What is the Velocity of the Last Piece of a Chain Falling on a Quarter Circle?

[member 428835]

Homework Statement

Consider a circle with radius $r$ diagrammed as the unit circle, but take only the second quadrant. On this quarter of the circle lies a chain with mass per unit length $\rho$ (the length of the chain is $\pi r/2$). If $\theta$ is the angle made with the vertical axis at any point on the circle, determine the velocity $v$ of the last piece of chain that falls at any arbitrary point in $\theta$. Ignore friction. The chain starts at rest.

The Attempt at a Solution

I know using work/energy will make life easier here: $$\Delta V_g+\Delta T=0$$ looking for $\Delta V_g$ is the tough part (change in gravitational potential). my thoughts were to look at an infinitesimal piece of chain $\rho r d\theta$ and then try to figure out how the height changes as $\theta$ changes.

I think $\Delta H$, where $H$ is height of a piece of chain, is $cos\theta_1-cos\theta_2$. From here, modeling went sour. Hopefully someone can help me out!

Thanks!

 

[haruspex]

When at theta, how much of the chain is still in contact with the quadrant? Can you determine the y-coordinate of the mass centre of that part and of the other part?

 

[haruspex]

Having thought about this some more, I think the problem is extremely difficult to answer fully. I suspect you are supposed to assume that each part of the chain remains in contact with the arc until it falls below the bottom of the arc. But that would not happen. After descending through a certain angle (the solution of 4sin(θ)+2θ=π cos(θ), approx 0.48 radians) the trailing end of the chain would detach from the arc. Thereafter it becomes very complex, with an increasing length of chain describing some curve in mid air. The chain would become completely detached from the arc before the trailing end reaches the 90 degree mark.

 

[CWatters]

After descending through a certain angle (the solution of 4sin(θ)+2θ=π cos(θ), approx 0.48 radians) the trailing end of the chain would detach from the arc.

As I read it that is the θ they are asking for when they say

"last piece of chain that falls at any arbitrary point in θ"

 

[haruspex]

As I read it that is the θ they are asking for when they say

"last piece of chain that falls at any arbitrary point in θ"

I can't read it that way. They're asking for a velocity at an arbitrary theta, not a value of theta; namely, the velocity of the trailing end as it passes angle theta.

 

[CWatters]

I was hoping that by finding the value of theta that would allow the velocity to be calculated? Not sure how.

 

[haruspex]

I was hoping that by finding the value of theta that would allow the velocity to be calculated? Not sure how.

It's no problem to calculate the velocity for all theta up to that critical value. Thereafter it's pretty much impossible. I did wonder if there's a way to get the eventual horizontal component of momentum, but even that looks too hard.

 

[utkarshakash]

Just concentrate on the centre of mass of the chain. Try to find out velocity of CM of chain at an arbitrary theta. Since the mass is uniformly distributed it is obvious where the CM would lie.

 

[haruspex]

Just concentrate on the centre of mass of the chain. Try to find out velocity of CM of chain at an arbitrary theta. Since the mass is uniformly distributed it is obvious where the CM would lie.

The chain is changing shape. The velocity of the mass centre will not tell you the velocity of the trailing end of the chain, which is what the question is asking.
Up until the trailing end starts to peel away from the surface, every part of the chain will be moving at the same speed (not the same velocity) and that can be determined from work conservation.