A pulley P is attached to the ceiling at O by a piece of metal that can swing (b) 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 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 (b) 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 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![A pulley P is attached to the ceiling at O by a piece of metal that can swing
(b)
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
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+ vr2 – x² – v1–2x+r² .
(1? -x²)- x²(1–2x + r²)
vt -x?VI-2x+r°{NP=x²+ xvI-2r+r*)
dl
(ii) Show that
dx
(iii) You are given the factorisation
(* -x*)-x'(1-2x +r*)=(x – 1)(2x² -²x -r*).
(Do NOT prove this.)
Find the value of x for which M is closest to the floor. Justify your
answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7586eeb-dfd5-4c6c-a6a2-6f2bc67010eb%2F59aa7835-746f-47b0-8069-265dc458f06f%2Fmcbzf2m.png&w=3840&q=75)
Transcribed Image Text:A pulley P is attached to the ceiling at O by a piece of metal that can swing
(b)
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
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+ vr2 – x² – v1–2x+r² .
(1? -x²)- x²(1–2x + r²)
vt -x?VI-2x+r°{NP=x²+ xvI-2r+r*)
dl
(ii) Show that
dx
(iii) You are given the factorisation
(* -x*)-x'(1-2x +r*)=(x – 1)(2x² -²x -r*).
(Do NOT prove this.)
Find the value of x for which M is closest to the floor. Justify your
answer.
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