183_notes:grav_pe_graphs

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183_notes:grav_pe_graphs [2021/04/01 16:46] – [Graphs of Gravitational Potential Energy] stumptyl183_notes:grav_pe_graphs [2021/04/01 16:57] – [Graphing Kinetic Energy] stumptyl
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-==== How is $\Delta U = mgh$ an approximation? ====+===== How is $\Delta U = mgh$ an approximation? =====
  
 [{{183_notes:grav_potential_enlarged.png?500|The gravitational potential energy near the surface of the Earth (or any massive object) can be approximated as a linearly increasing function.}}] [{{183_notes:grav_potential_enlarged.png?500|The gravitational potential energy near the surface of the Earth (or any massive object) can be approximated as a linearly increasing function.}}]
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 As you have read, the [[183_notes:grav_accel#the_local_gravitational_acceleration_revisited|gravitational force near the surface of the Earth is an approximation]] of the Newtonian gravitational force. As you might suspect, the gravitational potential energy near the surface of the Earth (or any large object) can be approximated also. As you have read, this form of the [[183_notes:grav_and_spring_pe#near_earth_gravitational_potential_energy|gravitational potential energy]] increases linearly with distance (i.e., $\Delta U_{grav} = +mg\Delta y$).  As you have read, the [[183_notes:grav_accel#the_local_gravitational_acceleration_revisited|gravitational force near the surface of the Earth is an approximation]] of the Newtonian gravitational force. As you might suspect, the gravitational potential energy near the surface of the Earth (or any large object) can be approximated also. As you have read, this form of the [[183_notes:grav_and_spring_pe#near_earth_gravitational_potential_energy|gravitational potential energy]] increases linearly with distance (i.e., $\Delta U_{grav} = +mg\Delta y$). 
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 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).  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 proof of the approximation ===+==== Mathematical Proof of the Approximation ====
  
-Consider an object of mass $m$ at a distance $y$ above the Earth's surface (mass, $M_E$; radius, $R_E$). The potential energy of the object-Earth system is:+Consider an object of mass $m$ (kg) at a distance $y$ (m) above the Earth's surface (mass, $M_E$; radius, $R_E$). The potential energy of the object-Earth system is:
  
 $$U_{grav} = -G\dfrac{M_Em}{\left(R_E+y\right)} = -G\dfrac{M_Em}{R_E\left(1+\dfrac{y}{R_E}\right)} = -m\dfrac{GM_E}{R_E}\dfrac{1}{\left(1+\dfrac{y}{R_E}\right)}$$ $$U_{grav} = -G\dfrac{M_Em}{\left(R_E+y\right)} = -G\dfrac{M_Em}{R_E\left(1+\dfrac{y}{R_E}\right)} = -m\dfrac{GM_E}{R_E}\dfrac{1}{\left(1+\dfrac{y}{R_E}\right)}$$
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 [{{ 183_notes:grav_potential_kinetic.png?450|A graph of the potential, kinetic, and total energy of a gravitationally bound system. The kinetic energy is only for the less massive object in the system. The assumption is that it is much less massive than the larger object.}}] [{{ 183_notes:grav_potential_kinetic.png?450|A graph of the potential, kinetic, and total energy of a gravitationally bound system. The kinetic energy is only for the less massive object in the system. The assumption is that it is much less massive than the larger object.}}]
  
-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 ====  ==== Examples ==== 
  
   * [[:183_notes:examples:videoswk8|Video Example: Evaluating potential energy graphs (in an orbit)]]   * [[:183_notes:examples:videoswk8|Video Example: Evaluating potential energy graphs (in an orbit)]]
  • 183_notes/grav_pe_graphs.txt
  • Last modified: 2024/01/31 14:45
  • by hallstein