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183_notes:force_and_pe [2014/10/10 21:01] – caballero | 183_notes:force_and_pe [2021/04/01 12:47] – [Force and Potential Energy] stumptyl | ||
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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 [[183_notes: |
+ | ==== Lecture Video ==== | ||
+ | {{youtube> | ||
==== Force is the Negative Gradient of Potential Energy ==== | ==== Force is the Negative Gradient of Potential Energy ==== | ||
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$$\Delta U = -\int_{x_i}^{x_f} F_x dx$$ | $$\Delta U = -\int_{x_i}^{x_f} F_x dx$$ | ||
- | The above integral considers the change in potential energy over all path that takes the object from $x_i$ to $x_f$. The differential ($dx$) is really small. Consider | + | The above integral considers the change in potential energy over all path that takes the object from $x_i$ to $x_f$. The differential ($dx$) is really small. Consider |
$$dU = -F_x dx$$ | $$dU = -F_x dx$$ | ||
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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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[{{ 183_notes: | [{{ 183_notes: | ||
- | Consider the potential energy of a spring-mass system. Here, the potential energy is quadratic (bowl-shaped) function, | + | Consider the [[183_notes: |
$$U=\dfrac{1}{2}kx^2$$ | $$U=\dfrac{1}{2}kx^2$$ | ||
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The force associated with that potential is the spring force, | The force associated with that potential is the spring force, | ||
- | $$F_x=-\dfrac{dU}{dx} = \dfrac{d}{dx}\left(\dfrac{1}{2}kx^2\right) = -kx$$ | + | $$F_x=-\dfrac{dU}{dx} = -\dfrac{d}{dx}\left(\dfrac{1}{2}kx^2\right) = -kx$$ |
The force is zero at $x=0$. At that point, the slope of the potential energy graph is also zero. This point is stable because it is at the bottom of the " | The force is zero at $x=0$. At that point, the slope of the potential energy graph is also zero. This point is stable because it is at the bottom of the " | ||
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=== More general potential energy diagrams === | === More general potential energy diagrams === | ||
- | [{{ 183_notes: | + | [{{ 183_notes: |
- | In a more general potential energy diagram (to the left), you can determine the equilibrium points by finding where the slope is zero ($F_x=0$). The stability of those points can be classified as stable or unstable. 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 B, a small push on the bead would cause the bead to move up a bit, but it would come back -- location B is stable. At location A and C, a bead given a small push would run away from those locations -- both are unstable. | + | In a more general potential energy diagram (to the right), you can determine the equilibrium points by finding where the slope is zero ($F_x=0$). The stability of those points can be classified as stable or unstable. |
- | {{183_notes: | + | 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 B, a small push on the bead would cause the bead to move up a bit, but it would come back -- location B is stable. At location A and C, a bead given a small push would run away from those locations -- both are unstable. |
+ | ==== Kinetic and Potential Energy in Potential Energy Graphs ==== | ||
+ | |||
+ | 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 $K_2$). | ||
+ | |||
+ | 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: |