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October 27th, 2019, 03:48 AM  #1 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus  How do I find the average angular velocity between two points?
The problem is as follows: In the figure from below it is shown an observer who has put himself at the center of the coordinate system. He sees an object moving in a circular trajectory. If the average speed between $A$ and $B$ is $\left ( 2\hat{i}+\hat{j} \right )\frac{m}{s}$ and its position on point $A$ is $5\hat{i}\,m$. Find on $\frac{rad}{s}$ the average angular velocity between $A$ and $B$ if the time the object takes to get from $A$ to $B$ is $4\,s$. The alternatives on my book: $\begin{array}{ll} 1.&0.35\hat{k}\\ 2.&0.5\hat{k}\\ 3.&0.55\hat{k}\\ 4.&0.6\hat{k}\\ 5.&0.75\hat{k}\\ \end{array}$ For this particular problem I'm stuck at how to use the information given the average velocity and the position. However I recall that when the word average is mentioned it mean this formula? $\overline{v}=\frac{\vec{r}}{\Delta t}$ But other than that I'm not sure if it applies in this situation. Can somebody offer some help with this question?. Last edited by Chemist116; October 27th, 2019 at 03:51 AM. Reason: missing characters 
October 27th, 2019, 04:59 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,101 Thanks: 1677 
$\Delta r = (2i+j) \cdot 4 = 8i+4j$ $r(4)r(0)=8i+4j$ $r(4)5i=8i+4j \implies r(4)=3i+4j$ $\theta (4) = \pi + \arctan\left(\dfrac{4}{3}\right)$ $\theta (0) = 0$ $\bar{\omega} = \dfrac{\Delta \theta}{\Delta t} = \dfrac{\theta (4)  \theta (0)}{4} = \dfrac{\pi+\arctan\left(\frac{4}{3}\right)}{4} \approx 0.55k$ 
October 27th, 2019, 02:17 PM  #3 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus 
Wait a second. When you subtract $r(4)r(0)$ wouldn't it be (for the first component). This $\left(85\right)\hat{i}=13\hat{i}$ Or could it be that I'm not understanding it right the subtraction of vectors?. Can you help me with this part because the rest was obvioud but it is there where I'm stuck!. Last edited by Chemist116; October 27th, 2019 at 02:19 PM. Reason: fixed characters 
October 27th, 2019, 02:24 PM  #4 
Senior Member Joined: Jun 2017 From: Lima, Peru Posts: 188 Thanks: 5 Math Focus: Calculus 
Sorry I had to re read what you did. It turns out that we don't know in advance the position vector for the second point so that what you used is to build an equation for obtaining that vector and from then the angle. But I'd like a disclaimer I mean some theory behind on what is the difference between these concepts and how should be used or what do they mean? average velocity average speed average acceleration and just acceleration. Are these terms interchangeable?. 
October 27th, 2019, 06:41 PM  #5  
Math Team Joined: Jul 2011 From: Texas Posts: 3,101 Thanks: 1677  Quote:
You were given the average angular velocity and the time. We use these two pieces of info to determine the change in position in the xy plane. $\Delta r = \omega \cdot t = (2i+j) \cdot 4 = 8i + 4j$ $r(0) = 5i + 0j$ at point A is given ... all you know about $r(4)$ at point B is what you can see in the diagram. It is located in quadrant II, hence a negative xcomponent and positive ycomponent. Let $r(4) = ai + bj$ ... like I stated earlier, all we know now is $a<0$ and $b>0$. $\Delta r = r(4)  r(0)$ $8i + 4j = (ai + bj)  (5i + 0j)$ add $(5i+0j)$ to both sides ... $(8i+4j)+(5i+0j) = ai + bj$ $3i + 4j = ai + bj \implies a = 3 \text{ and } b=4$ ... a vector in quad II with distance of 5 from the origin.  Quote:
Angular Motion, General Notes  

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