c} = riangle KE $</p></li><li><p>When friction is present, adjust the equation accordingly: <br>$ W{net} = KEf - KE_i $</p></li></ul></li></ul><h4 id="1d3b78ab-b23d-45c8-b1e0-cfd2fbb1506e" data-toc-id="1d3b78ab-b23d-45c8-b1e0-cfd2fbb1506e" collapsed="false" seolevelmigrated="true">Gravitational Potential Energy</h4><ul><li><p><strong>Conceptualization</strong>:</p><ul><li><p>Potential energy (PE) as a system property, especially in relation to gravitational forces acting on objects.</p></li><li><p>Update the work-energy theorem to include gravitational potential energy:</p></li><li><p>When dealing with gravitational work: <br>$ Wg = F ext{cos}( heta) = mg(yi - y_f) ext{cos}(0°) $</p></li><li><p>Incorporate modifications when applying non-conservative forces leading to overall mechanical energy changes: <br>$ W{net} = W{net} + W_g = riangle KE $</p></li></ul></li></ul><h4 id="408ec461-7105-4d63-a285-337ea05eefb6" data-toc-id="408ec461-7105-4d63-a285-337ea05eefb6" collapsed="false" seolevelmigrated="true">Conservation of Mechanical Energy</h4><ul><li><p><strong>Conservative Force Definition</strong>: Pressure/forces that behave independently of the path taken.</p></li><li><p><strong>Key Concepts</strong>:</p><ul><li><p>Work done through a closed path leads to zero change in energy: <br>$ riangle MME = 0 $</p></li><li><p>Mechanical energy conservation relationship: <br>$ W_{net} = riangle KE + riangle PE

  • Where total mechanical energy $ riangle MME = KEf + PEf - (KEi + PEi)$ remains constant.

Elastic Potential Energy

• Spring Mechanics:

  • The strength of the spring's bounce-back relates to its compression, signifying increased potential energy.

  • Hooke's Law: States that restoring force of the spring, F_s$, is proportional to the displacement from its rest position:</p></li><li><p>$ F_s = -kx $</p></li><li><p><strong>Work-Energy Relation</strong>:</p></li><li><p>Average spring force impacts work done over a distance:<br>$ W_s = rac{1}{2}kx^2 $</p></li><li><p>The potential energy in the spring is defined as: <br>$ PE_s = rac{1}{2} kx^2 $$

Problem-Solving Approach

• Steps:

  1. Read and elucidate the question.

  2. Define starting conditions and set a point of reference for conservation of energy.

  3. Write the conservation equation while identifying unknowns.

  4. Solve for the unknown value(s).