- #1

Rikudo

- 120

- 26

- Homework Statement:
- see attachment

- Relevant Equations:
- impulse-momentum, newton's 2nd law

In order to be able to solve the problem, I think I must find the equation of ##h## with respect to ##\dot h##.

Assuming that ##F## is the action-reaction force between the stone and the end of chain, then the Newton's equation

For the stone:$$-F-mg=m \ddot h$$ $$-\int (F+mg)dh = \frac 1 2 m(\dot h^2 - Vo^2)...(1)$$

For the chain in the air:

a. Equation for the momentum: $$p=\lambda h \dot h...(2)$$ b. equation for impulse-momentum: $$\frac {dh}{dh} \frac {dp}{dt} = F-\lambda gh$$ $$\int \dot h dp = \int Fdh - \lambda gh^2/2...(3)$$

Substituting the value of ##\int F dh## from (1) into (3) will give us $$\int \dot h dp = -mgh -\frac 1 2 m(\dot h^2 - Vo^2) - \lambda gh^2/2...(4)$$

taking the derivative of p, then substitute it into (4) :

$$\int \dot h (\lambda \dot h dh+ \lambda h d \dot h) = -mgh -\frac 1 2 m(\dot h^2 - Vo^2) - \lambda gh^2/2$$

After that, what should I do next?