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183_notes:newton_grav_pe [2014/10/17 15:06] – [Newtonian Gravitational Potential Energy] caballero | 183_notes:newton_grav_pe [2021/04/01 12:54] (current) – [Newtonian Gravitational Potential Energy] stumptyl | ||
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+ | Section 6.8 in Matter and Interactions (4th edition) | ||
+ | |||
===== Gravitational Potential Energy ===== | ===== Gravitational Potential Energy ===== | ||
- | You have read about the [[183_notes: | + | You have read about the [[183_notes: |
+ | |||
+ | ==== Lecture Video ==== | ||
+ | |||
+ | {{youtube> | ||
==== (Near Earth) Gravitational Potential Energy ==== | ==== (Near Earth) Gravitational Potential Energy ==== | ||
- | Earlier, you read how the gravitational potential energy for a system consisting of two objects (The Earth and something on the surface of the Earth) is given by, | + | Earlier, you read how the gravitational potential energy |
$$\Delta U_{grav} = +mg\Delta y$$ | $$\Delta U_{grav} = +mg\Delta y$$ | ||
- | where the separation distance is measured from the surface of the Earth. | + | where the separation distance |
- | However, | + | However, |
- | ==== Newtonian Gravitational Potential Energy ==== | + | ===== Newtonian Gravitational Potential Energy |
In general, the gravitational force exerted on a object of mass $m_1$ due to an object of mass $m_2$ is non-constant, | In general, the gravitational force exerted on a object of mass $m_1$ due to an object of mass $m_2$ is non-constant, | ||
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So for this force, what is the gravitational potential energy? | So for this force, what is the gravitational potential energy? | ||
- | === Solve the 1-dimensional problem first === | + | ==== Solve the One-Dimensional Problem First ==== |
Remember that the potential energy change is the negative change in the internal work ($\Delta U = -W_{int}$). So, you can calculate what the work done by the gravitational force would be and use that to determine that change in potential energy in going from location 1 to location 2, | Remember that the potential energy change is the negative change in the internal work ($\Delta U = -W_{int}$). So, you can calculate what the work done by the gravitational force would be and use that to determine that change in potential energy in going from location 1 to location 2, | ||
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You can compute the work done by the gravitational force as the mass moves from $x=x_1$ to $x=\infty$. | You can compute the work done by the gravitational force as the mass moves from $x=x_1$ to $x=\infty$. | ||
- | The force on the little mass is given by | + | The force on the little mass at any location $x$ is given by |
$$F_{grav}(x) = -G\dfrac{Mm}{x^2}$$ | $$F_{grav}(x) = -G\dfrac{Mm}{x^2}$$ | ||
- | where the minus indicates the force points to the left. Because the displacement ($dx$) is to the right the work done by the gravitational force in this case is negative ($W_{grav} < 0$). This serves as a check for you as you do the calculation; | + | where the minus indicates the force points to the left. Because the displacement ($dx$) is to the right, the work done by the gravitational force in this case is negative ($W_{grav} < 0$). This serves as a check for you as you do the calculation; |
- | $$W_{grav} = \int_{x_1}^{\infty} F(x)\:dx = -\int_{x_1}^{\infty}G\dfrac{Mm}{x^2}\: | + | $$W_{grav} = \int_{x_1}^{\infty} F(x)\:dx = -\int_{x_1}^{\infty}G\dfrac{Mm}{x^2}\: |
This potential energy is definitely negative because $x_1$ is a positive value. You can now determine the potential energy change, | This potential energy is definitely negative because $x_1$ is a positive value. You can now determine the potential energy change, | ||
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$$\Delta U_{grav} = \underbrace{U(x=\infty)}_{0} - U(x_1) = +G\dfrac{Mm}{x_1}$$ | $$\Delta U_{grav} = \underbrace{U(x=\infty)}_{0} - U(x_1) = +G\dfrac{Mm}{x_1}$$ | ||
$$U(x_1) = -G\dfrac{Mm}{x_1}$$ | $$U(x_1) = -G\dfrac{Mm}{x_1}$$ | ||
+ | |||
+ | ==== General form of the gravitational potential energy ==== | ||
Thus, in general, if we measure the radial distance from an object the gravitational potential energy varies inversely with the distance, | Thus, in general, if we measure the radial distance from an object the gravitational potential energy varies inversely with the distance, | ||
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$$U(r) = -G\dfrac{Mm}{r}$$ | $$U(r) = -G\dfrac{Mm}{r}$$ | ||
- | First, notice that as the distance gets very large the potential energy goes to zero ($U\rightarrow 0$ as $r\rightarrow \intfy$). | + | First, notice that as the distance gets very large the potential energy goes to zero ($U\rightarrow 0$ as $r\rightarrow \infty$). |
Second, notice that this is a slight notational change. The distance $r$ is the radial distance from the origin (or the object for which we consider to be at the origin). This value of $r$ is always positive. | Second, notice that this is a slight notational change. The distance $r$ is the radial distance from the origin (or the object for which we consider to be at the origin). This value of $r$ is always positive. | ||
As you will read, [[183_notes: | As you will read, [[183_notes: |