41.3: The Conical Pendulum
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A conical pendulum is also similar to a simple plane pendulum, except that the pendulum is constrained to move along the surface of a cone, so that the mass m moves in a horizontal circle of radius r, maintaining a constant angle θ from the vertical.
For a conical pendulum, we might ask: what speed v must the pendulum bob have in order to maintain an angle θ from the vertical? To solve this problem, let the pendulum have length L, and let the bob have mass m. A general approach to solving problems involving circular motion like this is to identify the force responsible for keeping the mass moving in a circle, then set that equal to the centripetal force mv2/r. In this case, the force keeping the mass moving in a circle is the horizontal component of the tension T, which is Tsinθ. Setting that equal to the centripetal force, we have
Tsinθ=mv2r.
The vertical component of the tension is
Tcosθ=mg
Dividing Eq. 41.3.1 by Eq. 41.3.2,
tanθ=v2gr
From geometry, the radius r of the circle is Lsinθ. Making this substitution, we have
tanθ=v2gLsinθ
Solving for the speed v, we finally get
v=√Lgsinθtanθ.