Question

an object moves in simple harmonic motion described by the equation d = - 6 cos(π/3)t, 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 max - displacement formula for $y = A\cos(\omega t)$

For a simple - harmonic motion equation of the form $d = A\cos(\omega t)$, the maximum displacement is given by $|A|$. In the equation $d=-6\cos(\frac{\pi}{3}t)$, $A = - 6$.

Step2: Calculate the maximum displacement

The maximum displacement is $| - 6|=6$ inches.

Step3: Recall the formula for angular frequency and frequency

The angular frequency $\omega=\frac{\pi}{3}$, and the relationship between angular frequency $\omega$ and frequency $f$ is $\omega = 2\pi f$.

Step4: Solve for the frequency

We have $\frac{\pi}{3}=2\pi f$. Solving for $f$, we get $f=\frac{1}{6}$ cycles per second.

Step5: Recall the formula for the period

The period $T$ (time for one cycle) is the reciprocal of the frequency, and also $T=\frac{2\pi}{\omega}$. Since $\omega=\frac{\pi}{3}$, then $T = \frac{2\pi}{\frac{\pi}{3}}$.

Step6: Calculate the period

$T=\frac{2\pi}{\frac{\pi}{3}}=6$ seconds.

Answer:

a. 6 in.
b. $\frac{1}{6}$ cycles per second
c. 6 s