Four supporting cable segments give an ideal four-to-one block tackle

An ideal block-and-tackle has one moving block carrying a load $W$. Exactly four straight cable segments change length by the same amount when the moving block rises; all other cable portions have constant length. The cable is massless, the pulleys are frictionless, and the free end is pulled downward by an applied force $P$. Let $y$ be the upward displacement of the moving block and let $x$ be the downward displacement of the free end. For compatible positive virtual displacements, the cable geometry gives a constant-length relation. 1. Derive $\delta x=4\,\delta y$ from the cable-length constraint, making the signs explicit. 2. Write the virtual work of $P$ and $W$ and derive the equilibrium relation between them. 3. For $W=8.00\ \mathrm{kN}$, compute the ideal effort $P$ and state how far the free end must move to raise the load $0.250\ \mathrm m$. 4. Check the result by equal cable tension and force equilibrium of the moving block. Explain why ideal pulley support reactions do no virtual work in the chosen system. Do not vary $x$ and $y$ independently; they are coupled by one cable.

Write cable length as constant terms plus four variable supporting lengths minus the downward free-end coordinate, then differentiate.