#$&*
Phy 231
Your 'cq_1_10.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** CQ_1_10.1_labelMessages **
A pendulum requires 2 seconds to complete a cycle, which consists of a complete back-and-forth oscillation (extreme point to equilibrium to opposite extreme point
back to equilibrium and finally to the original extreme point). As long as the amplitude of the motion (the amplitude is the distance from the equilibrium position to
the extreme point) is small compared to the length of the pendulum, the time required for a cycle is independent of the amplitude.
How long does it take to get from one extreme point to the other, how long from an extreme point to equilibrium, and how long to go from extreme point to
equilibrium to opposite extreme point and back to equilibrium?
#$&*
answer/question/discussion: ->->->->->->->->->->->-> scussion:
It should take 1 second to get from one extreme point to the other.
It should take less than half a second to get from one extreme point to equilirium, because there is a positive acceleration due to gravity from extreme point to equilibrium. (Then there is a negative acceleration due to gravity from equilirium back up to extreme point). So we deduce that the average velocity for the first half of the path from extreme to extreme is higher than for the second half of the path. Therefore, the time interval will be shorter for the first half than the second half.
By this logic, the time from extreme to extreme to equilibrium should be less than 1.5 seconds, because it includes one fast half, one slow half, and then one more fast half.
What reasonable assumption did you make to arrive at your answers?
#$&*
@&
Your logic is very good. Your conclusion follows from your premise flawlessly.
However your premise itself is flawed. You assume that a positive acceleration results in a shorter time interval than a negative acceleration.
A complete argument would justify this assumption, and would be based on the definitions of velocity and acceleration.
A simple example shows that your assumption is flawed. If is easy to show that if an object starting from rest has a uniform positive acceleration for a certain time interval, then the time interval requred for a subsequent negative acceleration of the same magnitude, acting for the same time interval, will be the identical to the original time interval.
The situation here is not one of uniform acceleration, but of continuously changing acceleration, with the magnitude of the acceleration decreasing with proximity to the equilbrium point. We will need to analyze this more deeply when we study the effects of forces and the nature of the forces involved here. But as it turns out the magnitude of the acceleration of an ideal pendulum at any position depends only on its distance from its equilibrium point.
This symmetry dictates a conclusion identical to that of the previous argument, in which equal and opposite uniform accelerations ensured equal time intervals.
*@
*#&!*#&!
@&
Very good.
You made an excellent argument on that last question, though your premise doesn't hold up. Be sure to check my note.
*@
#$&*
Phy 231
Your 'cq_1_10.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** CQ_1_10.1_labelMessages **
A pendulum requires 2 seconds to complete a cycle, which consists of a complete back-and-forth oscillation (extreme point to equilibrium to opposite extreme point
back to equilibrium and finally to the original extreme point). As long as the amplitude of the motion (the amplitude is the distance from the equilibrium position to
the extreme point) is small compared to the length of the pendulum, the time required for a cycle is independent of the amplitude.
How long does it take to get from one extreme point to the other, how long from an extreme point to equilibrium, and how long to go from extreme point to
equilibrium to opposite extreme point and back to equilibrium?
#$&*
answer/question/discussion: ->->->->->->->->->->->-> scussion:
It should take 1 second to get from one extreme point to the other.
It should take less than half a second to get from one extreme point to equilirium, because there is a positive acceleration due to gravity from extreme point to equilibrium. (Then there is a negative acceleration due to gravity from equilirium back up to extreme point). So we deduce that the average velocity for the first half of the path from extreme to extreme is higher than for the second half of the path. Therefore, the time interval will be shorter for the first half than the second half.
By this logic, the time from extreme to extreme to equilibrium should be less than 1.5 seconds, because it includes one fast half, one slow half, and then one more fast half.
What reasonable assumption did you make to arrive at your answers?
#$&*
@&
Your logic is very good. Your conclusion follows from your premise flawlessly.
However your premise itself is flawed. You assume that a positive acceleration results in a shorter time interval than a negative acceleration.
A complete argument would justify this assumption, and would be based on the definitions of velocity and acceleration.
A simple example shows that your assumption is flawed. If is easy to show that if an object starting from rest has a uniform positive acceleration for a certain time interval, then the time interval requred for a subsequent negative acceleration of the same magnitude, acting for the same time interval, will be the identical to the original time interval.
The situation here is not one of uniform acceleration, but of continuously changing acceleration, with the magnitude of the acceleration decreasing with proximity to the equilbrium point. We will need to analyze this more deeply when we study the effects of forces and the nature of the forces involved here. But as it turns out the magnitude of the acceleration of an ideal pendulum at any position depends only on its distance from its equilibrium point.
This symmetry dictates a conclusion identical to that of the previous argument, in which equal and opposite uniform accelerations ensured equal time intervals.
*@
*#&!*#&!
@&
Very good.
You made an excellent argument on that last question, though your premise doesn't hold up. Be sure to check my note.
*@
#$&*
Phy 231
Your 'cq_1_10.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** CQ_1_10.1_labelMessages **
A pendulum requires 2 seconds to complete a cycle, which consists of a complete back-and-forth oscillation (extreme point to equilibrium to opposite extreme point
back to equilibrium and finally to the original extreme point). As long as the amplitude of the motion (the amplitude is the distance from the equilibrium position to
the extreme point) is small compared to the length of the pendulum, the time required for a cycle is independent of the amplitude.
How long does it take to get from one extreme point to the other, how long from an extreme point to equilibrium, and how long to go from extreme point to
equilibrium to opposite extreme point and back to equilibrium?
#$&*
answer/question/discussion: ->->->->->->->->->->->-> scussion:
It should take 1 second to get from one extreme point to the other.
It should take less than half a second to get from one extreme point to equilirium, because there is a positive acceleration due to gravity from extreme point to equilibrium. (Then there is a negative acceleration due to gravity from equilirium back up to extreme point). So we deduce that the average velocity for the first half of the path from extreme to extreme is higher than for the second half of the path. Therefore, the time interval will be shorter for the first half than the second half.
By this logic, the time from extreme to extreme to equilibrium should be less than 1.5 seconds, because it includes one fast half, one slow half, and then one more fast half.
What reasonable assumption did you make to arrive at your answers?
#$&*
@&
Your logic is very good. Your conclusion follows from your premise flawlessly.
However your premise itself is flawed. You assume that a positive acceleration results in a shorter time interval than a negative acceleration.
A complete argument would justify this assumption, and would be based on the definitions of velocity and acceleration.
A simple example shows that your assumption is flawed. If is easy to show that if an object starting from rest has a uniform positive acceleration for a certain time interval, then the time interval requred for a subsequent negative acceleration of the same magnitude, acting for the same time interval, will be the identical to the original time interval.
The situation here is not one of uniform acceleration, but of continuously changing acceleration, with the magnitude of the acceleration decreasing with proximity to the equilbrium point. We will need to analyze this more deeply when we study the effects of forces and the nature of the forces involved here. But as it turns out the magnitude of the acceleration of an ideal pendulum at any position depends only on its distance from its equilibrium point.
This symmetry dictates a conclusion identical to that of the previous argument, in which equal and opposite uniform accelerations ensured equal time intervals.
*@
*#&!*#&!
@&
Very good.
You made an excellent argument on that last question, though your premise doesn't hold up. Be sure to check my note.
*@