# Physics Question!!!

A block of mass $m$ is attached to a spring and kept in horizontal position in a vertical plane. When the block is allowed to then, find the maximum extension in the spring.

Details--

1) Spring's constant=$k$

2) Natural length of spring= $l$

img

I was trying to make a problem using some concepts of Work Energy and Circular motion and I came up with this. Here is my approach, please tell me if I am right or not.

First of all, the extension in the spring will be maximum when radial velocity ($v_{r}$) will be zero. Now the work is only done by conservative forces in this situation so we can conserve energy.

$mg(l+x)cos \theta=\frac{mV_{0}^{2}}{2}+\frac{kx^{2}}{2}$

Now $V_{0}^{2}=V_{t}^{2}+V_{r}^{2}$

The extension will be maximum when $V_{r}=0$. So this does not implies that the velocity of the block is zero at the lower most position. Also we can see that only a component of $mg$ is acting in the tangential direction which is responsible for tangential velocity and tangential acceleration. So $V_{t}$ is not going to be zero at the lower most position.

If we consider torque about point $O$ then

$mgsin\theta (l+x)=\frac { dL }{ dt }$

$mgsin\theta (l+x)=\frac { dI }{ dt } \omega +I\frac { d\omega }{ dt }$

$I=m{ (l+x) }^{ 2 }$

$\frac { dI }{ dt } =m(2x{ v }_{ r }+2l{ v }_{ r })$

$mgsin\theta (l+x)=2m(x+l){ v }_{ r }\omega +I\frac { d\omega }{ dt }$

I got this equation. There is a term of $v_{r}$ in the equation that has to related with $\omega$ or $\theta$ to solve the problem.

Although the motion is not circular but if consider the motion of the block for a very small interval of time then we can assume that it's a circular motion so $v_{t}=(l+x) \omega$. Can we do so?

Using this we can express $V_{r}$ in term of angular velocity. But still there is term containing $x$ there. How can this be solved?

Am I overlooking some thing very obvious? Because using calculus in spring problem often gets very complicated.

Note by Satvik Pandey
4 years, 10 months ago

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There is a good video on youtube comparing Newtonian and Lagrangian mechanics on the example of the elastic pendulum, which the example here is.

- 1 year, 3 months ago

Try to use lagrangian mechanics. Find the total kinetic energy and take the total potential energy and apply the E-L equations. There are too many forces to equate. A spring pendulum is a hard mechanical system to crack.

- 1 year, 3 months ago

I think you cannot assume that the motion is circular, however small the interval you take. I for one think that maximum extension will occur at bottom-most position.

- 4 years, 9 months ago

- 4 years, 9 months ago

Thanks for the link! This proves that the extension is maximum at the bottom most position. :D

But what about the calculation and explanation??

- 4 years, 9 months ago

Is that simulation removed.

- 4 years, 9 months ago

You have to scroll down page to see it.

- 4 years, 9 months ago

Even if the motion is not completely circular then I think we can use $V=R \omega$. Like we did here. Can we ??

I too thought that the extension will be maximum at the bottom most position. But when I thought about that after watching Kushal comment, deeply then I don't see a way to justify that. :(

- 4 years, 9 months ago

- 4 years, 9 months ago

The total velocity is sum of the squares of radial and tangential velocities, just like x and y components in Cartesian co-ordinates.

Then use $V_{t}=R \omega$. :)

- 4 years, 9 months ago

What is $V_{\phi}$ ? What is it responsible for ?

- 4 years, 9 months ago

It is radial velocity. It is responsible for the radial movement of the particle.

- 4 years, 9 months ago

Thank you.

- 4 years, 10 months ago

You are welcome!!! :)

- 4 years, 10 months ago

In the sketch both $V_r$ look the same.

- 4 years, 10 months ago

My handwriting is not very good. Velocity of the block along the spring is $v_{r}$ and velocity of the block tangential to the spring is $v_{t}$.

- 4 years, 10 months ago

There are two $V_r$s ? Why should radial velocity be zero in all positions ?

- 4 years, 10 months ago

$V_{r}$ and $v_{r}$ both represents radial velocity. In order to achieve maximum extension in the spring the radial velocity of the block should be zero at a particular point. It is not zero through out the motion.

- 4 years, 10 months ago

We can use $g sin \theta =da_{t}/dt$ and substitute $a_{t}=(l+x) \alpha$ by approximation but still we get the same situation.

- 4 years, 10 months ago

I think that should be $\frac{dv_r}{dt} = gcos\theta-kx/m$

And $\frac{dv_t}{dt} = gsin \theta$

You should go with energy conservation , thats easier

- 4 years, 9 months ago

I wrote $\frac { { da }_{ t } }{ dt }$ by mistake.

I tried to use energy conservation. But here entire gravitational energy in not getting converted to spring potential energy only. A part of it is also converted to kinetic energy. I think the extension will be maximum when the radial velocity would be zero.right? But this does not implies that tangential velocity would also be zero at that moment.

Let $x$ be tha maximum extension in the spring then-

$mg(l+x)=\frac{mv_{t}^{2}+kx^{2}}{2}$. There are two variables $v_{t} and x$. And we have only one equation.

What do you think? @Kushal Patankar

- 4 years, 9 months ago

Cetripetal accleration equation will give the relation between $v_t$ and x %6

- 4 years, 9 months ago

I have one more confusion.

The motion of the block is not a pure circular motion which increases the intricacy of the problem.

I think we can not just equate $kx$ and $mgcos \theta$ to centripetal force. There is acceleration other that centripetal acceleration in radial direction because the block is performing radial motion also.

I think the equation should be

$mgcos\theta -kx=m\left\{ \frac { d\vec { { V }_{ r } } }{ dt } -\frac { { V }_{ t }^{ 2 } }{ (l+x) } \right\}$

Taking radially outward as +ve direction.

What do you think, Kushal??

- 4 years, 9 months ago

What is the need for that equation , I mean that mechanical energy conservation does all our work. And I think that the equation u made is correct for intermediate radial acceleration.

- 4 years, 9 months ago

How you got the second equation??

The radial velocity is zero at the lower most position but this doesn't implies that radial acceleration $\frac { \vec { { dV }_{ r } } }{ dt }$ is also zero at that point. Am I missing some thing very obvious? :(

- 4 years, 9 months ago

How could I miss that!!!!!!!!!!!

Thank you Kushal for helping.

- 4 years, 9 months ago

- 4 years, 10 months ago

@satvik pandey i suppose whatever you have said till now is correct..!!
It is a case of Coriolis force.. When the particle has radial speed as well then the acceleration is written as

Therefore we cannot write the equation of centripetal force as discussed before..!!

- 4 years, 8 months ago

Could you post whole analysis(solution) of the problem (along with calculation).

- 4 years, 8 months ago

Thank you ! Sir. Sorry for delay in replying. You got equation using polar co-ordinates. right?

- 4 years, 8 months ago

Yes Carioles component comes in to when the sliding part is curved. Can this problem be transformed in a such a mechanism ?

- 4 years, 8 months ago