Conical rotor and pendulum

Forces acting on a person inside the rotor

An exciting toy from some amusement parks is the so-called rotor . It consists of a vertical cylinder of radius r that rotates around its own axis. A person leaning against the inner wall of the toy in rotational motion does not slip vertically even when the floor is removed, as the force of static friction balances the force of weight. Thus, the normal force is the centripetal resultant. According to the force diagram, we have:

Since v = ω.r , we have: m.ω 2 .r = N .

When the cylinder rotates with a minimum angular velocity, the person is on the verge of slipping and, under these conditions, the friction force has a maximum value:

Remembering that the friction force must balance the weight force, we have:

Let us now look at the conical pendulum . When a body suspended by a string describes a circular motion in a horizontal plane, with constant angular velocity (ω), we have a conical pendulum. The figure below represents the two forces (weight and traction) that act on the pendulum mass and also the center of the curve.

The vector sum of the weight force and the pulling force gives us the centripetal resultant. From the force triangle we get:

We have to:

This last expression gives us the angular velocity of the conical pendulum defined by the angle θ .

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