Solve Falling Chain (3.11) Length & Speed

[Radarithm]

Homework Statement

A chain with length ℓ is held stretched out on a frictionless horizontal table, with a length y₀ hanging down through a hole in the table. The chain is released. As a function of time, find the length that hangs down through the hole (don't bother with t after the chain loses contact with the table). Also, find the speed of the chain right when it loses contact with the table.

Homework Equations

$$F_y=m\ddot{y}$$
$$y_0\ge y\ge l$$

The Attempt at a Solution

Conceptually, more of the chain must be through the hole so that y-naught is greater than y (initially). To cause an acceleration to the left (or towards the hole; however you picture the scenario), the tension T must be equal to mg (correct?). What I'm having trouble with is setting up an appropriate differential equation; should I just do the old: $$\ddot{y}=-g$$ because this would (should) hold true if I claim that the tension T is equal to mg. This means that something fundamental is getting past me, I just don't know what. Also this doesn't have to do with the question, but is K&K harder than the book I already have (Morin)?

 

[BvU]

The m in $\ddot y$ is different from the m in $mg$. Make a drawing.

 

[Radarithm]

http://postimg.org/image/vprzi0y6z/
The FBDiagram on the left is for a piece of the chain that is present on top of the table, whereas the one on the right is for a piece of the chain that is falling through the hole in the middle.

 

[BvU]

frictionless, so T isn't needed.
My point is that mg is not constant because that m = y x m/L

 

[Radarithm]

frictionless, so T isn't needed.
My point is that mg is not constant because that m = y x m/L

Would you mind proving how m = y x m/L? Is that a cross product?

 

[BvU]

This is one of these totally artificial exercises that confuse bright students. Just imagine what is needed to make the chain take a 90 degree turn with 0 radius at the rim of the hole. Yuch.

Never mind. Questions about your FBDs: On the left I see d M g. What is that ? T is the accelerating tension I suppose.
If there is an FBD for a (the ?) piece on top of the table, why is there no FBD for a (the) piece hanging vertically ?

On the right I see rather nothing. T is gone ?

 

[BvU]

No. If the whole chain has length L, mass m, then a piece with length y has mass $y\ m / L$

 

[Radarithm]

This is probably incorrect (I'm sure it is) but here I go:
$$dm\frac{dv}{dt}=-dm_2g$$
$$dmdv=-dm_2gdt$$
edit: Nevermind the diff eqs, that was before I saw your replies.
Can you at least give me a hint as to how I can relate the length to all of this? I can't seem to set up a diff eq.

 

[Radarithm]

In the left FBD, I now know that:
$$gdm=\frac{ym}{dL}$$
Forgot to include the tension in the FBD on the right, and it is the one for the vertical piece.

 

[Radarithm]

Ok so because:
$$m\frac{d^2y}{dt^2}=\frac{ym}{dL}g-T$$
for the vertical piece of the chain, then:
$$T=m\left(\frac{d^2y}{dt^2}-\frac{gy}{dL}\right)$$

Correct?
I don't know how to relate the diff eqs for the vertical and horizontal pieces of chain in order to find L(t).

I do know that the limits of integration (for the integral that I have to calculate) are $y_0$ and $l$

 

[Radarithm]

Ok so if what I got for T is correct then:
$$m\frac{d^2y}{dt^2}-m\frac{gy}{dL}=m\frac{d^2x}{dt^2}$$
$$\frac{d^2y}{dt^2}-\frac{gy}{dL}=\frac{d^2x}{dt^2}$$

I guess the only problem I have now is with setting up the diff eq. If I am right, does the $dL$ have to be on the left side with everything else on the right? (there are multiple dt's; I don't suppose that this is problematic, assuming that I can group them up together if they are similar).

 

[BvU]

This is going at a posts per few minutes. I can't follow. What is d in dL and what is x ?
You can't write $gdm=\frac{ym}{dL}$ if you take a differential in the lhs, you also need a d in the numerator on the rhs; it can't end up in the denominator.

We know the chain remains stretched. So vertical piece, length y(t) means horizontal piece length L - y(t).

Vertical: F = y(t) m/L g - T
Horizontal: F = T

Someone else will have to take over; I have to go to a training.

 

[BvU]

You may need to go back to F = dp/dt = dm/dt v + m dv/dt if you write a differential equation for only one of the sections (because for a section dm/dt is not 0). Either that , or only write the differential equation for the entire chain.

 

[jackarms]

Well, the accelerations for x and y must be equal -- otherwise the chain would stretch out or it would create extra slack. How about something like this?
$$\frac{d^{2}y}{dt^{2}} = \frac{g \cdot \lambda y}{M}$$ Where $\lambda$ is the linear mass density -- $\frac{M}{L}$, which reduces to: $$\frac{d^{2}y}{dt^{2}} = \frac{gy}{L}$$

This may be totally off though -- I'll think more about it.