Belt
A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
Problem
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| A length of thin wire ($L$ ) binds together two cylindrical welding rods, whose radii are $R$ and $r$ , by passing all the way around them both. Find $L$ in terms of $R$ and $r$. |
Getting Started
Where is the belt a tangent to the circles? Draw in radii at these points.
Student Solutions
Herbert of Sha Tin College, Hong Kong submitted the only correct solution to date which is given below. Can anyone give an alternative solution?
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| $\alpha = \sin^{-1}(R - r)/(R + r)$
$L_1 = R(\pi + 2\alpha)$
$L_4 = r(\pi - 2\alpha)$
$L_2 = L_3 = x$
$x^2 = (R + r)^2 - (R - r)^2$
$x^2 = 4Rr$
$x = 2\sqrt{Rr}$
$L_2 + L_3 = 4\sqrt{Rr}$ |
The total length $L$ is $4\sqrt{Rr} + \pi(R + r) + 2(R -r)\sin^{-1}(R - r)/(R + r).$
Teachers' Resources
Why do this problem?
It provides experience of geometrical thinking,applying only the geometry of right angles triangles and the formula for arc length.
Possible approach
Sugggest the learners draw a neat diagram and mark in everything they know and can deduce from the diagram.
Key question
How do we split the belt into sections for which the lengths can be calculated?