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183_notes:grav_pe_graphs [2015/10/09 19:50] – [Graphs of Gravitational Potential Energy] caballero | 183_notes:grav_pe_graphs [2021/05/25 16:43] – [Graphs of Gravitational Potential Energy] stumptyl | ||
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+ | Section 6.10 in Matter and Interactions (4th edition) | ||
+ | |||
===== Graphing Energy for Gravitationally Interacting Systems ===== | ===== Graphing Energy for Gravitationally Interacting Systems ===== | ||
- | Knowing the [[183_notes: | + | Knowing the [[183_notes: |
==== Lecture Video ==== | ==== Lecture Video ==== | ||
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==== Graphs of Gravitational Potential Energy ==== | ==== Graphs of Gravitational Potential Energy ==== | ||
- | [{{ 183_notes:grav_potential.png?400|A graph of the gravitational potential energy versus separation (solid red line); the zero of potential energy is marked with the solid black line.}}] | + | [{{ 183_notes:potentialgraph1_9.png?400|A graph of the gravitational potential energy versus separation (solid red line); the zero of potential energy is marked with the solid black line.}}] |
- | You can graph the gravitational potential energy as a function of the radial separation, | + | You can graph the gravitational potential energy |
$$U(r) = -G\dfrac{Mm}{r}$$ | $$U(r) = -G\dfrac{Mm}{r}$$ | ||
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=== Visualizing the kinetic energy === | === Visualizing the kinetic energy === | ||
- | The value of the potential energy is measured from the zero line down to the graph' | + | __**The value of the potential energy is measured from the zero line down to the graph' |
- | Notice that in the figure on the left, the total energy is negative and hence the less massive object cannot get any farther away then the location where the potential energy equals the system' | + | Notice that in the figure on the left, the total energy is negative and hence the less massive object cannot get any farther away then the location where the potential energy equals the system' |
- | For the figure on the right, the total energy is positive and hence, even at infinite distance, the less massive object has non-zero kinetic energy. This is an //unbound system// because the less massive object can move infinitely far away from the more massive object. | + | For the figure on the right, the total energy is positive and hence, even at infinite distance, the less massive object has non-zero kinetic energy. This is an **unbound system** because the less massive object can move infinitely far away from the more massive object. |
- | [{{183_notes: | + | [{{183_notes: |
- | [{{183_notes: | + | [{{183_notes: |
- | ==== How is $\Delta U = mgh$ an approximation? | + | ===== How is $\Delta U = mgh$ an approximation? |
- | [{{ 183_notes: | + | [{{183_notes: |
+ | |||
+ | \\ | ||
As you have read, the [[183_notes: | As you have read, the [[183_notes: | ||
- | If you zoom in on the graph of the gravitational potential energy, it looks like it increases linearly (figure to the right). You can show mathematically that this will produce the same expected result (with an additional constant term). | + | If you zoom in on the graph of the gravitational potential energy, it looks like it increases linearly (figure to the left). You can show mathematically that this will produce the same expected result (with an additional constant term). |
- | === Mathematical | + | ==== Mathematical |
- | Consider an object of mass $m$ at a distance $y$ above the Earth' | + | Consider an object of mass $m$ (kg) at a distance $y$ (m) above the Earth' |
$$U_{grav} = -G\dfrac{M_Em}{\left(R_E+y\right)} = -G\dfrac{M_Em}{R_E\left(1+\dfrac{y}{R_E}\right)} = -m\dfrac{GM_E}{R_E}\dfrac{1}{\left(1+\dfrac{y}{R_E}\right)}$$ | $$U_{grav} = -G\dfrac{M_Em}{\left(R_E+y\right)} = -G\dfrac{M_Em}{R_E\left(1+\dfrac{y}{R_E}\right)} = -m\dfrac{GM_E}{R_E}\dfrac{1}{\left(1+\dfrac{y}{R_E}\right)}$$ | ||
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[{{ 183_notes: | [{{ 183_notes: | ||
- | It is often the the kinetic energy of the less massive object which is graphed along side the potential energy of the system and the total energy. For a //bound system//, this graph looks like the one to the right (green line is the kinetic energy). | + | It is often the the kinetic energy of the less massive object which is graphed along side the potential energy of the system and the total energy. For **a bound system**, this graph looks like the one to the right (green line is the kinetic energy). |
The kinetic energy graph has the same characteristic shape as the potential energy graph, but it is a reflected version. As the potential energy gets larger (less negative), the kinetic gets smaller and vice versa. The kinetic energy cannot become negative, so its graph terminates at zero energy. This is the farthest location the less massive object can reach with the given total energy. | The kinetic energy graph has the same characteristic shape as the potential energy graph, but it is a reflected version. As the potential energy gets larger (less negative), the kinetic gets smaller and vice versa. The kinetic energy cannot become negative, so its graph terminates at zero energy. This is the farthest location the less massive object can reach with the given total energy. | ||
- | For an //unbound system// the kinetic energy levels off to the value of the total (positive) energy of the system. When the less massive object is infinitely far away, the potential energy of the system goes to zero. | + | For an **unbound system** the kinetic energy levels off to the value of the total (positive) energy of the system. When the less massive object is infinitely far away, the potential energy of the system goes to zero. |
+ | |||
+ | ==== Examples ==== | ||
+ | |||
+ | * [[: |