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183_notes:force_and_pe [2014/10/10 21:01] – caballero | 183_notes:force_and_pe [2014/10/24 20:46] – [Force is the Negative Gradient of Potential Energy] caballero | ||
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===== Force and Potential Energy ===== | ===== Force and Potential Energy ===== | ||
- | The work done by a force is the integral of the force along the path that the force acts. This definition of the work gives rise to a relationship between | + | The work done by a force is the integral of the force along the path that the force acts. This definition of the work gives rise to a relationship between the potential energy due to the interaction between the objects and the force responsible for that interaction. In these notes, you will read about the relationship between the force and the potential energy and how a graphical representation of the potential energy can also illustrate this force. |
+ | ==== Lecture Video ==== | ||
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+ | {{youtube> | ||
==== Force is the Negative Gradient of Potential Energy ==== | ==== Force is the Negative Gradient of Potential Energy ==== | ||
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The force in the x-direction is the negative derivative of the potential energy, | The force in the x-direction is the negative derivative of the potential energy, | ||
- | $$F_x = \dfrac{dU}{dx}$$ | + | $$F_x = - \dfrac{dU}{dx}$$ |
To find the force in three-dimensions, | To find the force in three-dimensions, | ||
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=== More general potential energy diagrams === | === More general potential energy diagrams === | ||
- | [{{ 183_notes: | + | [{{ 183_notes: |
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+ | In a more general potential energy diagram | ||
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+ | A way to think about stability is to think of a bead sitting at the equilibiurm location. Is it stable against small pushes? For example, at location | ||
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+ | ==== Kinetic and Potential Energy in Potential Energy Graphs ==== | ||
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+ | From these potential energy graphs, you are able to determine the kinetic energy of the system at any location along the graph if you know the total energy of the system. In graph below, the total energy is indicated with a dotted line. The potential energy at any point is measured from the $U=0$ line (e.g., $U_1$ and $U_2$). Because the total energy is the sum of kinetic and potential ($E_{tot} = K + U$), the kinetic energy is measured from the potential energy graph to the dotted line (e.g., $K_1$ and $K_2$). | ||
- | In a more general potential energy | + | From this diagram you can conclude that a particle with the given total energy will not make it past the location on the right where the dotted line crosses the solid line. It just doesn' |
- | {{183_notes: | + | {{ 183_notes: |