Energy loss in simple harmonic motion causes the time period to shorten?

[aspodkfpo]

Homework Statement

Energy loss in simple harmonic motion causes time period to shorten?

https://www.asi.edu.au/wp-content/uploads/2016/10/ASOEsolns2012.pdf

Relevant Equations

https://www.asi.edu.au/wp-content/uploads/2016/10/ASOEsolns2012.pdf

https://www.asi.edu.au/wp-content/uploads/2016/10/ASOEsolns2012.pdf
Q11 D) Markers comments: Few students reached part (d) and very few of those who did realized that the amplitude does affect the time taken for each of Mordred’s bounces. i.e. the energy losses results in shorter periods.

Never seen this before. Is this true and if so why is it true? Is this also true for pendulums in general?

 

[jbriggs444]

Energy loss in simple harmonic motion causes time period to shorten?

*sigh* We have to dig down to page 8 in the posted PDF to find the part d in the problem you are asking about.

d) Note: Use the axes supplied on p. 3 of the answer book to draw your answers to this part. DO NOT use those on p. 2, which are for part (b). Sketch the height, velocity and acceleration of Mordred the bungee jumper vs. time, if half the energy were dissipated in each bounce of Mordred. Use separate axes for each sketch, the same scale for time as you did in part (b), and mark all important points on your sketches. Solution: See over two pages for examples of sketches which would receive full marks. As Mordred reaches a lower height each bounce he spends less time with the rope slack and accelerating under gravity alone in each successive bounce.

Can you find any discrepancy between the problem and your interpretation of it?

 

[aspodkfpo]

*sigh* We have to dig down to page 8 in the posted PDF to find the part d in the problem you are asking about.

Can you find any discrepancy between the problem and your interpretation of it?

Don't see why he spends less time with the rope slack? Can see that they're argument is basically lesser speed, lesser distance travelled, less time. Not too convinced about its validity, nor am I able to figure out by what % the time period decreases.

Does this mean that pendulums also follow this rule?

 

[jbriggs444]

Don't see why he spends less time with the rope slack?

Have you graphed his motion? Maybe augment that graph with a horizontal line at the height where the rope goes slack.

 

[aspodkfpo]

Have you graphed his motion? Maybe augment that graph with a horizontal line at the height where the rope goes slack.

My graph didn't have a decreased time period.

 

[jbriggs444]

My graph didn't have a decreased time period.

Then that is where we need to work. When you drew the portion of the graph above the "slack line", what shape did you assume for the humps? What shape should they be? Sine wave, circle, parabola, loop-the-loop?

 

[aspodkfpo]

Then that is where we need to work. When you drew the portion of the graph above the "slack line", what shape did you assume for the humps? What shape should they be? Sine wave, circle, parabola, loop-the-loop?

I think where my thought process went wrong was that I was thinking that without losing energy, a greater force acts on the person which causes it to decelerate faster resulting in a same period with and without the energy loss. Why is this thought line wrong?

 

[jbriggs444]

I think where my thought process went wrong was that I was thinking that without losing energy, a greater force acts on the person which causes it to decelerate faster resulting in a same period with and without the energy loss. Why is this thought line wrong?

Lower energy means lower velocity which means less deceleration required to get the same period. Lower energy means lower displacement which means less deceleration provided by the bungee cord. If you understand simple harmonic motion, you should understand that those two effects exactly cancel.

Now then, with the bungee cord slack, do both of those two principles still apply? Which one(s) and why or why not?

 

[aspodkfpo]

Lower energy means lower velocity which means less deceleration required to get the same period. Lower energy means lower displacement which means less deceleration provided by the bungee cord. If you understand simple harmonic motion, you should understand that those two effects exactly cancel.

Now then, with the bungee cord slack, do both of those two principles still apply? Which one(s) and why or why not?

I get what you mean, but I was thinking gravity in the second scenario does not act in the same way as the first since energy is lost. I.e. the speed gained from gravity in the second scenario is lost in such a way that the time period remains the same.

 

[jbriggs444]

I get what you mean, but I was thinking gravity in the second scenario does not act in the same way as the first since energy is lost. I.e. the speed gained from gravity in the second scenario is lost in such a way that the time period remains the same.

The energy loss should be partially in air resistance and partially in lack of ideal elasticity in the bungee cord -- it will resist stretching more strongly than it recoils from stretching.

If this were [linearly damped] harmonic motion and if memory serves, then the period would indeed be unchanged. That fact might be demonstrated a course on differential equations (second order, linear, homogenous). You solve the characteristic equation (a quadratic) and substitute the solution(s) into a decaying exponential multiplied by a sine wave.

But this is not [damped] harmonic motion at all. We know that we do not have harmonic motion. The force profile when the bungee cord is slack does not obey $\sum F=-kx$. It obeys $\sum F=k$.

 

[aspodkfpo]

The energy loss should be partially in air resistance and partially in lack of ideal elasticity in the bungee cord -- it will resist stretching more strongly than it recoils from stretching.

If this were [linearly damped] harmonic motion and if memory serves, then the period would indeed be unchanged. That fact might be demonstrated a course on differential equations (second order, linear, homogenous).

However, in the case at hand, we know that we do not have harmonic motion. The force profile when the bungee cord is slack does not obey $\sum F=-kx$. It obeys $\sum F=k$.

And in this question it appears that loss of energy from air resistance was taken as near negligible? While the period for the spring stays largely the same due to linear damping?

 

[jbriggs444]

And in this question it appears that loss of energy from air resistance was taken as near negligible? While the period for the spring stays largely the same due to linear damping?

I see you continuing to resist looking at the motion when the spring is slack. It ain't harmonic.

 

[aspodkfpo]

I see you continuing to resist looking at the motion when the spring is slack.

?When the spring is slack we just ignore it no? And just consider gravity?

 

[jbriggs444]

?When the spring is slack we just ignore it no? And just consider gravity?

Yes

 

[aspodkfpo]

Yes

So this holds does it not?

"And in this question it appears that loss of energy from air resistance was taken as near negligible? While the period for the spring stays largely the same due to linear damping?"

Referring to the period where there is spring force. Oops, that might have seemed ambiguous.

 

[jbriggs444]

So this holds does it not?

Yes, with qualifications.

If we were considering the portion of the trajectory where the spring is not slack then this fits the force profile for simple harmonic motion. The sum of gravity plus restoring force from the spring will be directly proportional to displacement from the equilibrium position. Yes, this means that this portion of the graph will match a sine wave with a fixed frequency. That period of that wave will be unchanged regardless of amplitude.

If we consider linear damping, the above continues to hold. This portion of the graph would be a sine wave multiplied by a decaying exponential. The period of the wave form will be unchanged regardless of amplitude.

But...

There is another portion of the graph that does not fit this pattern -- the portion when the bungee is slack. If you consider the period of the oscillations of the person on the bungee for a full cycle, including both slack and not-slack time, it will NOT match the period of the wave form for the non-slack-only harmonic motion considered above.

 

[hutchphd]

If this were [linearly damped] harmonic motion and if memory serves, then the period would indeed be unchanged

If we consider linear damping, the above continues to hold. This portion of the graph would be a sine wave multiplied by a decaying exponential. The period of the wave form will be unchanged regardless of amplitude.

The presence of viscous (velocity proportional or electrical resistance) damping increases the (zero crossing) period for a simple harmonic oscillator (decreases the frequency).
This would seem to make the solution supplied to part (d) to be not quite accurate. Perhaps one can show the net effect to be a shorter interval but that is not clear to me immediately.

.

 

[jbriggs444]

The presence of viscous (velocity proportional or electrical resistance) damping increases the (zero crossing) period for a simple harmonic oscillator (decreases the frequency).
This would seem to make the solution supplied to part (d) to be not quite accurate. Perhaps one can show the net effect to be a shorter interval but that is not clear to me immediately.

I may have been unclear. Yes, I agree that damping increases the period. However amplitude does not change it: A low amplitude portion of the (simple harmonic motion with linear damping) wave is self-similar to a high amplitude portion shifted by a period or three.

Let us shift back to the situation at hand: Bungee that goes slack...

If one imagines a bungee in a low gravity environment with a high tension bungee, the situation may become more stark. Indeed, it may be useful to consider a basketball bouncing in the middle of a basketball court or a steel ball bearing bouncing on a slightly concave steel plate. You have simple harmonic motion while the ball is in contact with the floor. And you have parabolic motion while the ball is aloft.

If you've ever listened to or watched a basketball in this situation, the increase in frequency is dramatic.

 

[hutchphd]

I was also a little worried that the official solutions didn't reflect this change with viscosity but on close examination I guess they do. Good.