Question

an object moves in simple harmonic motion described by the equation d = 9 cos(π/2)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 a. in. b. cycles per second c. s

Explanation:

Step1: Recall max - displacement formula

For $d = A\cos(\omega t)$, the maximum displacement is $|A|$. Here $A = 9$, so the maximum displacement is 9 inches.

Step2: Recall angular - frequency formula

The general form of simple harmonic motion is $d=A\cos(\omega t)$. Given $d = 9\cos(\frac{\pi}{2}t)$, we have $\omega=\frac{\pi}{2}$. The frequency $f$ is related to $\omega$ by $\omega = 2\pi f$. Solving for $f$ gives $f=\frac{\omega}{2\pi}$. Substituting $\omega=\frac{\pi}{2}$ into the formula, we get $f=\frac{\frac{\pi}{2}}{2\pi}=\frac{1}{4}$ cycles per second.

Step3: Recall period formula

The time required for one cycle is the period $T$. The relationship between $\omega$ and $T$ is $\omega=\frac{2\pi}{T}$. Solving for $T$ gives $T = \frac{2\pi}{\omega}$. Substituting $\omega=\frac{\pi}{2}$ into the formula, we have $T=\frac{2\pi}{\frac{\pi}{2}}=4$ seconds.

Answer:

a. 9 in.
b. $\frac{1}{4}$ cycles per second
c. 4 s