Length of a belt around two pulleys
Length of a belt around two pulleys
IP 2
7 marks
A belt passes tightly around two pulleys with centres \(O_1\) and \(O_2\), \(6\) cm apart, and radii \(4\) cm and \(1\) cm. Each straight part of the belt, such as \(PQ\), touches both pulleys and is perpendicular to the radii \(O_1P\) and \(O_2Q\).
- By drawing a line through \(O_2\) parallel to \(PQ\), show that \(PQ=3\sqrt{3}\) cm and that \(\angle PO_1O_2=\dfrac{\pi}{3}\) radians.[3]
- Find the angles of the arcs of the belt in contact with each pulley.[2]
- Find the exact length of the belt.[2]
Solution:
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Answer:(a) \(PQ=\sqrt{6^2-3^2}=3\sqrt{3}\) cm; \(\cos\angle PO_1O_2=\dfrac{1}{2}\) (b) \(\dfrac{4\pi}{3}\) and \(\dfrac{2\pi}{3}\) (c) \((6\sqrt{3}+6\pi)\) cm
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