Spring Motion The position of a weight attached to a spring...

Spring Motion The position of a weight attached to a spring is $s(t)=-4 \cos 10 t$ inches after $t$ scconds. (a) Find the maximum height that the weight rises above the equilibrium position of $s(t)=0$ (b) What are the frequency and period? (c) When does the weight first reach its maximum height? (d) Calculate and interpret $s(1.466)$

Spring Motion The position of a weight attached to a spring is
$s(t)=-4 \cos 10 t$ inches after $t$ scconds.
(a) Find the maximum height that the weight rises above the equilibrium position of $s(t)=0$
(b) What are the frequency and period?
(c) When does the weight first reach its maximum height?
(d) Calculate and interpret $s(1.466)$

Key Concepts

Phase and Timing of Maximum Displacement

Phase refers to the initial angle or displacement at the start of the oscillatory motion, influencing when the maximum (or minimum) occurs within the cycle. For a cosine function, the maximum value is typically reached when the cosine argument is a multiple of 2?, allowing determination of the specific time when the maximum displacement is attained.

Period and Frequency

The period is the duration needed for one complete cycle of the motion and is calculated as 2? divided by the angular frequency. Frequency is the reciprocal of the period, representing the number of cycles per unit time. These concepts are fundamental in understanding the timing characteristics of oscillatory systems.

Function Evaluation and Interpretation

Evaluating the displacement function at a specific time gives the position of the oscillating object relative to the equilibrium. This process involves substituting the given time into the function and interpreting the result to understand the physical position of the object at that instance, such as whether it is above or below the equilibrium.

Amplitude

Amplitude is the maximum displacement from the equilibrium position. In the context of a cosine function modeling simple harmonic motion, the amplitude is the absolute value of the constant multiplying the cosine term, which represents how far the object moves from its central position during oscillations.

Simple Harmonic Motion

This is the study of oscillatory motion where an object moves back and forth through an equilibrium position under a restoring force that is proportional to its displacement. It is often represented using trigonometric functions like cosine or sine, which naturally model periodic behavior.

Angular Frequency

Angular frequency is the coefficient of time in the argument of the cosine or sine function in simple harmonic motion. It determines how quickly the oscillations occur and is used to calculate the period of the motion. A higher angular frequency results in a shorter period and more rapidly oscillating motion.

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