Question

an object moves in simple harmonic motion described by the equation d = 2 cos πt, where t is measured in seconds and d in inches. find the following. a. the maximum displacement b. the frequency c. the time required for one cycle

Explanation:

Step1: Recall maximum value of cosine function

The general form of a cosine - based simple harmonic motion is $d = A\cos(\omega t)$. For the cosine function $y=\cos(x)$, the maximum value of $\cos(x)$ is 1. Given $d = 2\cos(\pi t)$, when $\cos(\pi t)=1$, the maximum displacement occurs.

Step2: Calculate maximum displacement

Substitute $\cos(\pi t) = 1$ into the equation $d = 2\cos(\pi t)$. Then $d_{max}=2\times1 = 2$ inches.

Step3: Identify the angular frequency

For the equation $d = A\cos(\omega t)$, comparing $d = 2\cos(\pi t)$ with it, we have $\omega=\pi$.

Step4: Calculate the frequency

The formula for frequency $f$ in terms of angular frequency $\omega$ is $f=\frac{\omega}{2\pi}$. Substitute $\omega = \pi$ into the formula: $f=\frac{\pi}{2\pi}=\frac{1}{2}$ Hz.

Step5: Calculate the period

The period $T$ (time for one - cycle) is the reciprocal of the frequency. Since $f=\frac{1}{2}$ Hz, then $T=\frac{1}{f}=2$ seconds.

Answer:

a. 2 in.
b. $\frac{1}{2}$ Hz
c. 2 s