Question

The position of an object moving with simple harmonic motion is given by $x=4\cos \left(6\pi t\right)$, where x is in meters and t is in seconds. What is the period of the oscillating system? (a)$4s$ (b) $\frac{1}{6}s$(c)$\frac{1}{3}s$ (d)$6\pi \text{s}$ (e) impossible to determine from the information given.

Short answer

Option (c) is correct. The period of the oscillating system is$T=\frac{1}{3}\text{s}$ .

Step-by-step solution

The simple harmonic motion

When an object undergoes simple harmonic motion, the position as a function of time may be written as

$x=A\cos \left(\omega t\right)$

$A=$Amplitude

$\omega =$Angular frequency

$x=$Position of object

Find the period of the oscillating system

When an object undergoes simple harmonic motion, the position as a function of time may be written as $x=4\cos \left(6\pi t\right)$. Comparing this to the given relation, we see that the frequency of vibration is$f=3\text{Hz}$ , and the period is:

role="math" localid="1663742061871" $$\begin{gathered} T=\frac{1}{f} \\ T=\frac{1}{3\text{Hz}} \\ T=\frac{1}{3}\text{s} \end{gathered}$$

Hence option (c) is the correct answer for this question.