A piston in a gasoline engine is in simple harmonic motion....

A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3600 rev/min. Taking the extremes of its position relative to its center point as $\pm 5.00 \mathrm{~cm},$ find the magnitudes of the (a) maximum velocity and (b) maximum acceleration of the piston.

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from an equilibrium position. In SHM, the acceleration always acts towards the equilibrium point and is proportional to the negative of the displacement, which leads to sinusoidal motions over time.

Angular Frequency

Angular frequency, denoted by ?, quantifies the rate of oscillation in radians per second in simple harmonic motion. It is related to the period and frequency of the motion, and in contexts involving rotations, it requires converting given rotation rates (e.g., revolutions per minute) into radians per second to use in SHM formulas.

Maximum Velocity

In simple harmonic motion, the maximum velocity occurs as the oscillating object passes through the equilibrium position. It is given by the product of the angular frequency and the amplitude, reflecting the relationship between the motion’s speed and its oscillatory parameters.

Maximum Acceleration

The maximum acceleration in simple harmonic motion occurs at the points of maximum displacement from the equilibrium position. This acceleration is calculated as the product of the square of the angular frequency and the amplitude, indicating that both the speed of oscillation and the extent of displacement affect this maximum value.

Unit Conversions

Accurate calculations in problems involving rotational motion and oscillations require converting units to a consistent system, such as changing revolutions per minute to radians per second and centimeters to meters. This ensures that the equations used in the analysis of simple harmonic motion produce correct results.

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