quickQuiz #4 (solution)
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22 Nov 2007, 23:30
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Journals
For those who were interested yesterday about the solution to this cute little problem: http://crossfire.nu/?x=journal&mode=item&id=42481
M*x/L = the mass of the chain that already is on the scale
The small mass dM (can treat that as a delta) that is currently just touching the scale, has fallen for a length x (free fall), thus the work done by gravity is dM * g * x
Equate that for it's kinetic energy to get the speed:
dM * v^2 / 2 = dM * g * x
v^2 = 2 * g * x
v = sqrt (2 * g * x)
The duration dt (delta t) the small mass dM takes until it is at rest:
dt = dx / v = (dM * L / M ) / v = dM * L / (M * v)
Thus the force acting on the scale due to the falling dM:
F = dp / dt (impulse transfer over time period)
F = dM * v / dt = dM *v / (dM * L / (M*v)) = M v^2 / L = 2 M g x / L
Adding this force to the gravitational force of the piece of chain already on the scale:
Ftotal = g M x / L + 2 M g x / L = 3Mgx/L
That is, 3 times more than if piece of length x of the chain was stationary on the chain. Cool, huh?
M*x/L = the mass of the chain that already is on the scale
The small mass dM (can treat that as a delta) that is currently just touching the scale, has fallen for a length x (free fall), thus the work done by gravity is dM * g * x
Equate that for it's kinetic energy to get the speed:
dM * v^2 / 2 = dM * g * x
v^2 = 2 * g * x
v = sqrt (2 * g * x)
The duration dt (delta t) the small mass dM takes until it is at rest:
dt = dx / v = (dM * L / M ) / v = dM * L / (M * v)
Thus the force acting on the scale due to the falling dM:
F = dp / dt (impulse transfer over time period)
F = dM * v / dt = dM *v / (dM * L / (M*v)) = M v^2 / L = 2 M g x / L
Adding this force to the gravitational force of the piece of chain already on the scale:
Ftotal = g M x / L + 2 M g x / L = 3Mgx/L
That is, 3 times more than if piece of length x of the chain was stationary on the chain. Cool, huh?
on the contrary.. it's an answer ;)
Unless you were already wondering about it, in this case, you'd go to sleep as a happier person :)
If none of the two fit.. then... i guess nothing would happen :(
That means I'll have to find simpler ones :|
Where are you, rize? Your kombinatorik was nice :)