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# Ball Bearings

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Age 16 to 18

Challenge Level

The wheels of a car, or a bicycle, run smoothly because they are separated from the axle of the wheel by a ring of ball bearings as illustrated below. Of course, the wheel turns smoothly because the ball bearings fit exactly between the hub of the wheel and the axle with no room to move about except, of course, to rotate. It is this rotation that keeps the friction to a minimum, and so makes the wheel turn smoothly.

Suppose that $a$ is the radius of the axle, $b$ is the radius of each ball-bearing, and $c$ is the radius of the hub (see the figure).What are the ratios ${a\over b}$, ${b\over c}$ and ${c\over a}$ when there are exactly three ball-bearings? What are these ratios when there are exactly four ball-bearings? Try to explain why the number of ball bearings determines the ratio ${c\over a}$ exactly.
Can you find a formula for ${c\over a}$ in terms of $n$ when there are exactly $n$ ball-bearings?

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.