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A sector cut from a uniform disk of radius 12 cm and a uniform rod of the same mass bent into the shape of a circular arc (subtending the same angle as the sector) are placed facing each other with the same vertex, on opposite sides. If the centre of mass of the combination lies at the common centre (origin), what is the radius of the arc?

1. 8 cm
2. 9 cm
3. 12 cm
4. 18 cm

Correct answer: 8 cm

Solution

For equal masses on opposite sides, balance about the centre requires the sector COM distance (2R sin a)/(3a) to equal the arc COM distance (r sin a)/a; this gives r = 2R/3 = 2(12)/3 = 8 cm.

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