- #1
NTesla
- 264
- 27
- Homework Statement
- A chain of mass per unit length is hung from ceiling by both it's ends. At time t=0, one of it's end is allowed to fall freely. Find the force exerted at each moment of time, by the stationary portion of the chain to stop the elements of the falling portion of the chain, when the element of the falling portion of the chain comes to stop at the bottom of the stationary portion of the chain.
- Relevant Equations
- Force = ##\frac {dP}{dt}##
In case of a straight chain falling on a surface, the force required to stop an element of length ##\lambda dx## of the chain is = ##\frac{\lambda dxv}{dt}## = ##\lambda v^{2}##.
In case of U chain see fig. below, When the purely vertical portion(DE in pic below) falls a distance ##y##, the points D and B fall a distance ##\frac{y}{2}##. This means that the portion DE falls with velocity ##v## and portion DB falls with velocity ##\frac{v}{2}##. Also, different element of portion DB have different downward velocity.
I'm not sure what would be the velocity of the element of the chain, which is just before point B, and is about to come to stop at point B. Will it be ##\frac {v}{2}## or ##v## ? I'm sure the portion DB has downward velocity of ##\frac {v}{2}##, but it's also true that the velocity of different elements of portion DB is different.
In case of U chain see fig. below, When the purely vertical portion(DE in pic below) falls a distance ##y##, the points D and B fall a distance ##\frac{y}{2}##. This means that the portion DE falls with velocity ##v## and portion DB falls with velocity ##\frac{v}{2}##. Also, different element of portion DB have different downward velocity.
I'm not sure what would be the velocity of the element of the chain, which is just before point B, and is about to come to stop at point B. Will it be ##\frac {v}{2}## or ##v## ? I'm sure the portion DB has downward velocity of ##\frac {v}{2}##, but it's also true that the velocity of different elements of portion DB is different.