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=rac12mv2KE = 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:
        PEg=mghPE_g = mgh

      • Elastic Potential Energy (SPE):

      • Stored in springs.

      • Formula: PEs=rac12kx2PE_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=riangleKE=KE</em>f−KEiW<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=riangleKEW<em>{net} + W</em>{c} = riangle KE

    • Where $W_c$ is work done by non-conservative forces.
      } mv^2 </p></li></ul></li></ul><h4id="f652910d−1dce−4908−b244−987c7b60d789"data−toc−id="f652910d−1dce−4908−b244−987c7b60d789"collapsed="false"seolevelmigrated="true">Work−EnergyTheorem</h4><ul><li><p><strong>ScalarQuantity</strong>:</p><ul><li><p>KineticenergyismeasuredinJoules(J).</p></li><li><p><strong>Work−EnergyTheoremStatement</strong>:</p></li><li><p>Thenetworkonanobjectisequaltothechangeintheobject′skineticenergy.</p></li><li><p>Importanttoconsidernonconservativeforces:<br></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>Whenfrictionispresent,adjusttheequationaccordingly:<br></p></li><li><p>When friction is present, adjust the equation accordingly: <br> W{net} = KEf - KE_i </p></li></ul></li></ul><h4id="1d3b78ab−b23d−45c8−b1e0−cfd2fbb1506e"data−toc−id="1d3b78ab−b23d−45c8−b1e0−cfd2fbb1506e"collapsed="false"seolevelmigrated="true">GravitationalPotentialEnergy</h4><ul><li><p><strong>Conceptualization</strong>:</p><ul><li><p>Potentialenergy(PE)asasystemproperty,especiallyinrelationtogravitationalforcesactingonobjects.</p></li><li><p>Updatethework−energytheoremtoincludegravitationalpotentialenergy:</p></li><li><p>Whendealingwithgravitationalwork:<br></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>Incorporatemodificationswhenapplyingnon−conservativeforcesleadingtooverallmechanicalenergychanges:<br></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><h4id="408ec461−7105−4d63−a285−337ea05eefb6"data−toc−id="408ec461−7105−4d63−a285−337ea05eefb6"collapsed="false"seolevelmigrated="true">ConservationofMechanicalEnergy</h4><ul><li><p><strong>ConservativeForceDefinition</strong>:Pressure/forcesthatbehaveindependentlyofthepathtaken.</p></li><li><p><strong>KeyConcepts</strong>:</p><ul><li><p>Workdonethroughaclosedpathleadstozerochangeinenergy:<br></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>Mechanicalenergyconservationrelationship:<br></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:

    • F_s = -kx </p></li><li><p><strong>Work−EnergyRelation</strong>:</p></li><li><p>Averagespringforceimpactsworkdoneoveradistance:<br></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>Thepotentialenergyinthespringisdefinedas:<br></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).