http://www.physicsbook.gatech.edu/api.php?action=feedcontributions&feedformat=atom&user=Tschiavo3 2026-05-11T16:28:40Z User contributions MediaWiki 1.42.7 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48265 2026-04-29T00:06:23Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:Screenshot 2026-04-28 at 8.05.36 PM.png|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> [[File:Screenshot 2026-04-28 at 8.01.36 PM.png|center]]<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> [[File:Screenshot 2026-04-28 at 8.03.19 PM.png|center]]<br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_8.05.36_PM.png&diff=48264 2026-04-29T00:05:58Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48263 2026-04-29T00:04:32Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:1Screenshot 2026-04-28 at 7.49.50 PM.png|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> [[File:Screenshot 2026-04-28 at 8.01.36 PM.png|center]]<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> [[File:Screenshot 2026-04-28 at 8.03.19 PM.png|center]]<br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_8.03.19_PM.png&diff=48262 2026-04-29T00:03:58Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48261 2026-04-29T00:02:40Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:1Screenshot 2026-04-28 at 7.49.50 PM.png|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> [[File:Screenshot 2026-04-28 at 8.01.36 PM.png|center]]<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_8.01.36_PM.png&diff=48260 2026-04-29T00:02:11Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48259 2026-04-29T00:00:38Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:1Screenshot 2026-04-28 at 7.49.50 PM.png|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48258 2026-04-29T00:00:19Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:1Screenshot 2026-04-28 at 7.49.50 PM|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:1Screenshot_2026-04-28_at_7.49.50_PM.png&diff=48257 2026-04-28T23:59:38Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_7.49.50_PM.png&diff=48256 2026-04-28T23:58:57Z

<p>Tschiavo3: Tschiavo3 uploaded a new version of File:Screenshot 2026-04-28 at 7.49.50 PM.png</p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48255 2026-04-28T23:57:33Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:Potential_Energy_Curve.png|500px|center|frameless]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48254 2026-04-28T23:56:49Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:Screenshot_2026-04-28_at_7.49.50_PM.png|center]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48253 2026-04-28T23:55:56Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> [[File:Screenshot_2026-04-28_at_7.49.50_PM.png|400px|center|Potential Energy Curve]]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_7.49.50_PM.png&diff=48252 2026-04-28T23:50:01Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_7.49.12_PM.png&diff=48251 2026-04-28T23:49:25Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=File:Screenshot_2026-04-28_at_7.47.48_PM.png&diff=48250 2026-04-28T23:47:57Z

<p>Tschiavo3: </p> <hr /> <div></div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48249 2026-04-28T23:44:59Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> If U slopes up → force points left<br /> If U slopes down → force points right<br /> Steeper slope → stronger force<br /> <br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> <br /> Types:<br /> <br /> Minimum of U(x) → stable equilibrium<br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> <br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> K = 0<br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> High K → fast motion<br /> Low K → slow motion<br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> Parabola opening upward<br /> Minimum at x = 0 → stable equilibrium<br /> Motion is oscillatory<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> Linear with height<br /> Used for ramps and hills<br /> Speed depends only on height difference, not slope.<br /> <br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> Negative potential energy<br /> Stronger interaction at small r<br /> <br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> Positive potential energy<br /> Objects are pushed apart<br /> <br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> Object is trapped<br /> Motion occurs between turning points<br /> Example: orbiting planet<br /> <br /> <br /> Unbound System<br /> <br /> E &gt; 0<br /> <br /> Object escapes<br /> Example: spacecraft leaving a planet<br /> <br /> <br /> Escape Energy<br /> E = 0<br /> <br /> Object barely escapes<br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> 7. How to Read Any Energy Graph<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> Where U is high → speed is low<br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> Steeper slope → stronger force<br /> Minimum → stable equilibrium<br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always</div>

Tschiavo3 http://www.physicsbook.gatech.edu/index.php?title=Energy_Graphs&diff=48248 2026-04-28T23:38:26Z

<p>Tschiavo3: </p> <hr /> <div>THOMAS SCHIAVO FALL 2026<br /> <br /> 1. What Is an Energy Graph?<br /> Energy graphs typically plot:<br /> <br /> <br /> Energy vs. position → U(x), K(x), E(x)<br /> <br /> <br /> Energy vs. time → U(t), K(t), E(t)<br /> <br /> <br /> They allow you to:<br /> visualize where forces act<br /> determine where motion is possible<br /> identify equilibrium points<br /> find turning points<br /> compare speeds instantly<br /> <br /> <br /> <br /> 2. Potential Energy Graphs U(x)<br /> Potential energy graphs contain the most information.<br /> Force from Potential Energy<br /> Force is the negative slope of the graph:<br /> F(x) = – dU/dx<br /> <br /> <br /> If U slopes up → force points left<br /> <br /> <br /> If U slopes down → force points right<br /> <br /> <br /> Steeper slope → stronger force<br /> <br /> <br /> <br /> [INSERT IMAGE: U(x) curve with slope arrows showing force direction]<br /> <br /> Equilibrium Points<br /> Equilibrium occurs where:<br /> slope = 0 → F = 0<br /> Types:<br /> <br /> <br /> Minimum of U(x) → stable equilibrium<br /> <br /> <br /> Maximum of U(x) → unstable equilibrium<br /> <br /> <br /> <br /> [INSERT IMAGE: potential well showing stable vs unstable equilibrium]<br /> <br /> 3. Total Mechanical Energy<br /> Total energy is:<br /> E = K + U<br /> For conservative systems, total energy is constant → horizontal line on graphs.<br /> <br /> Allowed Motion<br /> Motion is only possible where:<br /> E ≥ U(x)<br /> <br /> <br /> If U &gt; E → forbidden region<br /> <br /> <br /> If U = E → turning point<br /> <br /> <br /> <br /> Turning Points<br /> At turning points:<br /> <br /> <br /> K = 0<br /> <br /> <br /> velocity = 0 (object reverses direction)<br /> <br /> <br /> <br /> [INSERT IMAGE: horizontal energy line intersecting U curve at turning points]<br /> <br /> 4. Kinetic Energy Graphs K(x)<br /> Kinetic energy is:<br /> K(x) = E – U(x)<br /> Since:<br /> K = ½mv²<br /> <br /> <br /> High K → fast motion<br /> <br /> <br /> Low K → slow motion<br /> <br /> <br /> K = 0 → object stops<br /> <br /> <br /> Important:<br /> <br /> <br /> K is always ≥ 0<br /> <br /> <br /> <br /> 5. Most Important Potential Shapes<br /> <br /> A. Spring Potential (Harmonic Oscillator)<br /> U(x) = ½kx²<br /> <br /> <br /> Parabola opening upward<br /> <br /> <br /> Minimum at x = 0 → stable equilibrium<br /> <br /> <br /> Motion is oscillatory<br /> <br /> <br /> <br /> [INSERT IMAGE: parabola with horizontal energy line and oscillation region]<br /> <br /> B. Gravitational Potential (Near Earth)<br /> U = mgh<br /> <br /> <br /> Linear with height<br /> <br /> <br /> Used for ramps and hills<br /> <br /> <br /> Key idea:<br /> Speed depends only on height difference, not slope.<br /> <br /> [INSERT IMAGE: different slopes with same height drop]<br /> <br /> C. Attractive Potentials (Gravity / Electric)<br /> U(r) = –k/r<br /> <br /> <br /> Negative potential energy<br /> <br /> <br /> Stronger interaction at small r<br /> <br /> <br /> <br /> [INSERT IMAGE: attractive potential curve approaching zero from below]<br /> <br /> D. Repulsive Potentials<br /> U(r) = +k/r<br /> <br /> <br /> Positive potential energy<br /> <br /> <br /> Objects are pushed apart<br /> <br /> <br /> <br /> [INSERT IMAGE: repulsive potential curve approaching zero from above]<br /> <br /> 6. Bound vs Unbound Systems<br /> <br /> Bound System<br /> <br /> <br /> E &lt; 0<br /> <br /> <br /> Object is trapped<br /> <br /> <br /> Motion occurs between turning points<br /> <br /> <br /> Example: orbiting planet<br /> <br /> [INSERT IMAGE: energy line below zero inside potential well]<br /> <br /> Unbound System<br /> <br /> <br /> E &gt; 0<br /> <br /> <br /> Object escapes<br /> <br /> <br /> Example: spacecraft leaving a planet<br /> <br /> [INSERT IMAGE: energy line above potential curve]<br /> <br /> Escape Energy<br /> E = 0<br /> <br /> <br /> Object barely escapes<br /> <br /> <br /> Final velocity approaches 0 at infinity<br /> <br /> <br /> <br /> [INSERT IMAGE: escape energy diagram]<br /> <br /> 7. How to Read Any Energy Graph (Exam Checklist)<br /> <br /> <br /> Where U is low → speed is high<br /> <br /> <br /> Where U is high → speed is low<br /> <br /> <br /> U = E → turning point<br /> <br /> <br /> Slope of U → direction of force<br /> <br /> <br /> Steeper slope → stronger force<br /> <br /> <br /> Minimum → stable equilibrium<br /> <br /> <br /> Maximum → unstable equilibrium<br /> <br /> <br /> K(x) = E – U(x) always<br /> <br /> <br /> <br /> 8. Common Mistakes<br /> <br /> <br /> Thinking steeper hill means faster (wrong)<br /> <br /> <br /> Letting kinetic energy be negative (impossible)<br /> <br /> <br /> Ignoring forbidden regions<br /> <br /> <br /> Confusing force with value of U (it’s the slope, not the height)<br /> <br /> <br /> <br /> 9. Example Problems<br /> <br /> Problem 1: Two hills, same height<br /> Which is faster at the bottom?<br /> Answer: Same speed<br /> Only height difference matters.<br /> <br /> Problem 2: Where is the object fastest?<br /> Answer: Where U is minimum.<br /> <br /> Problem 3: Direction of force<br /> <br /> <br /> Negative slope → force right<br /> <br /> <br /> Positive slope → force left<br /> <br /> <br /> <br /> Problem 4: Where can the object move?<br /> Answer: Only where E ≥ U(x)<br /> <br /> 10. Advanced Insight<br /> Energy graphs act like a “map of motion.”<br /> From one graph, you can determine:<br /> <br /> <br /> speed (from kinetic energy)<br /> <br /> <br /> acceleration (from slope)<br /> <br /> <br /> direction (from slope sign)<br /> <br /> <br /> This connects energy concepts directly to Newton’s Laws.<br /> <br /> 11. Interactive Simulation<br /> &lt;iframe src=&quot;https://trinket.io/glowscript/31d0f9ad9e&quot; width=&quot;100%&quot; height=&quot;600&quot;&gt;&lt;/iframe&gt;<br /> <br /> Final Takeaway<br /> Energy graphs let you solve problems by reading instead of calculating.<br /> If you can:<br /> <br /> <br /> read slopes<br /> <br /> <br /> compare U and E<br /> <br /> <br /> find turning points<br /> <br /> <br /> → you can solve most Physics 1 energy problems quickly.<br /> <br /> If you want next, I can make you a practice test that looks exactly like your exam (graphs + multiple choice traps).</div>

Tschiavo3