Predict/Calculate A vertical hollow tube is connected to a...

Predict/Calculate A vertical hollow tube is connected to a speaker, which vibrates vertically with simple harmonic motion (FIGURE $13-45$ ). The speaker operates with constant amplitude, $A$ but variable frequency, $f .$ A slender pencil is placed inside the tube. (a) At low frequencies the pencil stays in contact with the speaker at all times; at higher frequencies the pencil begins to rattle. Explain the reason for this behavior. (b) Find an expression for the frequency at which rattling begins.

Predict/Calculate A vertical hollow tube is connected to a
speaker, which vibrates vertically with simple harmonic motion
(FIGURE $13-45$ ). The speaker operates with constant amplitude, $A$
but variable frequency, $f .$ A slender pencil is placed inside the
tube. (a) At low frequencies the pencil stays in contact with the
speaker at all times; at higher frequencies the pencil begins to rattle. Explain the reason for this behavior. (b) Find an expression for
the frequency at which rattling begins.

Key Concepts

Inertial Effects and Loss of Contact

In oscillating systems, objects in contact with a vibrating base can lose contact when the acceleration of the base exceeds the acceleration due to gravity. This occurs because the object’s inertia prevents it from following the base's rapid acceleration changes perfectly, leading to temporary detachment. Understanding the interplay of inertial forces with gravitational forces is crucial in analyzing when and how an object might begin to 'rattle' or become airborne during vibration.

Critical Acceleration Condition

The detachment of an object from a vibrating surface is fundamentally determined by comparing the maximum acceleration of the surface with gravitational acceleration. When the vibratory acceleration, dictated by the amplitude and frequency of the oscillation, exceeds gravity, the normal force acting on the object becomes zero, prompting a transition from continuous contact to intermittent contact. This critical condition can be used to derive expressions for the onset of separation based on the parameters defining the harmonic motion.

Simple Harmonic Motion

This concept describes the type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. The motion is characterized by parameters such as amplitude, frequency, and acceleration, and its mathematical description involves sinusoidal functions. It forms the basis for understanding the behavior of systems where vibration and oscillation are present.

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