Question

an object moves in simple harmonic motion described by the equation d = 4cos(π/2 t+π/2), where t is measured in seconds and d in inches. answer the following questions a through e about the equation. a. what is the maximum displacement of the object? in. b. what is the frequency? cycle per sec. (type an integer or a fraction.) c. what is the time required for one cycle? 4 sec. (type an integer or a fraction.) d. what is the phase - shift of the motion? in. (type an integer or a fraction.)

Explanation:

Step1: Recall the general form of cosine - function for SHM

The general form of a simple - harmonic motion equation is $d = A\cos(\omega t+\varphi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\varphi$ is the phase shift. The given equation is $d = 4\cos(\frac{\pi}{2}t+\frac{\pi}{2})$.

Step2: Find the maximum displacement (amplitude)

For the cosine function $y = A\cos(x)$, the maximum value of $y$ is $|A|$. In the equation $d = 4\cos(\frac{\pi}{2}t+\frac{\pi}{2})$, $A = 4$. So the maximum displacement is 4 inches.

Step3: Calculate the frequency

The angular frequency $\omega=\frac{\pi}{2}$. The relationship between angular frequency $\omega$ and frequency $f$ is $\omega = 2\pi f$. Solving for $f$, we get $f=\frac{\omega}{2\pi}$. Substituting $\omega=\frac{\pi}{2}$ into the formula, we have $f=\frac{\frac{\pi}{2}}{2\pi}=\frac{1}{4}$ cycle per sec.

Step4: Calculate the period

The period $T$ (time for one cycle) is the reciprocal of the frequency. Since $f=\frac{1}{4}$ cycle per sec, then $T = 4$ sec.

Step5: Determine the phase - shift

For the function $d = A\cos(\omega t+\varphi)$, the phase - shift is $\varphi$. In the equation $d = 4\cos(\frac{\pi}{2}t+\frac{\pi}{2})$, the phase - shift is $\frac{\pi}{2}$.

Answer:

a. 4 inches
b. $\frac{1}{4}$ cycle per sec
c. 4 sec
d. $\frac{\pi}{2}$