An instrument package A is bolted to a shaker table as...

An instrument package $A$ is bolted to a shaker table as shown. The table moves vertically in simple harmonic motion at the same frequency as the variable-speed motor that drives it. The package is to be tested at a peak acceleration of $150 \mathrm{ft} / \mathrm{s}^{2} .$ Knowing that the amplitude of the shaker table is $2.3 \mathrm{in.}$, determine (a) the required speed of the motor in rpm, $(b)$ the maximum velocity of the table.

Key Concepts

Simple Harmonic Motion (SHM)

SHM describes a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. This concept is fundamental in analyzing systems such as vibrating tables or oscillating masses. The motion is typically represented by sine or cosine functions, which help determine properties like displacement, velocity, and acceleration over time.

Relationship between Acceleration, Velocity, and Amplitude in SHM

In SHM, the maximum acceleration is given by the expression a_max = ?²A, where ? is the angular frequency and A is the amplitude. Similarly, the maximum velocity is given by v_max = ?A. These relationships are essential for connecting the desired acceleration or velocity to the system's operating parameters and are commonly used in designing and testing oscillatory systems.

Angular Frequency and Frequency Conversion

Angular frequency, denoted by ?, is crucial in describing how quickly an oscillatory system moves through its cycle and is measured in radians per second. It is directly related to the regular frequency (f) by the formula ? = 2?f. Converting between these measures, and subsequently to revolutions per minute (rpm) by using the relation rpm = f × 60, is necessary for practical applications, especially when interfacing with mechanical motors and instrumentation.

Unit Conversion

Accurate conversion of measurement units, such as inches to feet or seconds to minutes, is vital when applying formulas that involve amplitude, acceleration, and velocity. Ensuring consistency in units allows for correct application of the mathematical relationships in SHM and avoids calculation errors, especially in real-world engineering problems.

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