A pulley P is attached to the ceiling at O by a piece of metal that can swing freely. One end of a rope is attached to the ceiling at A. The rope is passed through the pulley P and a weight is attached to the other end of the rope at M, as shown in the diagram. M The distance OA is 1 m, the length of the rope is 2 m, and the length of the piece of metal OP=r metres, where 0
A pulley P is attached to the ceiling at O by a piece of metal that can swing freely. One end of a rope is attached to the ceiling at A. The rope is passed through the pulley P and a weight is attached to the other end of the rope at M, as shown in the diagram. M The distance OA is 1 m, the length of the rope is 2 m, and the length of the piece of metal OP=r metres, where 0
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Transcribed Image Text:A pulley P is attached to the ceiling at O by a piece of metal that can swing
freely. One end of a rope is attached to the ceiling at A. The rope is passed
through the pulley P and a weight is attached to the other end of the rope at M,
as shown in the diagram.
A
M
The distance OA is 1 m, the length of the rope is 2 m, and the length of the piece
of metal OP=r metres, where 0<r<1. Let X be the point where the line MP
produced meets OA. Let OX=x metres and XM = l metres.
(i) By considering triangles OXP and AXP, show that
l=2+ v? -x² - VI-2x+r² .
-x² - V1-2x+p² .
de
(ii) Show that
dx
(P-x²)-x{(1–2x+r*)
2-x² +xv1-2x+r²
(iii)
You are given the factorisation
(1² – x²)– x²(1–2x + r²)=(x-1)(2x² – r²x – r²).
(Do NOT prove this.)
Find the value of x for which M is closest to the floor. Justify your
answer.
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