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A ball of mass m moves in a horizontal circle of radius r with speed v on the smooth inside surface of a cone (half-angle theta from the axis as shown). If N is the normal reaction from the cone on the ball, which relation is correct?

1. N = mg*cos(theta)
2. g*sin(theta) = (v²/r)*cos(theta)
3. N*sin(theta) - m*v²/r = 0
4. none of these

Correct answer: N*sin(theta) - m*v²/r = 0

Solution

For a ball on the inside of a cone moving in a horizontal circle, the horizontal component of N provides the centripetal force, so N*sin(theta) = m*v²/r, i.e. N*sin(theta) - m*v²/r = 0 (with theta measured appropriately).

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