In the diagram, ASRTB represents a piece of string passing over a pulley of radius 10cm in a vertical plane. O is the centre of the pulley and AMB is a hori...

Answer Details

From the diagram, the pulley has centre O and radius \(r = 10\text{ cm}\). The string runs \(A \to S\) (tangent on the left), over the arc \(S \to R \to T\) at the top, then \(T \to B\) (tangent on the right). \(AMB\) is horizontal, tangent to the pulley at M, with \(\angle SAB = 90^\circ\) and \(\angle TBA = 60^\circ\).

(a)(i) Obtuse angle SOT

From A, the tangents \(AS\) and \(AM\) meet the radii at right angles, so in quadrilateral \(ASOM\):

\[\angle SOM = 360^\circ - 90^\circ - 90^\circ - \angle SAM = 360^\circ - 90^\circ - 90^\circ - 90^\circ = 90^\circ\]

From B, the tangents \(BT\) and \(BM\) give, in quadrilateral \(BTOM\):

\[\angle TOM = 360^\circ - 90^\circ - 90^\circ - \angle TBM = 360^\circ - 90^\circ - 90^\circ - 60^\circ = 120^\circ\]

Going from S round the bottom through M to T gives \(90^\circ + 120^\circ = 210^\circ\). The angle on the top (through R), the obtuse \(\angle SOT\), is

\[\angle SOT = 360^\circ - 210^\circ = 150^\circ\]

(a)(ii) Arc SRT

\[\text{arc }SRT = \frac{150}{360}\times 2\pi r = \frac{150}{360}\times 2\times\frac{22}{7}\times 10 = \frac{5}{12}\times\frac{440}{7}\]\[= \frac{2200}{84} = 26.19\text{ cm} \approx 26\text{ cm}\]

(a)(iii) |BT|

In right triangle \(OTB\), \(OT = 10\text{ cm}\), \(\angle OTB = 90^\circ\), and \(OB\) bisects \(\angle TBM\) so \(\angle TBO = 30^\circ\):

\[\tan 30^\circ = \frac{OT}{BT} \;\Rightarrow\; BT = \frac{10}{\tan 30^\circ} = 10\sqrt{3} = 17.32\text{ cm}\]

(b) Length of the string

First find \(AS\). In right triangle \(OSA\), \(OS = 10\text{ cm}\), \(\angle OSA = 90^\circ\), and \(OA\) bisects \(\angle SAM\) so \(\angle SAO = 45^\circ\):

\[\tan 45^\circ = \frac{OS}{AS} \;\Rightarrow\; AS = \frac{10}{\tan 45^\circ} = 10\text{ cm}\]

The string length is

\[AS + \text{arc }SRT + BT = 10 + 26.19 + 17.32 = 53.51\text{ cm}\]

Length of the string \(\approx 54\text{ cm}\).

Summary: \(\angle SOT = 150^\circ\), arc \(SRT \approx 26\text{ cm}\), \(|BT| = 10\sqrt{3}\approx 17.32\text{ cm}\), string \(\approx 54\text{ cm}\).