My Math Forum  

Go Back   My Math Forum > Science Forums > Physics

Physics Physics Forum


Thanks Tree7Thanks
  • 1 Post By skipjack
  • 3 Post By skeeter
  • 2 Post By skipjack
  • 1 Post By skeeter
Reply
 
LinkBack Thread Tools Display Modes
October 26th, 2019, 10:33 PM   #1
Senior Member
 
Chemist116's Avatar
 
Joined: Jun 2017
From: Lima, Peru

Posts: 188
Thanks: 5

Math Focus: Calculus
Question How do I find the time from given velocities in a circular motion?

This problem has left me going in circles. No pun intended, but I'm confused at how to use the vectors given to find the time.

The problem is as follows:
A sphere has a uniform circular motion. This object passes through two points $A$ and $B$, when this happens its speed is $\left(-4\hat{\textrm{i}}+4\sqrt{3}\hat{\textrm{j}}\right )\frac{m}{s}$ and $\left(8\hat{\textrm{i}}\right)\frac{m}{s}$ respectively. The radius of the circle where this object is moving is $\textrm{3 meters}$. Find the time which will take the particle to move from $A$ to $B$.
The alternatives given are:

$\begin{array}{ll}
1.&1.57\\
2.&3.14\\
3.&6.28\\
4.&0.16\\
5.&0.31\\
\end{array}$

Typically I would try to show some effort into this. But I'm stuck at the beginning. What is it exactly that should I do to find the given time?

The only thing which I could come up with is that to find the time it can be obtained from the tangential speed as:

$v_{t}=\omega r = \frac{\theta}{t} r$

But apart from this I don't know how to link it with what is being asked. Can somebody give me a hand with this?

Last edited by skipjack; October 27th, 2019 at 03:26 AM.
Chemist116 is offline  
 
October 27th, 2019, 04:13 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 21,124
Thanks: 2332

Let's assume that "sphere" was supposed to be "particle", that the particle is moving along the circumference of a circle of radius 3 m at a constant speed, and that the particle's velocity vectors at points A and B on the circle are as given in the problem (albeit incorrectly called speeds).

The magnitude of the given vectors is 8 m/s, which is the particle's constant speed.

As the length of the circumference of the circle is 6$\pi$ m,
the time required for a complete circuit of the circle is $\frac34\pi$ s.

The distance to be covered in going from A to B may be 1/3 of the length of the circumference of the circle or 2/3 of that length (because of the angle between the given vectors). There is insufficient information to let you determine which applies unless you assume that one of the answers must be correct. You should choose the answer that is approximately $\frac14\pi$ s or $\frac12\pi$ s. Hence option 1 is the expected answer choice.
Thanks from topsquark
skipjack is offline  
October 27th, 2019, 05:35 AM   #3
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,101
Thanks: 1677

Assuming standard motion about the circle, the angular position of point B is $\theta = \dfrac{3\pi}{2}$ since its tangential velocity is purely in the positive x direction.

The tangential velocity components at point A indicate its position is in quadrant I. Since tangential velocity is perpendicular to the position vector, the position vector is in the direction $ \theta = \arctan\left(\dfrac{1}{\sqrt{3}}\right) = \dfrac{\pi}{6}$.

The angular distance from A to B is $\dfrac{4\pi}{3}$ radians. Given the radius of 3, the linear distance from A to B is $4\pi$ meters.

$t=\dfrac{d}{v} = \dfrac{4\pi}{8} = \dfrac{\pi}{2} \approx 1.57$ sec
Thanks from topsquark, idontknow and Chemist116
skeeter is offline  
October 27th, 2019, 05:57 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 21,124
Thanks: 2332

I assume that "standard motion" means "counterclockwise motion".
For clockwise motion, one gets $\displaystyle t = \frac{\pi}{4}$, which doesn't correspond to any of the listed answer choices.
Thanks from skeeter and Chemist116
skipjack is offline  
October 28th, 2019, 10:01 PM   #5
Senior Member
 
Chemist116's Avatar
 
Joined: Jun 2017
From: Lima, Peru

Posts: 188
Thanks: 5

Math Focus: Calculus
Lightbulb

Quote:
Originally Posted by skeeter View Post
Assuming standard motion about the circle, the angular position of point B is $\theta = \dfrac{3\pi}{2}$ since its tangential velocity is purely in the positive x direction.

The tangential velocity components at point A indicate its position is in quadrant I. Since tangential velocity is perpendicular to the position vector, the position vector is in the direction $ \theta = \arctan\left(\dfrac{1}{\sqrt{3}}\right) = \dfrac{\pi}{6}$.

The angular distance from A to B is $\dfrac{4\pi}{3}$ radians. Given the radius of 3, the linear distance from A to B is $4\pi$ meters.

$t=\dfrac{d}{v} = \dfrac{4\pi}{8} = \dfrac{\pi}{2} \approx 1.57$ sec
I believe that this answer would be benefited if it could be accompanied by a drawing or sketch. I did one. Not sure if it would help. But it seems to correspond to what you meant by your method.

If you go in one direction has half the speed from going in the other direction.



I'm not sure if this helps, but I used it as an aid to my understanding.
Chemist116 is offline  
October 29th, 2019, 07:36 AM   #6
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,101
Thanks: 1677

No, the speeds are the same ... the CW distance is 1/2 the CCW distance, hence the time for the CCW path takes twice the time.

1st screenshot is the CCW path,2nd is the CW path
Attached Images
File Type: png CCW.png (1.5 KB, 6 views)
File Type: png CW.png (1.4 KB, 6 views)
Thanks from Chemist116
skeeter is offline  
Reply

  My Math Forum > Science Forums > Physics

Tags
circular, find, ina, motion, time, velocities



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Circular Motion hyperbola Physics 1 May 12th, 2015 03:16 AM
Finding vertical displacement given velocities at time intervals? Wecoxepa Calculus 6 December 20th, 2014 09:06 AM
circular motion (hard) hoyy1kolko Physics 1 April 14th, 2011 08:54 AM
circular motion hoyy1kolko Physics 1 April 5th, 2011 08:32 AM
Circular Motion johnny Physics 3 December 16th, 2007 07:36 PM





Copyright © 2019 My Math Forum. All rights reserved.