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183_notes:constantf [2014/08/08 18:45] – caballero | 183_notes:constantf [2023/01/13 19:59] (current) – hallstein | ||
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+ | Section 2.5 in Matter and Interactions (4th edition) | ||
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===== Constant Force Motion ===== | ===== Constant Force Motion ===== | ||
- | You read previously how to (separately) [[183_notes: | + | You read previously how to (separately) [[183_notes: |
==== Lecture Video ==== | ==== Lecture Video ==== | ||
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==== A Constant Net Force ==== | ==== A Constant Net Force ==== | ||
- | A system that experiences a constant [[183_notes: | + | __//A system that experiences a constant [[183_notes: |
Depending on how you select your coordinate system, it might mean that more than one component of the momentum vector changes. Often, it is convenient to select a coordinate system where the net force is aligned with a coordinate direction, then only one momentum vector component changes in time. | Depending on how you select your coordinate system, it might mean that more than one component of the momentum vector changes. Often, it is convenient to select a coordinate system where the net force is aligned with a coordinate direction, then only one momentum vector component changes in time. | ||
+ | {{youtube> | ||
- | ==== Predicting the Motion ==== | + | |
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+ | ====== Predicting the Motion | ||
Consider a fan cart that is released on a low-friction track. Here's a video of the situation. | Consider a fan cart that is released on a low-friction track. Here's a video of the situation. | ||
{{ youtube> | {{ youtube> | ||
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+ | \\ | ||
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Notice that the fan cart's position changes more rapidly near the end of the video. The fan cart experiences (to a good approximation) a constant net force. The sum of all the forces acting on the fan cart give (roughly) a net force of constant magnitude and direction. Furthermore, | Notice that the fan cart's position changes more rapidly near the end of the video. The fan cart experiences (to a good approximation) a constant net force. The sum of all the forces acting on the fan cart give (roughly) a net force of constant magnitude and direction. Furthermore, | ||
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With this setup, you can predict the position of the fan cart given only information about its initial position, velocity (or momentum), and the net force acting on it. | With this setup, you can predict the position of the fan cart given only information about its initial position, velocity (or momentum), and the net force acting on it. | ||
- | === Deriving the Equation for Constant | + | \\ |
+ | ===== Deriving the Equation for Constant | ||
If you choose the horizontal direction to be the x-direction, | If you choose the horizontal direction to be the x-direction, | ||
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From this equation, you can determine the arithmetic average velocity, which in this case is equal to the average velocity. | From this equation, you can determine the arithmetic average velocity, which in this case is equal to the average velocity. | ||
+ | $$v_{avg,x} = \dfrac{v_{ix} + v_{fx}}{2} = \dfrac{ v_{ix} + v_{ix} + \dfrac{F_{net, | ||
+ | = \dfrac{2v_{ix}}{2}+ \dfrac{\dfrac{F_{net, | ||
- | $$v_{avg,x} = \dfrac{v_{xi} + v_{xf}}{2} = \dfrac{ v_{xi} + v_{ix} + \dfrac{F_{net,x}}{m} \Delta t}{2} | + | By using this average velocity in the [[183_notes: |
- | = \dfrac{2v_{xi}}{2}+ \dfrac{\dfrac{F_{net,x}}{m} \Delta t}{2} = v_{xi}+ \dfrac{1}{2}\dfrac{F_{net, | + | |
- | By using this average velocity in the [[183_notes: | + | $$x_{f} = x_{i} + v_{avg,x} \Delta t = x_{i} + v_{ix} \Delta t + \dfrac{1}{2}\dfrac{F_{net, |
- | + | ||
- | $$x_{f} = x_{i} + v_{avg,x} \Delta t = x_{i} + v_{xi} \Delta t + \dfrac{1}{2}\dfrac{F_{net, | + | |
In physics, the information about the system prior to predicting its motion is called the " | In physics, the information about the system prior to predicting its motion is called the " | ||
- | === Constant Force in 3D === | + | |
+ | \\ | ||
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+ | ==== Connection to Energy ==== | ||
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+ | As you will read, the motion of systems can also be predicted or explained by using the [[183_notes: | ||
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+ | For constant force motion in one dimension (e.g., x-direction), | ||
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+ | $$v_{xf}^2 = v_{xi}^2 + 2\dfrac{F_{net, | ||
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+ | Again, as you will read, this equation can also be derived from [[183_notes: | ||
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+ | \\ | ||
+ | ==== Summary of Constant Force ==== | ||
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+ | The relationship between force and acceleration (even for a variable net force): $\vec{F}_{net}=m\vec{a}$ OR $\vec{a}=\frac{\vec{F}_{net}}{m}$. | ||
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+ | The following 1D equations are valid ONLY if the net force (and therefore, the acceleration) is constant. | ||
+ | $$x_{f} = x_{i} + v_{avg,x} \Delta t$$ | ||
+ | $$v_{fx} = v_{ix} + \dfrac{F_{net, | ||
+ | $$v_{avg,x} = \dfrac{v_{ix} + v_{fx}}{2} = v_{ix}+ \dfrac{1}{2}\dfrac{F_{net, | ||
+ | $$x_{f} = x_{i} + v_{ix} \Delta t + \dfrac{1}{2}\dfrac{F_{net, | ||
+ | $$v_{xf}^2 = v_{xi}^2 + 2\dfrac{F_{net, | ||
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+ | \\ | ||
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+ | ==== Constant Force in 3D ==== | ||
The derivation for each dimension is similar (so long as the force is constant in each direction). The result is the following general equation, | The derivation for each dimension is similar (so long as the force is constant in each direction). The result is the following general equation, | ||
$$\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{i} \Delta t + \dfrac{1}{2}\dfrac{\vec{F}_{net}}{m} \Delta t^2$$ | $$\vec{r}_{f} = \vec{r}_{i} + \vec{v}_{i} \Delta t + \dfrac{1}{2}\dfrac{\vec{F}_{net}}{m} \Delta t^2$$ | ||
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===== Examples ===== | ===== Examples ===== | ||
[[: | [[: | ||
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+ | [[: |