An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.
𝒮(t) = 5 cos 2t
What is the frequency?

Verified step by step guidance

1. Identify the general form of the position function for simple harmonic motion, which is usually written as \(s(t) = A \cos(\omega t)\), where \(A\) is the amplitude and \(\omega\) is the angular frequency in radians per second.
2. From the given function \(s(t) = 5 \cos 2t\), recognize that the angular frequency \(\omega\) is 2 radians per second.
3. Recall the relationship between angular frequency \(\omega\) and frequency \(f\): \(\omega = 2\pi f\).
4. Rearrange the formula to solve for frequency: \(f = \frac{\omega}{2\pi}\).
5. Substitute the value of \(\omega = 2\) into the formula to express the frequency as \(f = \frac{2}{2\pi}\), which simplifies the expression for frequency.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes oscillatory motion where an object moves back and forth around an equilibrium point in a sinusoidal pattern. The position function s(t) typically involves sine or cosine functions, representing displacement over time.

Angular Frequency (ω)

Angular frequency ω is the rate of change of the phase of the sinusoidal function in radians per second. In the function s(t) = A cos(ωt), ω determines how fast the oscillations occur and is related to the frequency by ω = 2πf.

Frequency and its Relation to Angular Frequency

Frequency (f) is the number of oscillations per second, measured in Hertz (Hz). It is calculated from angular frequency using the formula f = ω / (2π). Knowing ω allows you to find how many complete cycles occur each second.