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--- 183_notes:ucm [2015/09/06 15:50] – caballero
+++ 183_notes:ucm [2021/02/18 21:12] (current) – [The Net Force for Uniform Circular Motion] stumptyl
@@ Line 1: / Line 1: @@
-===== Uniform Circular Motion =====
+====== Uniform Circular Motion ======
-There are times when you will observe systems that move around some central axis in a very regular fashion. For example, the Moon revolves around the Earth in an orbit that is nearly circular. In doing so, it moves with nearly the same speed (not velocity!) at every location in its orbit. A system whose motion can be modeled as moving in a circular orbit at constant speed is said to execute "uniform circular motion." It is called "uniform" because the speed of the system doesn't change. The velocity is always changing direction, but not size. In these notes, you will read about a special mathematical form that the net force takes when the motion of the system is uniform and circular.
+There are times when you will observe systems that move around some central axis in a very regular fashion. For example, the Moon revolves around the Earth in an orbit that is nearly circular. In doing so, it moves with nearly the same speed (not velocity!) at every location in its orbit. A system whose motion can be modeled as moving in a circular orbit at constant speed is said to execute "uniform circular motion." It is called "uniform" because the speed of the system doesn't change. The velocity is always changing direction, but not size. **In these notes, you will read about a special mathematical form that the net force takes when the motion of the system is uniform and circular.**
-=== The Net Force for Uniform Circular Motion ===
+==== The Net Force for Uniform Circular Motion ====
 [{{ 183_notes:ucm_setup.png?200|The Moon orbiting the Earth with a speed $v$ and at a distance $R$ from the Earth.}}]
@@ Line 33: / Line 33: @@
 $$ \vec{F}_{net} = \dfrac{mv^2}{R\theta} \langle - \sin \theta, \cos \theta - 1 \rangle $$
-In fact, this is the //average// net force in this situation. You cannot get a more accurate estimate on this average net force without considering shorter times steps. That is, situations where the angular distance is very small. If you do consider such situations, the average net force becomes the instantaneous net force at the location. To do this, we make the approximation that $\theta$ is very small. In calculus, you might have seen [[https://en.m.wikibooks.org/wiki/Trigonometry/Power_Series_for_Cosine_and_Sine|what happens to trig functions when their arguments get very small]],
+In fact, this is the //average// net force in this situation. You cannot get a more accurate estimate on this average net force without considering shorter times steps. That is, situations where the angular distance is very small. If you do consider such situations, the average net force becomes the instantaneous net force at the location. //To do this, we make the approximation that $\theta$ is very small.// In calculus, you might have seen [[https://en.m.wikibooks.org/wiki/Trigonometry/Power_Series_for_Cosine_and_Sine|what happens to trig functions when their arguments get very small]],
 \begin{eqnarray*}
@@ Line 50: / Line 50: @@
 $$F_{net,ucm} = \dfrac{mv^2}{R}$$
-and always points towards the inside of the circle. This is the direction that net force needs to be to keep the object moving in a circle. This [[http://paer.rutgers.edu/pt3/experiment.php?topicid=1&exptid=56|video demonstration]] illustrates applying a force inward to make a ball move in a circle. Sometimes this force is refereed to as the [[http://en.wikipedia.org/wiki/Centripetal_force|"centripetal force"]]((The word centripetal means "center seeking" and refers to the fact the the net force is the case of uniform circular motion always points towards the inside of the circle.)).
+and always points towards the inside of the circle. This is the direction that net force needs to be to keep the object moving in a circle. This [[http://paer.rutgers.edu/pt3/experiment.php?topicid=1&exptid=56|video demonstration]] illustrates applying a force inward to make a ball move in a circle. Sometimes this force is referred to as the [[http://en.wikipedia.org/wiki/Centripetal_force|"centripetal force"]]((The word centripetal means "center seeking" and refers to the fact the the net force is the case of uniform circular motion always points towards the inside of the circle.)).
-=== The Centripetal Force is not a Real Force ===
+==== The Centripetal Force is not a Real Force ====
 A force quantifies the interaction between pairs of objects. By this definition, the "centripetal force" is not a real force. It does not quantify the interaction between any pair of objects, it is a mathematical convenience when a system is moves in uniform circular motion. It is a conceptual and calculational tool.
-The real forces are the interactions (real pushes and pulls) that give rise to the net force. It is just for the case of uniform circular motion that the net force can also be calculated using the the change in momentum, which takes on the $mv^2/R$ form.
+The real forces are the interactions (real pushes and pulls) that give rise to the net force. It is just for the case of uniform circular motion that the net force can also be calculated using the the change in momentum, which takes on the $mv^2/R$ form. You might find many examples on the internet and (even in some books!) that claim otherwise, but the centripetal force does not result from the interaction of a pair of objects - it's not a real force.