Chain sliding off a table, find the function a(t)

[packers7]

Homework Statement

A chain lies on a frictionless table at rest, half off the edge, and half on. As soon as it is let go, it begins accelerating due to gravity only. Determine the acceleration of the chain as a function of time. The mass is m, gravity is g, and the length of the chain is L.

Homework Equations

F=ma
K=(1/2)mvv
Ug=mgh
W=integral(F*dr)
a=dv/dt , v=dd/dt

The Attempt at a Solution

Let x=the length of chain hanging vertically. Therefore x starts at L/2, and ends at L, the min and max accelerations are g/2 and g.
I figured out that a=x/L*g
I then tried to use energy, with my zero line level with the table, so I was dealing with negative values. I would get mgL/8 =-xxm/(2L)+mvv/2. Solving for v gives a messy square root, and I don't see how time fits into the equation.

I know that the function will be piecewise because once the acceleration reaches g, it will remain there. I just need to find the function as it grows from g/2 to g.

 

[tiny-tim]

Welcome to PF!

I figured out that a=x/L*g

Hi packers7! Welcome to PF!

Hint: x/L*g = a = dv/dt = dv/dx times … ?

 

[packers7]

oops, my energy equation is wrong, the first term, to the left of the = should be negative. And the first term to the right of the = should also have *g.

dv/dt=dv/dx*dx/dt
So a=dv/dx*dx/dt.
Taking the differential of my energy equation, simplified, yields dv/dx=x/(L*v). Should I substitute v with the energy equation, solved for v (sqrt of stuff)?
I tried using a=dv/dx*dx/dt. Which goes to x*g/L=x/(L*v)*dx/dt. That leads to dx/dt=v*g
Am I on the right track?

 

[tiny-tim]

I tried using a=dv/dx*dx/dt. Which goes to x*g/L=x/(L*v)*dx/dt. That leads to dx/dt=v*g

What happened to dv?

Start again:

you can do it two ways …

i] use Newton's second law F = ma and a = dv/dx to get a differential equation for v and x

ii] just use KE + PE = constant, to get an ordinary equation for v and x.

They come to the same thing in the end, but try both of them, just for practice.

 

[packers7]

sorry, I don't think I'm making any progress. I can't figure out when t comes into the answer.

I tried using F=m*dv/dx
m*a=m*dv/dx
a=dv/dx
x*g/L*dx=dv
integrate
x*x*g/(2*L)+C=v
C=-L*g/8
I can't use v=distance/time since that's avg velocity.

With energy, I tried
K+Ug=constant
.5*m*v*v+(x/L)*m*g*(-x/2)=-m/2*g*(-L/4)
.5*v*v-x*x*g/(2*L)=-g*L/8
Do I differentiate here?
v*dv-x*g/L*dx=0
v*dv=x*g/L*dx
dv/dx=x*g/(L*v)
I feel like I'm at a dead end

doesn't a=dv/dt, not dv/dx? x is not time, I made it a length that changes

 

[tiny-tim]

I tried using F=m*dv/dx
m*a=m*dv/dx
a=dv/dx
x*g/L*dx=dv

That's right, except a isn't dv/dx, is it?

With energy, I tried
K+Ug=constant
.5*m*v*v+(x/L)*m*g*(-x/2)=-m/2*g*(-L/4)
.5*v*v-x*x*g/(2*L)=-g*L/8
Do I differentiate here?

No! v = dx/dt … so you integrate!

 

[packers7]

then I'd get: .5[(dx/dt)^2]=x*x*g/(2*L)-g*L/8
(dx/dt)^2=x*x*g/L-g*L/4
(dx/dt)^2=(4*x*x*g-g*L*L)/(4*L)
move x so it's with dx
[(dx)^2]/(4*x*x*g-g*L*L)=4*L*[(dt)^2]
integrate
(dx)(-[tanh^-1[2*x/L]]/(2*g*L)=(dt)4*L*t
integrate again
-(x*[tanh^-1[2*x/L]]+(L/4)ln(L*L-4*x*x))/(2*g*L)=L*t*t/2
?
this seems like it's too complicated to be right...

 

[tiny-tim]

(have a square-root: √ and a squared: ² smile:)

Why are you integrating twice?

And where did tanh^-1 come from?

v = dx/dt = a√(x² - b²) …

 

[packers7]

So if I take the square root of (dx/dt)^2=(4*x*x*g-g*L*L)/(4*L)
I get dx/dt=√[(4x²g-gL²)/(4L)]
dx*√[4L/(4x²g-gL²)]=dt , integrate
ln(2gx+g√[4x²-L²])/√[g] + C = t
test x=L/2, t=0 yields C=-ln(gL)/√[g]
solve for x
x=L(1+e^(2t√[g]))/(4*e^(t√[g]))
plug into a=xg/L
a=g/(4*e^(t√[g]))+g*e^(t√[g])/4
is that correct? i tested t=0, and a=g/2, which is right!
a=g/2*cosh(t√[g])
therefore at t=ln(2+√[3])/√[g], a=g ?

 

[tiny-tim]

Hi packers7!

So if I take the square root of (dx/dt)^2=(4*x*x*g-g*L*L)/(4*L)
I get dx/dt=√[(4x²g-gL²)/(4L)]
dx*√[4L/(4x²g-gL²)]=dt , integrate
ln(2gx+g√[4x²-L²])/√[g] + C = t

No … where do you get ln from?

Hint: to make the integration easier, rewrite it as

(√(L/g))dx/√(x² - L²/4) = dt

 

[packers7]

I'm plugging in the equation into http://integrals.wolfram.com/index.jsp
using what you have, Sqrt[L/g]*Sqrt[1/(x*x-L*L/4)]
integrates into √[L/g]*ln(2x+√[4x²-L²]) (+C)
i checked by taking the derivative of that, and it works.
am I making a mathematical mistake?

 

[tiny-tim]

I'm plugging in the equation into http://integrals.wolfram.com/index.jsp
using what you have, Sqrt[L/g]*Sqrt[1/(x*x-L*L/4)]
integrates into √[L/g]*ln(2x+√[4x²-L²]) (+C)
i checked by taking the derivative of that, and it works.
am I making a mathematical mistake?

ok … bizarrely, that is right …

but that gives you t as a function of x, while the question asks for x as a function of t …

how are you going to "invert" something so complicated?

Hint: to solve dx/√(x² - b²) the easy way, make a substitution!

 

[packers7]

my logic is that if i solve that integrated equation for x, then i can plug that in for a=xg/L, getting an equation that gives acceleration as a function of time
is that what you're trying to tell me to do? x gives the amount of chain hanging vertically, not the acceleration.

 

[tiny-tim]

my logic is that if i solve that integrated equation for x, then i can plug that in for a=xg/L, getting an equation that gives acceleration as a function of time …

ok … show me!

 

[packers7]

i've already done it in an earlier post...

√[L/g]*ln(2x+√[4x²-L²]) (+C)=t
C=-√[L/g]*ln(L) (by using x=L/2 and t=0)
then solving for x yields x=(L²+L²*e^(2t√[g/L]))/(4L*e^(t√[g/L])
x=L*(1+e^(2t√[g/L]))/(4e^(t√[g/L])), which equals (L/2)*cosh(t√[g/L])

a=xg/L
substitute x=(L/2)*cosh(t√[g/L])
a=(g/2)*cosh(t√[g/L])

(at t=0, a=g/2 and if a=g, t=√[L/g]*ln(2+√[3]))