Stiffness constant of a spring

[upender singh]

1:question
A weight W is suspended from a rigid support by a hard spring with stiffness constant k . The spring is enclosed in a hard plastic sleeve, which prevents horizontal motion, but allows vertical oscillations. A simple pendulum of length l with a bob of
mass m (mg<<W) is suspended from the weight W and is set oscillating in a horizontal line with a small amplitude. After some time has passed, the weight W is observed to be oscillating up and down with a large amplitude (but not hitting the sleeve). It follows that the stiffness constant k ‬ must be:

1. k=4w/l
2. k=2w/l
3. k=w/l
4. k=w/2l

ans:1

2:relevant equations
w^2=k/m

3:attempt
Please hint me where to start. I tried balancing the forces but that brought me nowhere.

 

[lightgrav]

the pendulum mass has vertical acceleration component (v^2/r) as it swings.
The spring must pull the mass upward to provide that acceleration, so the mass pulls the spring downward.
If the timing of the pendulum pulls is appropriate, the spring's oscillations will increase in Amplitude (resonance).
(the pendulum's downward pull does positive Work to the spring if the spring is moving downward, meanwhile).

 

[upender singh]

thanks for the help lightgrav,

please reply if i am missing something

let Ω = oscillation frequency of spring mass system alone
let ω= oscillation frequency of the simple pendulum,
then mv^2/r =ml(Asinωt)^2 = mlA^2(1-cos2ωt)/2 , A being some constant
which says that driving force has a frequency= 2ω
equating both for resonance,
k*g/W = 4*g/l
k=4W/l

I assumed that mg won't contribute to the oscillations since mgcosθ≅mg for small θ.

 

[Rithishbarath]

@upender singh how did you get frequency =2w kindly post the steps .im trying for 4 days I am unable to so please post steps to get frquency of pendulum as 2 w

 

[wrobel]

looks like a problem from the perturbation theory and not like a home work for beginners

 

[haruspex]

@upender singh how did you get frequency =2w kindly post the steps .im trying for 4 days I am unable to so please post steps to get frquency of pendulum as 2 w

The pendulum exerts max tension on the spring twice each pendulum period, which must be once per spring period.

 

[wrobel]

The pendulum exerts max tension on the spring twice each pendulum period,

why do you think that the motion is periodic?

 

[Delta2]

why do you think that the motion is periodic?

It is given by the problem that the weight W oscillates up and down with a large amplitude (also it says that the spring is encased in a plastic sleeve which prevents horizontal motion or swinging). I suppose the pendulum swings left and right in a periodic way too, why wouldn't that be the case?

 

[wrobel]

You can not suppose anything after the system has been stated up. You must deduce all your assertions from the equations of motion.

 

[Delta2]

Ok i admit i cannot make the equation of motion of the pendulum, indeed it doesn't look so simple and its motion might not be periodic afterall.

 

[wrobel]

If you simulate this problem numerically you will see a chaotic behavior

 

[haruspex]

You can not suppose anything after the system has been stated up. You must deduce all your assertions from the equations of motion.

No, we are given that the mass oscillates, and I think it reasonable to assume they mean regularly. We have insufficient information to deduce that from the equations of motion.

 

[Delta2]

The chaotic behavior is kind of obvious if we assume that the weight W is free to swing left and right (together with the up-down motion) but are you sure we are getting chaotic behavior if we restrict W to move only up and down?

 

[wrobel]

Yes, roughly speaking, the chaotic behavior is a general case for Hamiltonian systems with $\ge 3/2$ degrees of freedom. A Hamiltonian system with complete set of the first integrals is a very exclusive case. I think that in this system the chaos can be proved with the help of the Poincare-Melnikov integral.
For details see http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf
or https://b-ok.cc/book/612281/eda574

 

[wrobel]

No, we are given that the mass oscillates, and I think it reasonable to assume they mean regularly. We have insufficient information to deduce that from the equations of motion.

By the existence and uniqueness theorem the equations of motion contain the complete information about the system.

 

[haruspex]

By the existence and uniqueness theorem the equations of motion contain the complete information about the system.

Yes, but we are not given the numbers, so the statement about steady oscillation is additional info. Or are you saying that no matter what the numbers such a system will always necessarily be chaotic, and therefore the question is unrealistic?

 

[wrobel]

Yes, but we are not given the numbers, so the statement about steady oscillation is additional info

The question is why does such an oscillation exist? To answer this question one must write down the equations of motion and substitute the hypothetical "steady oscillation" into these equations. Do that and make sure that the equations will not be satisfied.

 

[kuruman]

My interpretation is that this a variant of the Wilberforce pendulum. We have two coupled harmonic oscillators where the pendulum frequency is at resonance with the spring-mass frequency and mechanical energy is transferred back and forth from one to the other. See post #6 by @haruspex.

 

[wrobel]

$$L=\frac{m}{2}\Big(\dot y^2-2l\dot y\dot\varphi\sin\varphi+l^2\dot \varphi^2\Big)+\frac{M}{2}\dot y^2+mg(y+l\cos\varphi)+Mgy-\frac{ky^2}{2}$$
go ahead with considering this system as a linear one

 

[haruspex]

View attachment 264523$$L=\frac{m}{2}\Big(\dot y^2-2l\dot y\dot\varphi\sin\varphi+l^2\dot \varphi^2\Big)+\frac{M}{2}\dot y^2+mg(y+l\cos\varphi)+Mgy-\frac{ky^2}{2}$$
go ahead with considering this system as a linear one

I'm still not completely clear on what you are saying. Are you saying that a nonlinear system with two degrees of freedom is necessarily chaotic regardless of the details?

 

[wrobel]

I'm still not completely clear on what you are saying.

see #17

Are you saying that a nonlinear system with two degrees of freedom is necessarily chaotic regardless of the details?

No. This system is chaotic but it is just for infirmation.

 

[kuruman]

This reference treats the more general problem in which the spring is not constrained to move only vertically. A bit more than past the halfway mark, in section "Another type of motion?", the author summarizes the motion described by OP albeit in more detail:

• If the bob is started with almost entirely vertical oscillations, gradually the vertical oscillations subside and a swinging motion occurs.
• Swinging motion subsides and is replaced by a springing motion as before, and the process repeats.
• The motion appears planar (but is really elliptical).
• The “swing plane” rotates each time we return to swinging motion.

It is interesting to note that the spring constant chosen by the author for this study is equal to one of the four choices in this question. Farther below this section the author also discusses energy-dependent chaotic behavior in this system.

It is not clear to me what effects (if any) of constraining the spring to move only vertically would be on the motion of the pendulum bob. Perhaps the swing plane is prevented from rotating. Nevertheless, it seems to me that the motion as described by the OP is possible and that it is not necessarily chaotic.

 

[wrobel]

the author discusses the motion described by OP

what is it motion described by OP? please write the formulas
$y(t)=\ldots,\quad \varphi(t)=\ldots$ (for $y$ and $\varphi$ see the picture at post #19)

 

[kuruman]

The motion described by the OP is qualitative. I am not disputing the equations you posted in #19.

A simple pendulum of length l with a bob of mass m (mg<<W) is suspended from the weight W and is set oscillating in a horizontal line with a small amplitude. After some time has passed, the weight W is observed to be oscillating up and down with a large amplitude (but not hitting the sleeve).

 

[wrobel]

The motion described by the OP is qualitative.

Can this motion be expressed by mathematical formulas?

 

[kuruman]

Can this motion be expressed by mathematical formulas?

Yes, the formulas are shown in the link I provided in post #22.

 

[wrobel]

Yes, the formulas are shown in the link I provided in post #22.

please give the page