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 period of this motion?

Verified step by step guidance

1. Identify the general form of the position function for simple harmonic motion, which is usually given by \(s(t) = A \cos(\omega t)\) or \(s(t) = A \sin(\omega t)\), where \(\omega\) 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.
3. Recall the formula that relates the period \(T\) of the motion to the angular frequency: \(T = \frac{2\pi}{\omega}\).
4. Substitute the value of \(\omega = 2\) into the formula to express the period as \(T = \frac{2\pi}{2}\).
5. Simplify the expression to find the period \(T\) in seconds, which represents the time it takes for one complete cycle of the motion.

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 position function is typically sinusoidal, such as s(t) = A cos(ωt) or s(t) = A sin(ωt), where A is amplitude and ω is angular frequency.

Angular Frequency (ω)

Angular frequency ω represents how quickly the object oscillates in radians per second. It is the coefficient of t inside the cosine or sine function in SHM equations. The angular frequency relates to the period by the formula ω = 2π / T, where T is the period.

Period of Oscillation (T)

The period T is the time it takes for one complete cycle of motion. It is inversely related to angular frequency by T = 2π / ω. Knowing ω from the position function allows calculation of T, which answers how long one full oscillation lasts.