In Exercises 65–66, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in centimeters. In each exercise, find: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 20 cos π/4 t

Verified step by step guidance

1. Identify the given equation of simple harmonic motion: \(d = 20 \cos \left( \frac{\pi}{4} t \right)\), where \(d\) is displacement in centimeters and \(t\) is time in seconds.
2. To find the maximum displacement, recognize that the amplitude of the motion is the coefficient in front of the cosine function. So, the maximum displacement is the absolute value of this coefficient.
3. To find the frequency, recall that the general form of simple harmonic motion is \(d = A \cos(\omega t)\), where \(\omega\) is the angular frequency in radians per second. Identify \(\omega\) from the equation and use the relation \(f = \frac{\omega}{2\pi}\) to find the frequency in hertz (cycles per second).
4. To find the time required for one cycle (the period), use the formula \(T = \frac{1}{f}\), where \(T\) is the period in seconds and \(f\) is the frequency found in the previous step.
5. Summarize the results: maximum displacement is the amplitude, frequency is calculated from angular frequency, and period is the reciprocal of frequency.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes oscillatory motion where the restoring force is proportional to displacement and acts in the opposite direction. The displacement varies sinusoidally with time, typically modeled by equations like d = A cos(ωt), where A is amplitude and ω is angular frequency.

Amplitude and Maximum Displacement

Amplitude is the maximum displacement from the equilibrium position in SHM. It represents the peak value of the cosine or sine function in the motion equation, indicating how far the object moves from its central position.

Frequency and Period of Oscillation

Frequency is the number of complete oscillations per second, measured in hertz (Hz), and is related to angular frequency ω by f = ω/(2π). The period is the time for one full cycle, given by T = 1/f, representing how long the object takes to repeat its motion.