183_notes:force_and_pe

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183_notes:force_and_pe [2015/10/09 18:58] – [Equilibrium Points] caballero183_notes:force_and_pe [2023/11/30 20:35] (current) – [Force is the Negative Gradient of Potential Energy] hallstein
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 ===== Force and Potential Energy ===== ===== Force and Potential Energy =====
  
-The [[183_notes:work_by_nc_forces|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.+The [[183_notes:work_by_nc_forces|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 ==== ==== Lecture Video ====
  
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 ==== Force is the Negative Gradient of Potential Energy ==== ==== Force is the Negative Gradient of Potential Energy ====
  
-As [[183_notes:work_by_nc_forces|you have read]], the work done by a force is related to the integral along the path that the object takes. For forces where you can associate a potential energy, this integral is also related to the change in potential energy.+As [[183_notes:work_by_nc_forces|you have read]], the work (J) done by a force (N) is related to the integral along the path that the object takes. For forces where you can associate potential energy (J), this integral is also related to the change in potential energy.
  
 $$\Delta U = -W_{int} = -\int_i^f\vec{F}\cdot d\vec{r}$$ $$\Delta U = -W_{int} = -\int_i^f\vec{F}\cdot d\vec{r}$$
  
-The potential energy is the negative line integral of the force. In one-dimension this can be written as follows,+__**The potential energy is the negative line integral of the force.**__ In one-dimension this can be written as follows,
  
 $$\Delta U = -\int_{x_i}^{x_f} F_x dx$$ $$\Delta U = -\int_{x_i}^{x_f} F_x dx$$
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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 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 "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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