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183_notes:grav_and_spring_pe [2014/10/10 15:45] – [Spring Potential Energy] caballero | 183_notes:grav_and_spring_pe [2014/10/10 18:15] – caballero | ||
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===== Types of Potential Energy ===== | ===== Types of Potential Energy ===== | ||
+ | [[183_notes: | ||
==== (Near Earth) Gravitational Potential Energy ==== | ==== (Near Earth) Gravitational Potential Energy ==== | ||
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[{{ 183_notes: | [{{ 183_notes: | ||
- | To determine the (near Earth) gravitational potential energy associated with a system consisting of an object and the Earth, consider the work done by the Earth on the object (mass, $m$) that is being lifted to height $h$ above the surface of the Earth. | + | To determine the (near Earth) gravitational potential energy associated with a system consisting of an object and the Earth, consider the work done by the Earth on the object (mass, $m$) that is being lifted to height $h$ above the surface of the Earth. The displacement and the gravitational force are in opposite directions. |
- | To calculate the work that Earth does, consider the object as the system. | + | To calculate the work that the Earth does, consider the object as the system. |
- System: object; Surroundings: | - System: object; Surroundings: | ||
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==== Spring Potential Energy ==== | ==== Spring Potential Energy ==== | ||
- | [{{ 183_notes: | + | [{{ 183_notes: |
+ | |||
+ | To determine the potential energy associated with a spring-mass system, consider the work done by a spring on an object (mass, $m$) attached to its end. The spring is stretched through a displacement ($\Delta \vec{s}$). The displacement and the gravitational force are in opposite directions. | ||
+ | |||
+ | To calculate the work that the spring does, consider the object as the system. Remember that the [[183_notes: | ||
+ | |||
+ | - System: object; Surroundings: | ||
+ | - Initial state: object at $s_i=0$; Final state: object at $s_f=s$ | ||
+ | |||
+ | $$W_{s} = \int_0^s \vec{F}_{spring}\cdot d\vec{r}$$ | ||
+ | $$W_{s} = \int_0^s \left(-k_sx\right)dx = -k_s\int_0^s x\;dx = -k_sx^2\Big|_0^s = -\dfrac{1}{2}k_ss^2$$ | ||
+ | |||
+ | So more generally, the work done by a spring is given by, | ||
+ | |||
+ | $$W_{s} = -\dfrac{1}{2}k_s\left(s_f^2-s_i^2\right)$$ | ||
+ | |||
+ | If you include the spring in your system, so that the system is now the spring and the object, then potential energy shared spring-object system is given by, | ||
+ | |||
+ | - System: object+spring; | ||
+ | - Initial state: object at $s_i=0$; Final state: object at $s_f=s$ | ||
+ | |||
+ | $$\Delta U_{s} = - W_{s} = +\dfrac{1}{2}k_s\left(s_f^2-s_i^2\right)$$ | ||
+ | |||
+ | The spring potential energy depends on the spring constant ($k_s$) and how stretch changes ($s_f-s_i$). | ||
+ | |||
+ | ==== Conservative Forces ==== | ||
+ | |||
+ | Both of the examples above (local gravitational force and spring force) are examples of [[http:// | ||
+ | |||
+ | Dissipative forces such as friction and air drag are non-conservative forces. The path that an object takes matters very much when non-conservative forces are present. Moreover, these dissipative forces cannot be associated with any construct like potential energy. |