A penny rides on top of a piston as it undergoes vertical...

A penny rides on top of a piston as it undergoes vertical simple harmonic motion with an amplitude of $4.0 \mathrm{cm} .$ If the frequency is low, the penny rides up and down without difficulty. If the frequency is steadily increased, there comes a point at which the penny leaves the surface. a. At what point in the cycle does the penny first lose contact with the piston? b. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

A penny rides on top of a piston as it undergoes vertical simple harmonic motion with an amplitude of $4.0 \mathrm{cm} .$ If the frequency is low, the penny rides up and down without difficulty. If the frequency is steadily increased, there comes a point at which the penny leaves the surface.
a. At what point in the cycle does the penny first lose contact with the piston?
b. What is the maximum frequency for which the penny just barely remains in place for the full cycle?

Key Concepts

Maximum Frequency Condition in Oscillatory Systems

In oscillatory systems like the piston, the maximum frequency at which an object can remain in contact occurs when the peak acceleration of the piston equals gravitational acceleration. By equating the maximum acceleration from SHM (which depends on the square of the angular frequency and the amplitude) to g, one can determine the threshold frequency beyond which the object (the penny) will lose contact during part of the cycle.

Gravitational Acceleration

Gravitational acceleration (g) is the constant acceleration (approximately 9.8 m/s² near Earth’s surface) that all objects experience due to gravity. It acts as a benchmark in problems involving oscillatory motion on a vertical plane, as the object can only remain in contact with an accelerating surface if the surface’s acceleration does not exceed g in the downward direction.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic oscillatory motion where the restoring force is proportional to the displacement but directed opposite to it. This leads to motions that can be described by sinusoidal functions, characterized by amplitude, angular frequency, and phase. SHM is central to understanding how the piston moves and how its acceleration varies during the cycle.

Normal Force and Loss of Contact

The normal force is the force exerted by a surface to support the weight of an object in contact with it. In scenarios where an object sits on a moving surface, the object will lose contact when the surface accelerates downward faster than gravity, causing the normal force to drop to zero. Recognizing when the normal force becomes zero is key to predicting the moment the penny first loses contact with the piston.

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