183_notes:force_and_pe

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183_notes:force_and_pe [2021/04/01 12:49] – [Force is the Negative Gradient of Potential Energy] stumptyl183_notes:force_and_pe [2023/11/30 20:35] (current) – [Force is the Negative Gradient of Potential Energy] hallstein
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 To find the force in three-dimensions, this derivative of the potential becomes the [[http://en.wikipedia.org/wiki/Gradient|gradient]] of the potential, To find the force in three-dimensions, this derivative of the potential becomes the [[http://en.wikipedia.org/wiki/Gradient|gradient]] of the potential,
  
-$$\vec{F} = - \nabla U = \left\langle \dfrac{dU}{dx},  -\dfrac{dU}{dy},  -\dfrac{dU}{dz}\right\rangle$$+$$\vec{F} = - \nabla U = \left\langle -\dfrac{dU}{dx},  -\dfrac{dU}{dy},  -\dfrac{dU}{dz}\right\rangle$$
 $$\vec{F} = -\dfrac{dU}{dx}\hat{x} -\dfrac{dU}{dy} \hat{y} -\dfrac{dU}{dz} \hat{z}$$ $$\vec{F} = -\dfrac{dU}{dx}\hat{x} -\dfrac{dU}{dy} \hat{y} -\dfrac{dU}{dz} \hat{z}$$
  
-==== Equilibrium Points ====+===== Equilibrium Points =====
  
 That the force is the spatial derivative of the potential energy is a helpful way of thinking about equilibria -- locations in space where the force acting on the particle is zero. Some equilibria are stable -- if the particle is located at that point, it will stay near it even when given a small push. Some are unstable -- given a small push, the particle will run away.  That the force is the spatial derivative of the potential energy is a helpful way of thinking about equilibria -- locations in space where the force acting on the particle is zero. Some equilibria are stable -- if the particle is located at that point, it will stay near it even when given a small push. Some are unstable -- given a small push, the particle will run away. 
  
-=== Spring-Mass System ===+==== Spring-Mass System ====
  
-[{{ 183_notes:potential_energy.007.png?300|The potential energy of the spring-mass system is plotted as a function of the stretch.}}]+[{{ 183_notes:potential_energy.007.png?300|The potential energy of the spring-mass system is plotted as a function of the stretch in order to highlight the equilibrium points.}}]
  
 Consider the [[183_notes:grav_and_spring_pe|potential energy of a spring-mass system]]. Here, the potential energy is quadratic (bowl-shaped) function, Consider the [[183_notes:grav_and_spring_pe|potential energy of a spring-mass system]]. Here, the potential energy is quadratic (bowl-shaped) function,
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 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 "bowl-shaped" potential energy. Also, the force to the right side of the equilibrium point is pointing to the left ($F=-kx<0$ because $x>0$) and the force to the left side of the equilibrium point is pointing to the right ($F=-kx>0$ because $x<0$). 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 "bowl-shaped" potential energy. Also, the force to the right side of the equilibrium point is pointing to the left ($F=-kx<0$ because $x>0$) and the force to the left side of the equilibrium point is pointing to the right ($F=-kx>0$ because $x<0$).
  
-=== More general potential energy diagrams ===+==== More general potential energy diagrams ====
  
 [{{ 183_notes:potential_energy.004.png?350|For this potential energy graph, the equilibria are marked.}}] [{{ 183_notes:potential_energy.004.png?350|For this potential energy graph, the equilibria are marked.}}]
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  • Last modified: 2021/04/01 12:49
  • by stumptyl